Convergence of a Sequence, Monotone Sequences
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Limit of a Sequence - Limit Arithmetic
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- The Limit of a Sequence - Limit Arithmetic
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24

Proving the Limit of a Sequence Using the Definition of a Sequence
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- Limit of a Sequence, Definition - Part 1
- Limit of a Sequence, Definition - Part 2
- Limit of a Sequence, Definition - Part 3
- Limit of a Sequence, Definition - Part 4
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13

Sandwich Theorem
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Euler Limit
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Ratio Test
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Root Test
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Recursive Sequences
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Proving Divergence of a Sequence Using Subsequences
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Disproving Limits of Sequences Using the Definition of a Limit
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Cauchy Criterion
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Bolzano-Weierstrass Theorem
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Stolz-Cesaro Convergence Theorem
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Heine Limit Definition and Continuity
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Advanced Theory Exercises
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Convergence of a Sequence, Monotone Sequences","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Sequence","Duration":"9m 42s","ChapterTopicVideoID":30342,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this clip, we\u0027re starting a new topic, sequences."},{"Start":"00:04.725 ","End":"00:09.945","Text":"A sequence is an ordered list of numbers and those numbers are called its terms."},{"Start":"00:09.945 ","End":"00:14.550","Text":"But in some places they\u0027re called members or elements of"},{"Start":"00:14.550 ","End":"00:19.500","Text":"the list and the list could be finite or infinite."},{"Start":"00:19.500 ","End":"00:24.645","Text":"But in this course, we\u0027re only going to be concerned with infinite sequences."},{"Start":"00:24.645 ","End":"00:28.370","Text":"Anyway, let\u0027s look at some examples."},{"Start":"00:28.370 ","End":"00:30.095","Text":"Here\u0027s the 1st one."},{"Start":"00:30.095 ","End":"00:33.605","Text":"The first term is 1 over 1,"},{"Start":"00:33.605 ","End":"00:35.870","Text":"the second term is 1.5,"},{"Start":"00:35.870 ","End":"00:41.410","Text":"then 1/4, 1/4 and you can guess the pattern and so on."},{"Start":"00:41.410 ","End":"00:43.935","Text":"In the second example,"},{"Start":"00:43.935 ","End":"00:47.190","Text":"the first term is 1 over 2,"},{"Start":"00:47.190 ","End":"00:49.160","Text":"2^11, 1 over 2 squared,"},{"Start":"00:49.160 ","End":"00:50.270","Text":"1 over 2 cubed,"},{"Start":"00:50.270 ","End":"00:52.520","Text":"1 over 2^4, and again,"},{"Start":"00:52.520 ","End":"00:53.870","Text":"you can probably guess the pattern,"},{"Start":"00:53.870 ","End":"00:58.740","Text":"the next element term would be 1 over 2 to the fifth."},{"Start":"00:59.170 ","End":"01:03.530","Text":"Third example, just alternating minus 1,"},{"Start":"01:03.530 ","End":"01:05.525","Text":"1, minus 1, 1,"},{"Start":"01:05.525 ","End":"01:08.600","Text":"minus 1, 1, and so on."},{"Start":"01:08.600 ","End":"01:11.555","Text":"Our last example here,"},{"Start":"01:11.555 ","End":"01:17.015","Text":"this is what it looks like and you can probably guess the pattern."},{"Start":"01:17.015 ","End":"01:23.450","Text":"For example, the next one in the sequence would be root 5 over 6."},{"Start":"01:23.450 ","End":"01:25.610","Text":"I think you get the idea."},{"Start":"01:25.610 ","End":"01:29.990","Text":"Now there\u0027s a notation for sequences on the terms of the sequence."},{"Start":"01:29.990 ","End":"01:33.890","Text":"I just want to remind you though that terms sometimes called members or"},{"Start":"01:33.890 ","End":"01:38.480","Text":"elements and we denote them as follows."},{"Start":"01:38.480 ","End":"01:42.350","Text":"The first term of the sequence is usually called a_1,"},{"Start":"01:42.350 ","End":"01:44.435","Text":"the second term a_2,"},{"Start":"01:44.435 ","End":"01:48.410","Text":"the third term a_3, and so on."},{"Start":"01:48.410 ","End":"01:51.185","Text":"But it doesn\u0027t have to be the letter a."},{"Start":"01:51.185 ","End":"01:54.650","Text":"You could have b_1 for the first term,"},{"Start":"01:54.650 ","End":"01:56.240","Text":"b_2, for the second term,"},{"Start":"01:56.240 ","End":"01:59.045","Text":"b_3, and so on."},{"Start":"01:59.045 ","End":"02:07.099","Text":"Some examples, and actually these are the same examples that we had above."},{"Start":"02:07.099 ","End":"02:10.040","Text":"I just added the notation so,"},{"Start":"02:10.040 ","End":"02:11.375","Text":"in the first sequence,"},{"Start":"02:11.375 ","End":"02:13.080","Text":"the first term is a_1,"},{"Start":"02:13.080 ","End":"02:14.340","Text":"next one is a_2,"},{"Start":"02:14.340 ","End":"02:16.395","Text":"a_3, a_4, and so on."},{"Start":"02:16.395 ","End":"02:18.620","Text":"If we have several sequences at once,"},{"Start":"02:18.620 ","End":"02:20.030","Text":"we don\u0027t want to mix the letters up,"},{"Start":"02:20.030 ","End":"02:21.470","Text":"so I\u0027ll give each one a different letter."},{"Start":"02:21.470 ","End":"02:24.969","Text":"This would be say, b_1, b_2, b_3, b_4,"},{"Start":"02:24.969 ","End":"02:27.965","Text":"and then here I used the letter c,"},{"Start":"02:27.965 ","End":"02:34.770","Text":"and here the letter d and the subscript indicates what the position is in the sequence."},{"Start":"02:35.000 ","End":"02:42.020","Text":"Next, I\u0027ll be talking about something called the general term of a sequence."},{"Start":"02:42.020 ","End":"02:44.270","Text":"Now we need to talk about a general term"},{"Start":"02:44.270 ","End":"02:46.220","Text":"because we don\u0027t if we just want to talk about the 1st,"},{"Start":"02:46.220 ","End":"02:47.945","Text":"2nd, 3rd, 4th terms."},{"Start":"02:47.945 ","End":"02:52.490","Text":"I want to talk about the 17th term, the 119th term."},{"Start":"02:52.490 ","End":"02:54.740","Text":"In general, for a number n,"},{"Start":"02:54.740 ","End":"03:00.605","Text":"I want to know what is the element or term in the nth place."},{"Start":"03:00.605 ","End":"03:02.420","Text":"Perhaps to have consistency,"},{"Start":"03:02.420 ","End":"03:10.605","Text":"I should just stick with the same term which is term member element term yeah."},{"Start":"03:10.605 ","End":"03:12.540","Text":"In the nth place,"},{"Start":"03:12.540 ","End":"03:15.110","Text":"and it\u0027s the nth element,"},{"Start":"03:15.110 ","End":"03:20.680","Text":"and it\u0027s usually denoted as a_n."},{"Start":"03:20.680 ","End":"03:25.850","Text":"In the nth place I have a and then a subscript n. Of course,"},{"Start":"03:25.850 ","End":"03:28.100","Text":"if I\u0027m not using a in this sequence,"},{"Start":"03:28.100 ","End":"03:34.190","Text":"so general term would be b_n or c_n or d_n, and so on."},{"Start":"03:34.190 ","End":"03:39.500","Text":"But the subscript denotes what place in the sequence we are."},{"Start":"03:39.500 ","End":"03:44.110","Text":"Now, I\u0027ll give some examples which are actually these examples."},{"Start":"03:44.110 ","End":"03:46.575","Text":"In this example here,"},{"Start":"03:46.575 ","End":"03:49.815","Text":"for example, the 4th term is 1 over 4."},{"Start":"03:49.815 ","End":"03:51.320","Text":"If you look at the pattern,"},{"Start":"03:51.320 ","End":"04:00.030","Text":"the nth term will be 1 over n. So a_n is 1 over n. Then the 2nd example here,"},{"Start":"04:00.030 ","End":"04:07.870","Text":"which you only went up to 4, you can see that in place n in the nth term is 1 over 2^n."},{"Start":"04:07.870 ","End":"04:13.895","Text":"Now given the other couple of examples which correspond to these two."},{"Start":"04:13.895 ","End":"04:17.420","Text":"For this sequence, which is an alternating sequence,"},{"Start":"04:17.420 ","End":"04:20.015","Text":"meaning it goes minus plus minus plus."},{"Start":"04:20.015 ","End":"04:26.240","Text":"The general term would be minus 1^n."},{"Start":"04:26.240 ","End":"04:30.140","Text":"Now, note that if n is odd,"},{"Start":"04:30.140 ","End":"04:35.260","Text":"then minus 1^n is minus 1 and if it\u0027s even,"},{"Start":"04:35.260 ","End":"04:40.480","Text":"it\u0027s plus 1 and luckily turned out right that the 1st place was minus,"},{"Start":"04:40.480 ","End":"04:42.010","Text":"the 2nd place was plus."},{"Start":"04:42.010 ","End":"04:46.430","Text":"On a remark, if you wanted to take it the other way around, 1,"},{"Start":"04:46.430 ","End":"04:51.440","Text":"minus 1,1 minus 1, et cetera."},{"Start":"04:51.440 ","End":"04:57.775","Text":"Then the way you do it is you would say that the general term is minus 1^n."},{"Start":"04:57.775 ","End":"05:03.450","Text":"You would add another minus and then it would reverse the odd and the even."},{"Start":"05:03.450 ","End":"05:06.830","Text":"The odd places would get the plus 1,"},{"Start":"05:06.830 ","End":"05:09.670","Text":"and the even places will get the minus 1."},{"Start":"05:09.670 ","End":"05:12.810","Text":"This happens a lot so often you want to adjust"},{"Start":"05:12.810 ","End":"05:17.895","Text":"minus 1^n to be minus 1^n plus 1 and reverse the sign."},{"Start":"05:17.895 ","End":"05:19.970","Text":"In this example also,"},{"Start":"05:19.970 ","End":"05:23.150","Text":"we can see the pattern under the square root."},{"Start":"05:23.150 ","End":"05:25.730","Text":"We see that we have 1, 2, 3, 4,"},{"Start":"05:25.730 ","End":"05:30.060","Text":"so in place n it would be n. On the denominator,"},{"Start":"05:30.060 ","End":"05:32.820","Text":"in place 1 we have 2, 2nd term is 3,"},{"Start":"05:32.820 ","End":"05:34.615","Text":"3rd term is 4, 4th term is 5,"},{"Start":"05:34.615 ","End":"05:36.995","Text":"so the nth term would be an n plus 1."},{"Start":"05:36.995 ","End":"05:44.630","Text":"This would be the nth term or the general term of this sequence."},{"Start":"05:44.630 ","End":"05:46.520","Text":"N is a convenient letter,"},{"Start":"05:46.520 ","End":"05:51.920","Text":"but sometimes it\u0027s convenient to use i or j or m,"},{"Start":"05:51.920 ","End":"05:53.900","Text":"unless we say otherwise we choose n,"},{"Start":"05:53.900 ","End":"05:58.825","Text":"and usually if a is free we\u0027ll use letter a."},{"Start":"05:58.825 ","End":"06:05.240","Text":"Next, I\u0027ll show you how to use the general term to describe the sequence."},{"Start":"06:05.240 ","End":"06:10.430","Text":"How do we describe a sequence by means of its general element?"},{"Start":"06:10.430 ","End":"06:12.230","Text":"In the previous examples,"},{"Start":"06:12.230 ","End":"06:18.070","Text":"we found the formula for the general term a_n in terms of n and from it,"},{"Start":"06:18.070 ","End":"06:21.020","Text":"you can compute a_1, a_2, a_3,"},{"Start":"06:21.020 ","End":"06:24.590","Text":"or any particular a by just letting n equal 1,"},{"Start":"06:24.590 ","End":"06:26.620","Text":"2, 3, or whatever."},{"Start":"06:26.620 ","End":"06:29.840","Text":"If I just give you the formula or describe"},{"Start":"06:29.840 ","End":"06:34.715","Text":"the general element a_n then I can figure out the whole sequence."},{"Start":"06:34.715 ","End":"06:38.250","Text":"For example, and this was our first example,"},{"Start":"06:38.250 ","End":"06:39.960","Text":"instead of writing 1,"},{"Start":"06:39.960 ","End":"06:42.185","Text":"1/2, 1/3, 1/4 or 1/5, and so on."},{"Start":"06:42.185 ","End":"06:44.750","Text":"All you have to do is say that a_n is"},{"Start":"06:44.750 ","End":"06:47.780","Text":"1 over n. Because then you can figure out all of these."},{"Start":"06:47.780 ","End":"06:51.590","Text":"You would say, the 1st element, I put an equals 1,"},{"Start":"06:51.590 ","End":"06:53.390","Text":"1 over 1, 2nd element,"},{"Start":"06:53.390 ","End":"06:55.535","Text":"n is 2,1 over 2, and so on."},{"Start":"06:55.535 ","End":"06:58.295","Text":"This is much more compact."},{"Start":"06:58.295 ","End":"07:04.525","Text":"Our 2nd example which was this is written more simply as,"},{"Start":"07:04.525 ","End":"07:07.170","Text":"I\u0027ll use a different letter than a is b."},{"Start":"07:07.170 ","End":"07:12.435","Text":"So b_n, the nth element is 1 over 2^n."},{"Start":"07:12.435 ","End":"07:16.530","Text":"Like the 4th element is 1 over 2^4 and so on."},{"Start":"07:16.530 ","End":"07:19.620","Text":"In our 3rd example is minus 1,1,"},{"Start":"07:19.620 ","End":"07:20.960","Text":"minus 1,1, and so on."},{"Start":"07:20.960 ","End":"07:23.059","Text":"We found that the general element,"},{"Start":"07:23.059 ","End":"07:27.415","Text":"general term c_n is minus1^n."},{"Start":"07:27.415 ","End":"07:34.580","Text":"If I just say this is more of a shorthand way in writing this whole thing with a dot,"},{"Start":"07:34.580 ","End":"07:36.575","Text":"dot, dot at the end."},{"Start":"07:36.575 ","End":"07:43.715","Text":"Now there are some other notations that we can use instead of writing a_n is 1 over n,"},{"Start":"07:43.715 ","End":"07:45.980","Text":"there are other possibilities,"},{"Start":"07:45.980 ","End":"07:50.390","Text":"and I\u0027ll illustrate the notations on the series here,"},{"Start":"07:50.390 ","End":"07:53.335","Text":"the 1/5, 1/3, and so on."},{"Start":"07:53.335 ","End":"07:58.235","Text":"What we wrote simply as a_n equals 1 over n,"},{"Start":"07:58.235 ","End":"08:01.775","Text":"can be written in some minor variations."},{"Start":"08:01.775 ","End":"08:06.170","Text":"The most similar one to this is this,"},{"Start":"08:06.170 ","End":"08:12.080","Text":"which is just putting this in a bracket to emphasize that we\u0027re talking about"},{"Start":"08:12.080 ","End":"08:18.720","Text":"the whole sequence and not about the particular general term, the nth term."},{"Start":"08:19.000 ","End":"08:25.310","Text":"This one is a bit more precise because we assume"},{"Start":"08:25.310 ","End":"08:31.130","Text":"always that we start from the first element and keep going to infinity,"},{"Start":"08:31.130 ","End":"08:33.905","Text":"1, 2, 3, 4, and so on."},{"Start":"08:33.905 ","End":"08:40.609","Text":"This just makes it precise that we do start at element 1 and keep going to infinity."},{"Start":"08:40.609 ","End":"08:43.880","Text":"Sometimes we don\u0027t start at number 1,"},{"Start":"08:43.880 ","End":"08:48.390","Text":"sometimes we start at 0 or at any other number."},{"Start":"08:48.390 ","End":"08:52.600","Text":"The 1st one, we could call it a_3, a_4, a_5."},{"Start":"08:53.030 ","End":"08:55.670","Text":"If we want to be really precise,"},{"Start":"08:55.670 ","End":"08:58.390","Text":"then this will tell us where we start and where we end,"},{"Start":"08:58.390 ","End":"09:01.920","Text":"and if up its infinity it means never-ending."},{"Start":"09:02.620 ","End":"09:06.005","Text":"This is the same thing as this one,"},{"Start":"09:06.005 ","End":"09:11.190","Text":"except that here I use curly brackets instead of round brackets."},{"Start":"09:12.350 ","End":"09:18.350","Text":"This is another variation of this one,"},{"Start":"09:18.350 ","End":"09:21.305","Text":"where instead of saying n goes from 1 to infinity,"},{"Start":"09:21.305 ","End":"09:24.605","Text":"I say, n bigger or equal to 1."},{"Start":"09:24.605 ","End":"09:28.550","Text":"We always assume that n is a whole number,"},{"Start":"09:28.550 ","End":"09:32.165","Text":"but it tells us it\u0027s all the whole numbers from 1 onwards."},{"Start":"09:32.165 ","End":"09:35.390","Text":"These are all variations you might see."},{"Start":"09:35.390 ","End":"09:43.080","Text":"In fact, I\u0027ll be using some of these in the rest of this topic."}],"ID":32373},{"Watched":false,"Name":"Examples of Sequences","Duration":"6m 19s","ChapterTopicVideoID":30344,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.300","Text":"In this clip, we\u0027ll be solving some example exercises and here\u0027s the first 1."},{"Start":"00:06.300 ","End":"00:11.010","Text":"We have to write the first 5 terms of the sequence, which is this."},{"Start":"00:11.010 ","End":"00:13.349","Text":"Notice I\u0027ve used the curly brace notation,"},{"Start":"00:13.349 ","End":"00:15.525","Text":"that\u0027s 1 of the several notations."},{"Start":"00:15.525 ","End":"00:21.345","Text":"Basically what it says is that the nth term is n plus 4 over n squared."},{"Start":"00:21.345 ","End":"00:25.350","Text":"What I want to do is plug in successively n equals 1, 2,"},{"Start":"00:25.350 ","End":"00:29.085","Text":"3, 4, 5 and here we are."},{"Start":"00:29.085 ","End":"00:31.980","Text":"Instead of n here and here,"},{"Start":"00:31.980 ","End":"00:35.250","Text":"I put 1 or 2 or 3 or 4 or 5,"},{"Start":"00:35.250 ","End":"00:37.620","Text":"then this is what I get for a_1,"},{"Start":"00:37.620 ","End":"00:40.410","Text":"a_2, a_3, a_4, a_5, first 5 terms."},{"Start":"00:40.410 ","End":"00:42.705","Text":"If I just simplify them,"},{"Start":"00:42.705 ","End":"00:47.025","Text":"then here\u0027s what we get like 1 plus 4 is 5 over 1,"},{"Start":"00:47.025 ","End":"00:50.100","Text":"2 plus 4, 6 over 4,"},{"Start":"00:50.100 ","End":"00:52.655","Text":"7 over 9, and so on."},{"Start":"00:52.655 ","End":"00:55.609","Text":"Now let\u0027s go on to the next example."},{"Start":"00:55.609 ","End":"00:57.590","Text":"In this next example,"},{"Start":"00:57.590 ","End":"01:00.560","Text":"we have to add the first 5 terms of this sequence."},{"Start":"01:00.560 ","End":"01:02.120","Text":"It\u0027s not a_n, it\u0027s b_n."},{"Start":"01:02.120 ","End":"01:08.005","Text":"That\u0027s okay and it\u0027s written with the notation n goes from 1 to infinity."},{"Start":"01:08.005 ","End":"01:10.470","Text":"We just want from 1 to 5,"},{"Start":"01:10.470 ","End":"01:14.240","Text":"so we just take this expression and plug in successively 1,"},{"Start":"01:14.240 ","End":"01:17.440","Text":"2, 3, 4, and 5. Here\u0027s what we get."},{"Start":"01:17.440 ","End":"01:19.430","Text":"Instead of the n, which is here and here,"},{"Start":"01:19.430 ","End":"01:21.395","Text":"I put 1 and then 2,"},{"Start":"01:21.395 ","End":"01:23.350","Text":"3, and so on."},{"Start":"01:23.350 ","End":"01:25.685","Text":"Now we just simplify a little bit."},{"Start":"01:25.685 ","End":"01:31.910","Text":"Note that minus 1 to an even power is plus 1."},{"Start":"01:31.910 ","End":"01:33.305","Text":"So we\u0027ll start with plus,"},{"Start":"01:33.305 ","End":"01:35.870","Text":"and minus 1 to an odd power will be minus."},{"Start":"01:35.870 ","End":"01:37.765","Text":"It will alternate."},{"Start":"01:37.765 ","End":"01:39.705","Text":"This is what we get."},{"Start":"01:39.705 ","End":"01:41.540","Text":"Of course the plus is unnecessary,"},{"Start":"01:41.540 ","End":"01:43.550","Text":"but it\u0027s just for emphasis that we have plus,"},{"Start":"01:43.550 ","End":"01:46.805","Text":"minus, plus, minus, plus."},{"Start":"01:46.805 ","End":"01:55.920","Text":"In this example, we\u0027re given a sequence c_n n bigger or equal to 1,"},{"Start":"01:55.920 ","End":"01:58.410","Text":"meaning n goes 1, 2, 3, 4, 5,"},{"Start":"01:58.410 ","End":"02:02.955","Text":"and so on and we\u0027re given c_n descriptively."},{"Start":"02:02.955 ","End":"02:05.760","Text":"We\u0027re told that c_n is the nth prime number."},{"Start":"02:05.760 ","End":"02:07.830","Text":"It\u0027s not a formula, but it\u0027s"},{"Start":"02:07.830 ","End":"02:11.180","Text":"definitely well defined because we know what the prime numbers are,"},{"Start":"02:11.180 ","End":"02:13.380","Text":"at least the first few."},{"Start":"02:13.570 ","End":"02:16.885","Text":"Assuming you know your primes,"},{"Start":"02:16.885 ","End":"02:20.265","Text":"c_1 is the first prime number is 2."},{"Start":"02:20.265 ","End":"02:22.640","Text":"I know some people think that 1 is a prime,"},{"Start":"02:22.640 ","End":"02:24.650","Text":"but it\u0027s not considered to be."},{"Start":"02:24.650 ","End":"02:26.360","Text":"Next prime number is 3,"},{"Start":"02:26.360 ","End":"02:28.825","Text":"then 5, then 7, then 11."},{"Start":"02:28.825 ","End":"02:31.490","Text":"If they ask for the 6th member,"},{"Start":"02:31.490 ","End":"02:35.180","Text":"for example, I could say c_6 is 13,"},{"Start":"02:35.180 ","End":"02:37.630","Text":"c_7 is the 7th prime,"},{"Start":"02:37.630 ","End":"02:42.975","Text":"it is 17, and so on but we\u0027re just asked for 5 so that\u0027ll do."},{"Start":"02:42.975 ","End":"02:47.420","Text":"This exercise is a little bit different to the previous ones."},{"Start":"02:47.420 ","End":"02:51.080","Text":"We also have a sequence and we want to write the first 5 terms."},{"Start":"02:51.080 ","End":"02:55.250","Text":"But this time, the sequence is defined recursively."},{"Start":"02:55.250 ","End":"02:57.725","Text":"I\u0027ll just highlight that term."},{"Start":"02:57.725 ","End":"03:02.840","Text":"What it means is that we don\u0027t have an explicit formula for"},{"Start":"03:02.840 ","End":"03:07.860","Text":"each n. What we do have is 1 explicit 1,"},{"Start":"03:07.860 ","End":"03:11.400","Text":"usually the 1st 1 is given explicitly a_1."},{"Start":"03:11.400 ","End":"03:17.120","Text":"But subsequent a_n\u0027s are given in terms of previous a_n\u0027s,"},{"Start":"03:17.120 ","End":"03:23.325","Text":"so we have to build up the sequence 1 by 1 and you\u0027ll see in a moment."},{"Start":"03:23.325 ","End":"03:26.220","Text":"Okay. As I said, the 1st 1 is just given to us,"},{"Start":"03:26.220 ","End":"03:27.960","Text":"so we just copy that."},{"Start":"03:27.960 ","End":"03:30.460","Text":"Now, for the next 1,"},{"Start":"03:30.460 ","End":"03:32.910","Text":"how do we find a_2?"},{"Start":"03:32.910 ","End":"03:39.525","Text":"We can get a_2 if we put n equals 1 in this formula, and we get."},{"Start":"03:39.525 ","End":"03:43.370","Text":"Notice that everywhere there\u0027s n I\u0027m putting 1 and it\u0027s colored in blue."},{"Start":"03:43.370 ","End":"03:45.785","Text":"a_1 plus 1, which is a_2,"},{"Start":"03:45.785 ","End":"03:51.450","Text":"will be given from a_1 plus 2 times 1."},{"Start":"03:51.450 ","End":"03:55.780","Text":"Now, we know a_1 because it\u0027s written here."},{"Start":"03:55.940 ","End":"04:00.780","Text":"So a_1 is 1 and 2 times 1 is 2,"},{"Start":"04:00.780 ","End":"04:03.825","Text":"so we have 1 plus 2 is 3."},{"Start":"04:03.825 ","End":"04:06.790","Text":"So a_2 is 3."},{"Start":"04:06.790 ","End":"04:08.360","Text":"But we didn\u0027t find it directly."},{"Start":"04:08.360 ","End":"04:12.739","Text":"We found it in terms of the previous n. In fact, this,"},{"Start":"04:12.739 ","End":"04:15.540","Text":"and I\u0027ll highlight it,"},{"Start":"04:16.090 ","End":"04:20.795","Text":"is a special case of a recursive formula."},{"Start":"04:20.795 ","End":"04:25.220","Text":"Sometimes we say recursive formula,"},{"Start":"04:25.220 ","End":"04:29.165","Text":"but sometimes we say recursive rule."},{"Start":"04:29.165 ","End":"04:30.800","Text":"Anyway, don\u0027t worry about the name,"},{"Start":"04:30.800 ","End":"04:33.665","Text":"that\u0027s just learn how to use such a thing."},{"Start":"04:33.665 ","End":"04:35.150","Text":"Let\u0027s go on to the next 1."},{"Start":"04:35.150 ","End":"04:37.415","Text":"If I want a_3,"},{"Start":"04:37.415 ","End":"04:43.025","Text":"what I have to do is let n equals 2 because then 2 plus 1 is 3."},{"Start":"04:43.025 ","End":"04:47.059","Text":"So we get, if n is 2 in the recursive formula,"},{"Start":"04:47.059 ","End":"04:52.595","Text":"a_2 plus 1 equals a_2 plus twice 2."},{"Start":"04:52.595 ","End":"04:56.075","Text":"Now a_2 we found already is 3,"},{"Start":"04:56.075 ","End":"04:57.350","Text":"so we put that here,"},{"Start":"04:57.350 ","End":"04:59.300","Text":"and twice 2 is 4 is 7."},{"Start":"04:59.300 ","End":"05:03.740","Text":"We got a_3 from previous value a_2."},{"Start":"05:03.740 ","End":"05:07.190","Text":"We still have to do 2 more of these to get a_4 and a_5."},{"Start":"05:07.190 ","End":"05:09.590","Text":"To get a_4, I think you get the idea,"},{"Start":"05:09.590 ","End":"05:11.965","Text":"we let n equals 3."},{"Start":"05:11.965 ","End":"05:14.490","Text":"So n equals 3,"},{"Start":"05:14.490 ","End":"05:19.095","Text":"a_3 plus 1 is a_3 plus twice 3."},{"Start":"05:19.095 ","End":"05:22.050","Text":"We have a_3 because it\u0027s here, it\u0027s 7."},{"Start":"05:22.050 ","End":"05:30.280","Text":"I should have really been indicating that I get the 7 from here like a_2 was 3."},{"Start":"05:30.280 ","End":"05:37.890","Text":"That was what gave me this and the 1st 1 is what gave me this here."},{"Start":"05:37.890 ","End":"05:43.200","Text":"Okay. So a_4 is 13,"},{"Start":"05:43.200 ","End":"05:47.410","Text":"here is 7 plus 6, and we have 1 more."},{"Start":"05:47.810 ","End":"05:50.910","Text":"This time we put n equals 4."},{"Start":"05:50.910 ","End":"05:52.605","Text":"If we want to get a_5,"},{"Start":"05:52.605 ","End":"05:56.175","Text":"so a_4 plus 1 is a_n, which is a_4,"},{"Start":"05:56.175 ","End":"06:01.500","Text":"plus twice 4 and a_ 4 is 13,"},{"Start":"06:01.500 ","End":"06:03.240","Text":"which I got from here."},{"Start":"06:03.240 ","End":"06:07.050","Text":"Twice 4 is 8, 13 and 8 is 21."},{"Start":"06:07.050 ","End":"06:12.120","Text":"Now we have all the first 5 elements,"},{"Start":"06:12.120 ","End":"06:16.485","Text":"terms, members of the sequence."},{"Start":"06:16.485 ","End":"06:20.020","Text":"That\u0027s it for this exercise."}],"ID":32374},{"Watched":false,"Name":"Defining a Sequence as a Function","Duration":"3m 21s","ChapterTopicVideoID":30345,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.750","Text":"Up to now, we\u0027ve been using a sequence as an ordered list of numbers."},{"Start":"00:07.750 ","End":"00:11.485","Text":"There is another mathematical way of defining a sequence."},{"Start":"00:11.485 ","End":"00:16.010","Text":"A sequence can be considered as a function of sorts."},{"Start":"00:16.310 ","End":"00:19.030","Text":"We\u0027ll illustrate it on an example."},{"Start":"00:19.030 ","End":"00:27.120","Text":"Let\u0027s suppose we have this sequence where the nth term a_n is n plus 4 over n squared."},{"Start":"00:27.120 ","End":"00:31.989","Text":"We can actually view it as a function from the natural numbers."},{"Start":"00:31.989 ","End":"00:33.820","Text":"N is a natural number."},{"Start":"00:33.820 ","End":"00:35.760","Text":"After I\u0027ve made the computation,"},{"Start":"00:35.760 ","End":"00:38.280","Text":"I get a real number, so I have a function."},{"Start":"00:38.280 ","End":"00:40.495","Text":"The domain is natural numbers,"},{"Start":"00:40.495 ","End":"00:42.430","Text":"the range is real numbers,"},{"Start":"00:42.430 ","End":"00:46.985","Text":"and we can write it as f of n equals n plus 4 over n squared."},{"Start":"00:46.985 ","End":"00:54.450","Text":"We define instead of a_n as the nth member of the sequence we, in this notation,"},{"Start":"00:54.450 ","End":"00:57.225","Text":"would say f of n."},{"Start":"00:57.225 ","End":"01:03.065","Text":"This interpretation of a sequence as a function will be useful later on."},{"Start":"01:03.065 ","End":"01:07.460","Text":"For example, it will help us to determine if a sequence is"},{"Start":"01:07.460 ","End":"01:12.965","Text":"increasing or decreasing based on the derivative of this function."},{"Start":"01:12.965 ","End":"01:17.059","Text":"Just point out that it\u0027s not necessarily the natural numbers"},{"Start":"01:17.059 ","End":"01:22.445","Text":"because this notation doesn\u0027t tell us where a runs from."},{"Start":"01:22.445 ","End":"01:24.365","Text":"If I said a_n."},{"Start":"01:24.365 ","End":"01:27.955","Text":"N goes from 1 to infinity,"},{"Start":"01:27.955 ","End":"01:30.200","Text":"then it would be the natural numbers."},{"Start":"01:30.200 ","End":"01:31.579","Text":"But as I mentioned,"},{"Start":"01:31.579 ","End":"01:34.850","Text":"the series doesn\u0027t necessarily start at 1 even if it\u0027s infinite,"},{"Start":"01:34.850 ","End":"01:36.469","Text":"it could start at 0,"},{"Start":"01:36.469 ","End":"01:40.560","Text":"or at 3, or something else in which case we just modify it"},{"Start":"01:40.560 ","End":"01:42.720","Text":"slightly and say not the natural numbers but"},{"Start":"01:42.720 ","End":"01:45.390","Text":"the natural numbers from 3 onwards or something."},{"Start":"01:45.390 ","End":"01:46.710","Text":"I don\u0027t want to get too technical,"},{"Start":"01:46.710 ","End":"01:49.620","Text":"we\u0027ll just stick to the natural numbers to the real numbers,"},{"Start":"01:49.620 ","End":"01:56.090","Text":"and I want to move on and talk about the sketch or graph of such a function."},{"Start":"01:56.090 ","End":"01:58.585","Text":"I\u0027ll just show it to you right away."},{"Start":"01:58.585 ","End":"02:01.940","Text":"We see it\u0027s just a set of isolated points."},{"Start":"02:01.940 ","End":"02:04.385","Text":"That\u0027s what the graph of a sequence looks like."},{"Start":"02:04.385 ","End":"02:10.710","Text":"We only have values for n which is a whole number 1, 2, 3, 4."},{"Start":"02:10.710 ","End":"02:12.410","Text":"We don\u0027t have anything in between,"},{"Start":"02:12.410 ","End":"02:16.600","Text":"so we don\u0027t join the points with a curve."},{"Start":"02:16.600 ","End":"02:19.200","Text":"The way we get it, it\u0027s just regular."},{"Start":"02:19.200 ","End":"02:21.180","Text":"We plug in values."},{"Start":"02:21.180 ","End":"02:23.170","Text":"Let\u0027s say when n is 1,"},{"Start":"02:23.170 ","End":"02:27.455","Text":"then we\u0027d get 1 plus 4 over 1 squared is 5."},{"Start":"02:27.455 ","End":"02:30.080","Text":"I\u0027ll just write the y value is 5."},{"Start":"02:30.080 ","End":"02:31.985","Text":"Of course, the point is 1, 5."},{"Start":"02:31.985 ","End":"02:36.155","Text":"I want to plug in 2. 2 plus 6 over 4 that\u0027s 1/2."},{"Start":"02:36.155 ","End":"02:39.950","Text":"Plug in 3 I would get 7/9,"},{"Start":"02:39.950 ","End":"02:43.020","Text":"and so on. That\u0027s what it looks like."},{"Start":"02:43.020 ","End":"02:49.690","Text":"However, if you did join it with a curve that would give you something else."},{"Start":"02:49.690 ","End":"02:53.165","Text":"That would give you a function from the real numbers to the real numbers."},{"Start":"02:53.165 ","End":"02:56.940","Text":"f of x is x plus 4 over x squared,"},{"Start":"02:56.940 ","End":"02:59.705","Text":"and that\u0027s something totally different."},{"Start":"02:59.705 ","End":"03:05.930","Text":"But they are related and we will be using the properties of the derivative,"},{"Start":"03:05.930 ","End":"03:12.965","Text":"for example, to show that a sequence is decreasing if the function is decreasing."},{"Start":"03:12.965 ","End":"03:16.340","Text":"Anyway, you\u0027ll see more about that later."},{"Start":"03:16.340 ","End":"03:19.250","Text":"That\u0027s assuming you\u0027ve studied derivatives."},{"Start":"03:19.250 ","End":"03:22.320","Text":"I\u0027m just going to end the clip here."}],"ID":32375},{"Watched":false,"Name":"Limit of a Sequence","Duration":"3m 5s","ChapterTopicVideoID":30346,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this clip, we\u0027ll be talking about the limit of a sequence and it\u0027s"},{"Start":"00:04.440 ","End":"00:08.430","Text":"got some similarity with limits of functions, as we shall see."},{"Start":"00:08.430 ","End":"00:10.665","Text":"Anyway, we\u0027ll start with an example."},{"Start":"00:10.665 ","End":"00:16.440","Text":"The familiar 1 over n sequence in the nth place,"},{"Start":"00:16.440 ","End":"00:18.300","Text":"we\u0027ll write a few terms,"},{"Start":"00:18.300 ","End":"00:21.330","Text":"starts off 1, 1/2, 1/3 and so on,"},{"Start":"00:21.330 ","End":"00:23.370","Text":"and very quickly we get to small numbers,"},{"Start":"00:23.370 ","End":"00:25.950","Text":"1 over 400 ,1 over 10,000,"},{"Start":"00:25.950 ","End":"00:28.215","Text":"1 over a billion."},{"Start":"00:28.215 ","End":"00:30.180","Text":"If we keep writing them,"},{"Start":"00:30.180 ","End":"00:31.230","Text":"keep adding more terms."},{"Start":"00:31.230 ","End":"00:38.340","Text":"I think it\u0027s intuitively obvious anyway that they approach 0,"},{"Start":"00:38.340 ","End":"00:41.230","Text":"get closer and closer to 0."},{"Start":"00:42.110 ","End":"00:46.730","Text":"We say that the limit of this sequence is 0,"},{"Start":"00:46.730 ","End":"00:51.725","Text":"and we use a notation similar to functions."},{"Start":"00:51.725 ","End":"00:57.880","Text":"We say that the limit as n goes to infinity of 1 over n is 0,"},{"Start":"00:57.880 ","End":"01:01.880","Text":"and there\u0027s an alternative notation."},{"Start":"01:01.880 ","End":"01:06.050","Text":"Sorry, this arrow doesn\u0027t have to be quite so long."},{"Start":"01:06.050 ","End":"01:11.555","Text":"1 over n tends to 0 as n tends to infinity."},{"Start":"01:11.555 ","End":"01:14.940","Text":"This notation, you don\u0027t need the word lim."},{"Start":"01:15.330 ","End":"01:19.940","Text":"I want to introduce another term, converges."},{"Start":"01:19.940 ","End":"01:23.570","Text":"We say that the sequence converges to 0."},{"Start":"01:23.570 ","End":"01:26.540","Text":"If it has a limit of 0, then it converges to 0."},{"Start":"01:26.540 ","End":"01:28.520","Text":"Nothing special about 0."},{"Start":"01:28.520 ","End":"01:30.200","Text":"We could say if the limit was 3,"},{"Start":"01:30.200 ","End":"01:32.030","Text":"then it would converge to 3."},{"Start":"01:32.030 ","End":"01:34.850","Text":"But if we don\u0027t care what it converges to,"},{"Start":"01:34.850 ","End":"01:38.705","Text":"we just say that the sequence is convergent."},{"Start":"01:38.705 ","End":"01:41.040","Text":"It has a limit."},{"Start":"01:41.160 ","End":"01:44.710","Text":"Now, not every sequence has a limit,"},{"Start":"01:44.710 ","End":"01:48.920","Text":"this 1 did, but there are examples which don\u0027t."},{"Start":"01:48.920 ","End":"01:52.450","Text":"For example, look at the following sequence,"},{"Start":"01:52.450 ","End":"01:54.430","Text":"the 1 that alternates 1 minus 1,"},{"Start":"01:54.430 ","End":"01:56.965","Text":"1 minus 1, and so on."},{"Start":"01:56.965 ","End":"02:00.835","Text":"I claim that it does not have a limit,"},{"Start":"02:00.835 ","End":"02:02.920","Text":"and why it doesn\u0027t it have a limit, although,"},{"Start":"02:02.920 ","End":"02:05.875","Text":"I think it\u0027s fairly intuitive?"},{"Start":"02:05.875 ","End":"02:09.640","Text":"Well, it keeps alternating between 1 and minus 1,"},{"Start":"02:09.640 ","End":"02:13.420","Text":"so it doesn\u0027t approach any 1 single number."},{"Start":"02:13.420 ","End":"02:21.070","Text":"It approaches 2 different numbers so it doesn\u0027t have a limit."},{"Start":"02:21.070 ","End":"02:23.980","Text":"If it doesn\u0027t have a limit, It\u0027s not convergent."},{"Start":"02:23.980 ","End":"02:26.360","Text":"There\u0027s a name for not convergent."},{"Start":"02:26.360 ","End":"02:29.600","Text":"Instead of that, we say divergent."},{"Start":"02:29.600 ","End":"02:32.240","Text":"If it\u0027s not convergent, it\u0027s divergent,"},{"Start":"02:32.240 ","End":"02:36.170","Text":"I\u0027d like to mention that there are a whole bunch of techniques you"},{"Start":"02:36.170 ","End":"02:40.160","Text":"learned for computing the limit of a function at infinity."},{"Start":"02:40.160 ","End":"02:41.600","Text":"In the case of functions,"},{"Start":"02:41.600 ","End":"02:44.560","Text":"you would have x goes to infinity."},{"Start":"02:44.560 ","End":"02:48.170","Text":"Pretty much with only minor modifications,"},{"Start":"02:48.170 ","End":"02:51.965","Text":"all that theory applies for limit of sequences,"},{"Start":"02:51.965 ","End":"02:54.040","Text":"so we\u0027re not going to repeat it."},{"Start":"02:54.040 ","End":"02:59.929","Text":"But basically, limits of sequences are like limits of functions when x goes to infinity."},{"Start":"02:59.929 ","End":"03:05.520","Text":"That\u0027s all I want to say about limits of sequences of this stage, so we\u0027re done."}],"ID":32376},{"Watched":false,"Name":"Increasing and Decreasing Sequences","Duration":"7m 40s","ChapterTopicVideoID":30343,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"Our next topic as far as sequences go,"},{"Start":"00:04.035 ","End":"00:07.455","Text":"is increasing and decreasing sequences."},{"Start":"00:07.455 ","End":"00:11.100","Text":"These occur quite often and they\u0027re particularly"},{"Start":"00:11.100 ","End":"00:14.490","Text":"useful in theorems on limits of sequences."},{"Start":"00:14.490 ","End":"00:18.210","Text":"Many theorems require a sequence to be increasing or decreasing,"},{"Start":"00:18.210 ","End":"00:20.265","Text":"then we can deduce a lot from it."},{"Start":"00:20.265 ","End":"00:22.560","Text":"Let\u0027s give some examples."},{"Start":"00:22.560 ","End":"00:28.845","Text":"Here are 3 different sequences to consider."},{"Start":"00:28.845 ","End":"00:32.010","Text":"In the first sequence, 2, 4, 6,"},{"Start":"00:32.010 ","End":"00:34.925","Text":"8 and assuming this pattern continues,"},{"Start":"00:34.925 ","End":"00:43.780","Text":"each term in the sequence is bigger than the previous 1 and so in this case,"},{"Start":"00:43.780 ","End":"00:50.320","Text":"we say that the sequence is increasing and in the second 1,"},{"Start":"00:50.320 ","End":"00:51.350","Text":"it\u0027s going to be the opposite."},{"Start":"00:51.350 ","End":"00:59.345","Text":"If you look at it, then each term is smaller than the previous 1 and so it\u0027s decreasing."},{"Start":"00:59.345 ","End":"01:00.950","Text":"But in the third case,"},{"Start":"01:00.950 ","End":"01:03.170","Text":"it\u0027s going down then up."},{"Start":"01:03.170 ","End":"01:06.950","Text":"It\u0027s neither increasing nor decreasing."},{"Start":"01:06.950 ","End":"01:12.825","Text":"Now let\u0027s see if we can write something a little more formal than that."},{"Start":"01:12.825 ","End":"01:16.775","Text":"Suppose we have sequence a_n,"},{"Start":"01:16.775 ","End":"01:21.094","Text":"it\u0027s called increasing if each term"},{"Start":"01:21.094 ","End":"01:26.435","Text":"is less than or equal to the following term or if you look at the other way around,"},{"Start":"01:26.435 ","End":"01:30.860","Text":"the following term is bigger or equal to the current term."},{"Start":"01:30.860 ","End":"01:33.055","Text":"If this is true for all n,"},{"Start":"01:33.055 ","End":"01:36.145","Text":"then it\u0027s increasing and"},{"Start":"01:36.145 ","End":"01:41.060","Text":"decreasing means that each term is bigger than the following term."},{"Start":"01:41.060 ","End":"01:47.510","Text":"Or I\u0027d rather say the following term is always less than or equal to the current term."},{"Start":"01:47.510 ","End":"01:50.460","Text":"If it\u0027s increasing, if n is 1,"},{"Start":"01:50.460 ","End":"01:55.350","Text":"we have here a_1 and less than or equal to a_2."},{"Start":"01:55.350 ","End":"01:56.790","Text":"If put n equals 2,"},{"Start":"01:56.790 ","End":"02:04.425","Text":"we get that a_2 is less than or equal to a_3, and so on."},{"Start":"02:04.425 ","End":"02:07.700","Text":"Here the decreasing, if it at n equal 1,"},{"Start":"02:07.700 ","End":"02:10.625","Text":"we get that a_1 bigger or equal to a_2,"},{"Start":"02:10.625 ","End":"02:18.300","Text":"put an equals 2 you get a_2 bigger or equal to a_3, and so on."},{"Start":"02:18.650 ","End":"02:24.140","Text":"Note that in each of these inequalities are used less than or equal,"},{"Start":"02:24.140 ","End":"02:26.030","Text":"greater than or equal."},{"Start":"02:26.030 ","End":"02:30.080","Text":"We\u0027re going to say that a sequence is"},{"Start":"02:30.080 ","End":"02:35.340","Text":"strictly increasing if we have the strict inequality here."},{"Start":"02:35.340 ","End":"02:40.924","Text":"Similarly, if it\u0027s a strict inequality that each successive term gets smaller,"},{"Start":"02:40.924 ","End":"02:43.070","Text":"then we say strictly decreasing."},{"Start":"02:43.070 ","End":"02:48.499","Text":"That\u0027s the word strictly if the inequalities are strict."},{"Start":"02:48.499 ","End":"02:53.960","Text":"If a sequence is either increasing or decreasing,"},{"Start":"02:53.960 ","End":"02:55.865","Text":"there\u0027s a common name for that."},{"Start":"02:55.865 ","End":"02:59.170","Text":"It\u0027s called a monotonic sequence."},{"Start":"02:59.170 ","End":"03:02.000","Text":"I think you can even say strictly"},{"Start":"03:02.000 ","End":"03:05.675","Text":"monotonic if it\u0027s strictly increasing or strictly decreasing,"},{"Start":"03:05.675 ","End":"03:08.510","Text":"but this is getting too subtle."},{"Start":"03:08.510 ","End":"03:13.040","Text":"Now I want to know if you\u0027ve learned differentiation already,"},{"Start":"03:13.040 ","End":"03:17.870","Text":"then there are quite a few tools to decide if"},{"Start":"03:17.870 ","End":"03:20.855","Text":"a sequence is increasing or decreasing"},{"Start":"03:20.855 ","End":"03:24.395","Text":"using the derivative or what is called differentiation,"},{"Start":"03:24.395 ","End":"03:27.580","Text":"this will turn up in the following example."},{"Start":"03:27.580 ","End":"03:30.695","Text":"Here\u0027s our example exercise."},{"Start":"03:30.695 ","End":"03:32.240","Text":"We have to determine if"},{"Start":"03:32.240 ","End":"03:37.955","Text":"the following sequence is increasing or decreasing or it could be neither."},{"Start":"03:37.955 ","End":"03:40.885","Text":"Let\u0027s write a few terms and see what\u0027s going on."},{"Start":"03:40.885 ","End":"03:44.300","Text":"The numerator is always 1 more than the denominator,"},{"Start":"03:44.300 ","End":"03:48.025","Text":"2/1, 3/2, 4/3, 5/4,"},{"Start":"03:48.025 ","End":"03:50.300","Text":"looks like it\u0027s decreasing."},{"Start":"03:50.300 ","End":"03:53.465","Text":"We\u0027ll prove it in 2 ways."},{"Start":"03:53.465 ","End":"03:58.740","Text":"Let\u0027s first of all do the first way which is straight from the definition."},{"Start":"04:01.820 ","End":"04:07.445","Text":"Actually I didn\u0027t say strictly so we can have bigger or equal to."},{"Start":"04:07.445 ","End":"04:10.490","Text":"This is actually going to be strictly decreasing."},{"Start":"04:10.490 ","End":"04:14.965","Text":"Anyway, don\u0027t worry about the greater than or greater than or equal to."},{"Start":"04:14.965 ","End":"04:20.239","Text":"This is what we have to prove and then just rewrite it in terms of the definition."},{"Start":"04:20.239 ","End":"04:22.805","Text":"A_n is this, I just copied it,"},{"Start":"04:22.805 ","End":"04:27.155","Text":"A_n plus 1 means replace n by n plus 1."},{"Start":"04:27.155 ","End":"04:29.580","Text":"We have n plus 1 plus 1,"},{"Start":"04:29.580 ","End":"04:32.490","Text":"which is I\u0027m plus 2 and here m plus 1."},{"Start":"04:32.490 ","End":"04:34.215","Text":"If we can show this,"},{"Start":"04:34.215 ","End":"04:36.315","Text":"then this is true."},{"Start":"04:36.315 ","End":"04:41.160","Text":"All these quantities are positive, especially the denominators."},{"Start":"04:41.160 ","End":"04:45.935","Text":"We can say that this is bigger than this if and only if"},{"Start":"04:45.935 ","End":"04:50.960","Text":"the cross multiplication this times this is bigger than this times this,"},{"Start":"04:50.960 ","End":"04:57.190","Text":"it\u0027s equivalent and let\u0027s see if we can show that."},{"Start":"04:57.190 ","End":"05:02.450","Text":"Expanding this is equivalent to showing that this is bigger than this."},{"Start":"05:02.450 ","End":"05:07.475","Text":"Subtract m squared plus 2n from both sides and get 1 is bigger than 0."},{"Start":"05:07.475 ","End":"05:10.945","Text":"Anyway, it\u0027s obvious and that concludes the proof."},{"Start":"05:10.945 ","End":"05:12.230","Text":"Because 1 is bigger than 0,"},{"Start":"05:12.230 ","End":"05:14.600","Text":"this is bigger than this, so this is bigger than this."},{"Start":"05:14.600 ","End":"05:19.925","Text":"We trace it backwards we\u0027ve got a_n is actually strictly bigger than a_n plus 1."},{"Start":"05:19.925 ","End":"05:24.875","Text":"We\u0027ve even proved that the sequence is strictly decreasing."},{"Start":"05:24.875 ","End":"05:27.995","Text":"Now let\u0027s do the solution another way,"},{"Start":"05:27.995 ","End":"05:31.760","Text":"but only if you\u0027ve learned derivatives and differentiation."},{"Start":"05:31.760 ","End":"05:37.250","Text":"If you haven\u0027t learned derivatives yet then skip the rest of this clip."},{"Start":"05:37.250 ","End":"05:47.510","Text":"I just want to remind you which is scrolled off that a_n was n plus 1."},{"Start":"05:47.510 ","End":"05:52.205","Text":"Now remember we talked about sequences of functions."},{"Start":"05:52.205 ","End":"05:57.200","Text":"This n plus 1 is a function of n. What we\u0027re going to do"},{"Start":"05:57.200 ","End":"06:02.820","Text":"is extend it to a function of real numbers."},{"Start":"06:02.820 ","End":"06:07.335","Text":"All x\u0027s or at least x bigger or equal to 1,"},{"Start":"06:07.335 ","End":"06:11.270","Text":"we\u0027ll define this function and if"},{"Start":"06:11.270 ","End":"06:15.515","Text":"x is bigger or equal to 1 and we won\u0027t have any problem with the denominator either,"},{"Start":"06:15.515 ","End":"06:22.390","Text":"and then a_n is just f of n. Now we\u0027re going to use derivatives to"},{"Start":"06:22.390 ","End":"06:29.210","Text":"show that f is decreasing as a function of x for x bigger or equal to 1."},{"Start":"06:29.210 ","End":"06:31.045","Text":"To show it\u0027s decreasing,"},{"Start":"06:31.045 ","End":"06:35.705","Text":"we show that the derivative is negative."},{"Start":"06:35.705 ","End":"06:39.745","Text":"Here\u0027s a computation of the derivative."},{"Start":"06:39.745 ","End":"06:42.805","Text":"Hope you remember the quotient rule."},{"Start":"06:42.805 ","End":"06:44.859","Text":"This is a quotient,"},{"Start":"06:44.859 ","End":"06:49.675","Text":"so we take the derivative of the numerator times denominator"},{"Start":"06:49.675 ","End":"06:56.935","Text":"minus the derivative of the denominator times the numerator over the denominator squared."},{"Start":"06:56.935 ","End":"07:00.235","Text":"Anyway, it comes out to be minus 1/x squared."},{"Start":"07:00.235 ","End":"07:02.430","Text":"This thing is positive."},{"Start":"07:02.430 ","End":"07:04.850","Text":"Minus 1 is negative,"},{"Start":"07:04.850 ","End":"07:06.410","Text":"negative and positive is negative."},{"Start":"07:06.410 ","End":"07:08.330","Text":"It\u0027s always negative."},{"Start":"07:08.330 ","End":"07:13.125","Text":"So the derivative of the function is negative."},{"Start":"07:13.125 ","End":"07:15.800","Text":"The function of x is decreasing."},{"Start":"07:15.800 ","End":"07:20.600","Text":"If f is decreasing for all x bigger or equal to 1,"},{"Start":"07:20.600 ","End":"07:22.955","Text":"it\u0027s also decreasing if I just take"},{"Start":"07:22.955 ","End":"07:29.360","Text":"whole numbers and the sequence is decreasing for n equals 1,"},{"Start":"07:29.360 ","End":"07:31.710","Text":"2, 3, and so on."},{"Start":"07:33.040 ","End":"07:41.290","Text":"That\u0027s that for increasing and decreasing and that concludes this clip."}],"ID":32377},{"Watched":false,"Name":"Bounded Sequences","Duration":"15m 38s","ChapterTopicVideoID":30341,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this clip, we\u0027ll talk about bounded sequences."},{"Start":"00:03.720 ","End":"00:06.050","Text":"We\u0027ve already learned increasing sequences,"},{"Start":"00:06.050 ","End":"00:09.090","Text":"decreasing sequences, limits of the sequence."},{"Start":"00:09.090 ","End":"00:12.195","Text":"Now it\u0027s time for bounded sequences."},{"Start":"00:12.195 ","End":"00:16.440","Text":"First I need to talk about upper and lower bounds."},{"Start":"00:16.440 ","End":"00:18.480","Text":"To help me illustrate this concept,"},{"Start":"00:18.480 ","End":"00:20.670","Text":"let\u0027s consider the following 2 sequences."},{"Start":"00:20.670 ","End":"00:21.990","Text":"This one, 2, 4, 6,"},{"Start":"00:21.990 ","End":"00:23.940","Text":"8, the even numbers,"},{"Start":"00:23.940 ","End":"00:25.560","Text":"and here 1, 1/2,"},{"Start":"00:25.560 ","End":"00:28.600","Text":"1/3, 1/4, we\u0027ve seen this before."},{"Start":"00:29.870 ","End":"00:33.420","Text":"We\u0027ll consider the first sequence."},{"Start":"00:33.420 ","End":"00:37.535","Text":"First, I\u0027m going to use it to illustrate the concept of a lower bound."},{"Start":"00:37.535 ","End":"00:40.460","Text":"Notice that all the elements,"},{"Start":"00:40.460 ","End":"00:42.320","Text":"no matter how far I go,"},{"Start":"00:42.320 ","End":"00:44.840","Text":"they\u0027re all bigger or equal to 2,"},{"Start":"00:44.840 ","End":"00:47.010","Text":"there are at least 2."},{"Start":"00:47.260 ","End":"00:49.340","Text":"This being the case,"},{"Start":"00:49.340 ","End":"00:56.850","Text":"we say that the sequence is bounded below, that\u0027s general."},{"Start":"00:58.640 ","End":"01:03.980","Text":"Specifically, 2 is a lower bound of the sequence."},{"Start":"01:03.980 ","End":"01:10.355","Text":"Because 2 is less than or equal to all the elements of the sequence, all the terms."},{"Start":"01:10.355 ","End":"01:13.130","Text":"Now, lower bounds are not unique."},{"Start":"01:13.130 ","End":"01:15.775","Text":"If I have a lower bound in this case 2"},{"Start":"01:15.775 ","End":"01:18.800","Text":"anything smaller than 2 will also be a lower bound."},{"Start":"01:18.800 ","End":"01:21.740","Text":"For example, 1 is also a lower bound because it\u0027s"},{"Start":"01:21.740 ","End":"01:25.340","Text":"also less than or equal to all the elements in the sequence."},{"Start":"01:25.340 ","End":"01:27.595","Text":"That\u0027s just something to note."},{"Start":"01:27.595 ","End":"01:36.080","Text":"I\u0027m going to use the second sequence to illustrate the concept of an upper bound."},{"Start":"01:36.080 ","End":"01:40.895","Text":"Notice that in the second sequence,"},{"Start":"01:40.895 ","End":"01:44.270","Text":"all the elements are less than or equal to 1,"},{"Start":"01:44.270 ","End":"01:47.880","Text":"they\u0027re all at most 1."},{"Start":"01:47.990 ","End":"01:52.235","Text":"In this case, we say the sequence is bounded from above"},{"Start":"01:52.235 ","End":"01:56.590","Text":"and 1 is an upper bound of the sequence."},{"Start":"01:56.590 ","End":"02:00.380","Text":"Here too, we don\u0027t have uniqueness because anything"},{"Start":"02:00.380 ","End":"02:04.400","Text":"that\u0027s bigger than an upper bound is also going to be an upper bound."},{"Start":"02:04.400 ","End":"02:06.620","Text":"7 is bigger than 1."},{"Start":"02:06.620 ","End":"02:08.180","Text":"It\u0027s also going to be an upper bound."},{"Start":"02:08.180 ","End":"02:12.840","Text":"7 is bigger or equal to all the elements of the sequence."},{"Start":"02:13.120 ","End":"02:20.630","Text":"Now it just so happens that this second sequence also has a lower bound."},{"Start":"02:20.630 ","End":"02:22.190","Text":"It\u0027s bounded from below."},{"Start":"02:22.190 ","End":"02:24.605","Text":"Because if I take 0,"},{"Start":"02:24.605 ","End":"02:32.390","Text":"all these terms are positive and they\u0027re all bigger than or equal to 0."},{"Start":"02:32.390 ","End":"02:35.590","Text":"It also has a lower bound."},{"Start":"02:35.590 ","End":"02:40.450","Text":"This has an upper bound and a lower bound."},{"Start":"02:40.450 ","End":"02:44.405","Text":"Because this sequence has both an upper and a lower bound,"},{"Start":"02:44.405 ","End":"02:45.620","Text":"we say it\u0027s bounded."},{"Start":"02:45.620 ","End":"02:49.655","Text":"In general, the sequence is bounded above and below,"},{"Start":"02:49.655 ","End":"02:51.650","Text":"then we just say it\u0027s bounded."},{"Start":"02:51.650 ","End":"02:53.930","Text":"Here we had an example that was bounded,"},{"Start":"02:53.930 ","End":"02:56.420","Text":"but the previous one, what was it?"},{"Start":"02:56.420 ","End":"03:01.415","Text":"It was 2, 4, 6, 8, etc."},{"Start":"03:01.415 ","End":"03:04.370","Text":"This one isn\u0027t bounded."},{"Start":"03:04.370 ","End":"03:07.685","Text":"We showed that it\u0027s bounded from below,"},{"Start":"03:07.685 ","End":"03:10.160","Text":"but it has no bound from above,"},{"Start":"03:10.160 ","End":"03:11.944","Text":"and no upper bound."},{"Start":"03:11.944 ","End":"03:15.170","Text":"It can\u0027t be just bounded."},{"Start":"03:15.170 ","End":"03:19.195","Text":"It\u0027s half bounded, if you want it bounded from below."},{"Start":"03:19.195 ","End":"03:24.230","Text":"Now I\u0027m going to introduce 2 more concepts."},{"Start":"03:24.230 ","End":"03:29.600","Text":"One of them is the greatest lower bound and the other one is the least upper bound."},{"Start":"03:29.600 ","End":"03:31.610","Text":"They have alternative names,"},{"Start":"03:31.610 ","End":"03:35.705","Text":"Latin-sounding names, more international."},{"Start":"03:35.705 ","End":"03:41.270","Text":"The greatest lower bound is called the infimum."},{"Start":"03:41.270 ","End":"03:45.515","Text":"The least upper bound will also be called the supremum."},{"Start":"03:45.515 ","End":"03:47.525","Text":"Let\u0027s start with one of them,"},{"Start":"03:47.525 ","End":"03:49.675","Text":"the greatest lower bound."},{"Start":"03:49.675 ","End":"03:52.240","Text":"What is this?"},{"Start":"03:52.240 ","End":"03:54.530","Text":"Let\u0027s go back to that example."},{"Start":"03:54.530 ","End":"03:55.790","Text":"We had the sequence 2,"},{"Start":"03:55.790 ","End":"03:58.540","Text":"4, 6, 8 and so on."},{"Start":"03:58.540 ","End":"04:02.390","Text":"We found that 2 was a lower bound,"},{"Start":"04:02.390 ","End":"04:07.310","Text":"but that wasn\u0027t the only one that anything less than 2 is also a lower bound."},{"Start":"04:07.310 ","End":"04:09.680","Text":"It has lots of lower bounds."},{"Start":"04:09.680 ","End":"04:11.515","Text":"In fact, infinitely many."},{"Start":"04:11.515 ","End":"04:16.215","Text":"For example, 2, 1, minus 17,"},{"Start":"04:16.215 ","End":"04:22.770","Text":"minus a million, 0, lots of them."},{"Start":"04:22.770 ","End":"04:25.440","Text":"But there\u0027s 1 special 1."},{"Start":"04:25.440 ","End":"04:31.640","Text":"In some sense 2 is a special lower bound out of all the lower bounds."},{"Start":"04:31.640 ","End":"04:35.350","Text":"You might ask what\u0027s so special about it?"},{"Start":"04:35.350 ","End":"04:40.405","Text":"Well, it\u0027s the greatest from all the lower bounds of this sequence,"},{"Start":"04:40.405 ","End":"04:43.285","Text":"2 is actually the greatest."},{"Start":"04:43.285 ","End":"04:48.830","Text":"There\u0027s no other lower bound that\u0027s greater than 2."},{"Start":"04:49.140 ","End":"04:53.920","Text":"Because anything greater than 2"},{"Start":"04:53.920 ","End":"04:59.255","Text":"will no longer be greater than all the terms of the sequence."},{"Start":"04:59.255 ","End":"05:06.940","Text":"It\u0027s not surprising that we give it the name greatest lower bound and often abbreviated."},{"Start":"05:06.940 ","End":"05:12.655","Text":"In fact, I\u0027ll typically be abbreviating it as GLB in this context."},{"Start":"05:12.655 ","End":"05:21.030","Text":"As I mentioned, it\u0027s also called the infimum of the sequence."},{"Start":"05:21.030 ","End":"05:23.280","Text":"For our sequence, it was 2,"},{"Start":"05:23.280 ","End":"05:26.740","Text":"but this property exists in general."},{"Start":"05:26.770 ","End":"05:32.285","Text":"Very similar to the greatest lower bound will be the least upper bound."},{"Start":"05:32.285 ","End":"05:36.140","Text":"It\u0027s just going to work on the opposite side."},{"Start":"05:36.140 ","End":"05:40.650","Text":"Instead of taking lower bounds,"},{"Start":"05:40.650 ","End":"05:42.495","Text":"we\u0027re going to take upper bound."},{"Start":"05:42.495 ","End":"05:45.980","Text":"Let\u0027s return to the sequence we had before."},{"Start":"05:45.980 ","End":"05:47.510","Text":"This one, 1, 1/2, 1/3,"},{"Start":"05:47.510 ","End":"05:49.405","Text":"1/4, and so on."},{"Start":"05:49.405 ","End":"05:53.525","Text":"We said that 1 is an upper bound,"},{"Start":"05:53.525 ","End":"05:57.710","Text":"but anything greater than 1 is also an upper bound."},{"Start":"05:57.710 ","End":"06:00.415","Text":"There\u0027s infinitely many."},{"Start":"06:00.415 ","End":"06:06.225","Text":"For example, 1, 2, 13, 100,"},{"Start":"06:06.225 ","End":"06:09.660","Text":"245, they are all upper bounds,"},{"Start":"06:09.660 ","End":"06:14.035","Text":"they are all bigger or equal to everything in the sequence."},{"Start":"06:14.035 ","End":"06:20.285","Text":"Amongst all these infinitely many upper bounds is 1 special 1,"},{"Start":"06:20.285 ","End":"06:22.805","Text":"and that is the number 1."},{"Start":"06:22.805 ","End":"06:25.475","Text":"What\u0027s special about it?"},{"Start":"06:25.475 ","End":"06:33.365","Text":"Well, earlier we talked about the greatest lower bound and it was the greatest."},{"Start":"06:33.365 ","End":"06:40.530","Text":"This 1 has the opposite property that it\u0027s the smallest or the least."},{"Start":"06:40.530 ","End":"06:43.845","Text":"There\u0027s no other upper bound less than it."},{"Start":"06:43.845 ","End":"06:52.909","Text":"Anything that\u0027s less than 1 is not going to be an upper bound."},{"Start":"06:52.909 ","End":"06:57.650","Text":"Because an upper bound has to be bigger or equal to all the elements of the sequence,"},{"Start":"06:57.650 ","End":"07:00.390","Text":"so has to be at least 1."},{"Start":"07:01.710 ","End":"07:08.770","Text":"Such an upper bound that\u0027s the least 1 is just called the least upper bound,"},{"Start":"07:08.770 ","End":"07:12.230","Text":"and the abbreviated LUB,"},{"Start":"07:12.570 ","End":"07:16.705","Text":"also called the supremum of the sequence."},{"Start":"07:16.705 ","End":"07:26.590","Text":"Now we also noted earlier that this sequence also has a greatest lower bound."},{"Start":"07:26.590 ","End":"07:29.380","Text":"Well, we noted that it has a lower bound,"},{"Start":"07:29.380 ","End":"07:34.705","Text":"that 0 is a lower bound because all of these are positive and are big or equal to 0."},{"Start":"07:34.705 ","End":"07:41.980","Text":"In fact, 0 is the greatest lower bound."},{"Start":"07:41.980 ","End":"07:46.315","Text":"Nothing greater than 0 will be less than or equal to all of these."},{"Start":"07:46.315 ","End":"07:48.865","Text":"Can you think why?"},{"Start":"07:48.865 ","End":"07:51.625","Text":"Well, I\u0027ll tell you."},{"Start":"07:51.625 ","End":"07:54.475","Text":"Let\u0027s see. I want to keep it inside."},{"Start":"07:54.475 ","End":"07:58.690","Text":"Well, if something is greater than 0,"},{"Start":"07:58.690 ","End":"08:02.350","Text":"then at some point any number that\u0027s larger than 0,"},{"Start":"08:02.350 ","End":"08:08.125","Text":"there\u0027s going to be some n such that 1 over n is less than any positive number."},{"Start":"08:08.125 ","End":"08:11.770","Text":"There\u0027ll be some members of the sequence that will be smaller than it,"},{"Start":"08:11.770 ","End":"08:17.660","Text":"so it can\u0027t possibly be a lower bound,"},{"Start":"08:17.660 ","End":"08:20.080","Text":"so that\u0027s a proof by contradiction."},{"Start":"08:20.080 ","End":"08:24.025","Text":"Anyway, I don\u0027t want to dwell on that too much."},{"Start":"08:24.025 ","End":"08:30.685","Text":"What I want to do is make things a bit more formal."},{"Start":"08:30.685 ","End":"08:33.969","Text":"We define things pretty vaguely."},{"Start":"08:33.969 ","End":"08:38.365","Text":"Let\u0027s just go over these concepts again."},{"Start":"08:38.365 ","End":"08:43.430","Text":"We start off in general with the sequence a_n."},{"Start":"08:43.680 ","End":"08:54.400","Text":"If we have a number capital M such that a_n is less than or equal to M for all n,"},{"Start":"08:54.400 ","End":"09:00.610","Text":"in other words, M is bigger or equal to all the terms in the sequence,"},{"Start":"09:00.610 ","End":"09:04.460","Text":"then it\u0027s called an upper bound of the sequence."},{"Start":"09:04.500 ","End":"09:09.655","Text":"If the sequence has an upper bound and not all sequences do,"},{"Start":"09:09.655 ","End":"09:14.590","Text":"then it said to be bounded above or bounded from above,"},{"Start":"09:14.590 ","End":"09:17.500","Text":"optional the word from."},{"Start":"09:17.500 ","End":"09:20.365","Text":"Some upper bound are special."},{"Start":"09:20.365 ","End":"09:26.035","Text":"An upper bound of a sequence is called the least upper bound, LUB,"},{"Start":"09:26.035 ","End":"09:33.400","Text":"or supremum of the sequence if there is no smaller upper bound of the sequence."},{"Start":"09:33.400 ","End":"09:34.600","Text":"In other words, it\u0027s the smallest."},{"Start":"09:34.600 ","End":"09:36.740","Text":"There\u0027s nothing smaller than it."},{"Start":"09:37.440 ","End":"09:42.699","Text":"All this is pretty much repeated for the lower bounds."},{"Start":"09:42.699 ","End":"09:48.460","Text":"We take a little m such that a_n is bigger or equal to m for all n,"},{"Start":"09:48.460 ","End":"09:53.589","Text":"meaning that little m is less than or equal to all the members of the sequence,"},{"Start":"09:53.589 ","End":"09:55.810","Text":"so it\u0027s called the lower bound."},{"Start":"09:55.810 ","End":"09:58.030","Text":"If the sequence has a lower bound,"},{"Start":"09:58.030 ","End":"10:00.280","Text":"it\u0027s bounded from below."},{"Start":"10:00.280 ","End":"10:02.530","Text":"A lower bound might be special."},{"Start":"10:02.530 ","End":"10:10.224","Text":"It could be the greatest lower bound or infimum if it\u0027s the greatest,"},{"Start":"10:10.224 ","End":"10:14.275","Text":"if there\u0027s no other lower bound that\u0027s greater than it."},{"Start":"10:14.275 ","End":"10:17.020","Text":"Those are the formal definitions."},{"Start":"10:17.020 ","End":"10:20.210","Text":"Next, let\u0027s move on to an example."},{"Start":"10:20.280 ","End":"10:23.890","Text":"In this example exercise,"},{"Start":"10:23.890 ","End":"10:25.780","Text":"we\u0027re given the sequence a_n,"},{"Start":"10:25.780 ","End":"10:29.500","Text":"by this formula, minus 1^n plus 1 over n squared,"},{"Start":"10:29.500 ","End":"10:31.255","Text":"and these 3 parts."},{"Start":"10:31.255 ","End":"10:38.875","Text":"First of all, write a few upper bounds of the sequence and write its least upper bound."},{"Start":"10:38.875 ","End":"10:43.060","Text":"Then part b is to write a few lower bounds of"},{"Start":"10:43.060 ","End":"10:48.190","Text":"the sequence and what is its greatest lower bound."},{"Start":"10:48.190 ","End":"10:49.990","Text":"Finally, the question is,"},{"Start":"10:49.990 ","End":"10:52.580","Text":"is the sequence bounded?"},{"Start":"10:53.010 ","End":"10:55.675","Text":"For the solution."},{"Start":"10:55.675 ","End":"10:57.610","Text":"Before we get properly started,"},{"Start":"10:57.610 ","End":"10:59.379","Text":"let\u0027s just write a few terms,"},{"Start":"10:59.379 ","End":"11:00.880","Text":"see what\u0027s going on."},{"Start":"11:00.880 ","End":"11:03.850","Text":"Let n equal 1,"},{"Start":"11:03.850 ","End":"11:05.485","Text":"then 2, then 3,"},{"Start":"11:05.485 ","End":"11:11.170","Text":"and 4, 5, 6."},{"Start":"11:11.170 ","End":"11:14.560","Text":"You\u0027ll see that these are what we get."},{"Start":"11:14.560 ","End":"11:17.920","Text":"There\u0027s something that happens alternately."},{"Start":"11:17.920 ","End":"11:20.890","Text":"We have a minus 1, a minus 1, a minus 1,"},{"Start":"11:20.890 ","End":"11:25.340","Text":"and here we have a 1, a 1, and a 1."},{"Start":"11:25.340 ","End":"11:28.290","Text":"Now, if you just stare at it a while,"},{"Start":"11:28.290 ","End":"11:33.975","Text":"you\u0027ll see that this term is the least upper bound."},{"Start":"11:33.975 ","End":"11:36.435","Text":"First of all, it\u0027s an upper bound"},{"Start":"11:36.435 ","End":"11:43.540","Text":"because all the terms in the odd places are negative or 0."},{"Start":"11:43.540 ","End":"11:45.910","Text":"Well, this 1 is 0, but this 1 is negative,"},{"Start":"11:45.910 ","End":"11:48.370","Text":"negative, and so on."},{"Start":"11:48.370 ","End":"11:51.100","Text":"Then in the 2nd, 4th,"},{"Start":"11:51.100 ","End":"11:53.890","Text":"and 6th terms, it keeps getting smaller."},{"Start":"11:53.890 ","End":"11:58.810","Text":"We have 1 1/4, 1 1/16, 1 and 1/36."},{"Start":"11:58.810 ","End":"12:01.870","Text":"This is going to be bigger than all of them."},{"Start":"12:01.870 ","End":"12:04.900","Text":"Since it\u0027s 1 of the terms in the sequence,"},{"Start":"12:04.900 ","End":"12:08.365","Text":"it has to be the least upper bound."},{"Start":"12:08.365 ","End":"12:12.910","Text":"Because any upper bound has to be at least 1 1/4,"},{"Start":"12:12.910 ","End":"12:17.080","Text":"it\u0027s going to be bigger than all the terms in the sequence."},{"Start":"12:17.080 ","End":"12:19.180","Text":"Once we have the least upper bound,"},{"Start":"12:19.180 ","End":"12:25.225","Text":"we can write a few more by just taking any few numbers larger than this."},{"Start":"12:25.225 ","End":"12:28.150","Text":"For example, 2 is an upper bound,"},{"Start":"12:28.150 ","End":"12:34.435","Text":"then 100 and 13 1/2 and Pi and whatever."},{"Start":"12:34.435 ","End":"12:38.620","Text":"Just as long as it\u0027s bigger than 1 1/4."},{"Start":"12:38.620 ","End":"12:41.755","Text":"Now, in part b,"},{"Start":"12:41.755 ","End":"12:44.965","Text":"we want some lower bounds."},{"Start":"12:44.965 ","End":"12:49.764","Text":"Once again, it\u0027s easiest to start with the greatest lower bound."},{"Start":"12:49.764 ","End":"12:57.055","Text":"Notice that the terms in the even places are all positive."},{"Start":"12:57.055 ","End":"13:00.385","Text":"They\u0027re all 1 and something: 1 plus this 1, plus this 1, plus this."},{"Start":"13:00.385 ","End":"13:02.155","Text":"So let\u0027s leave those out."},{"Start":"13:02.155 ","End":"13:04.690","Text":"If we take the terms in odd places,"},{"Start":"13:04.690 ","End":"13:06.985","Text":"we have here minus 1 plus 1 is 0,"},{"Start":"13:06.985 ","End":"13:08.485","Text":"minus 1 plus 1/9,"},{"Start":"13:08.485 ","End":"13:11.125","Text":"minus 1 plus 1/25."},{"Start":"13:11.125 ","End":"13:16.240","Text":"It\u0027s always minus 1 plus a bit and that bit keeps getting smaller."},{"Start":"13:16.240 ","End":"13:21.580","Text":"It looks like that minus 1 is going to be the greatest lower bound."},{"Start":"13:21.580 ","End":"13:26.125","Text":"First of all, it\u0027s a lower bound because as I said,"},{"Start":"13:26.125 ","End":"13:28.060","Text":"these terms, this 1,"},{"Start":"13:28.060 ","End":"13:30.340","Text":"and this 1, and this 1 are positive."},{"Start":"13:30.340 ","End":"13:32.170","Text":"In the odd places,"},{"Start":"13:32.170 ","End":"13:35.470","Text":"it\u0027s minus 1 plus something positive,"},{"Start":"13:35.470 ","End":"13:41.300","Text":"so they\u0027re all bigger than minus 1."},{"Start":"13:42.480 ","End":"13:49.810","Text":"Now, why is it the greatest lower bound?"},{"Start":"13:49.810 ","End":"13:53.590","Text":"Because if I take something bigger than minus 1,"},{"Start":"13:53.590 ","End":"13:56.995","Text":"it\u0027s going to be minus 1 plus something positive."},{"Start":"13:56.995 ","End":"14:01.630","Text":"Sooner or later, 1/4,"},{"Start":"14:01.630 ","End":"14:04.465","Text":"1/16, 1/36, or get to 1 over something,"},{"Start":"14:04.465 ","End":"14:07.525","Text":"that\u0027s less than that something positive,"},{"Start":"14:07.525 ","End":"14:12.860","Text":"so it\u0027ll be less than anything that\u0027s bigger than minus 1."},{"Start":"14:12.860 ","End":"14:20.760","Text":"I think it\u0027s intuitively clear that minus 1 is the greatest lower bound."},{"Start":"14:21.310 ","End":"14:26.745","Text":"We were asked to write a few more lower bounds."},{"Start":"14:26.745 ","End":"14:32.950","Text":"Is there anything smaller than minus 1?"},{"Start":"14:32.950 ","End":"14:36.070","Text":"Minus 1 1/4, minus 2,"},{"Start":"14:36.070 ","End":"14:40.700","Text":"minus 17, minus 100, minus a zillion."},{"Start":"14:43.140 ","End":"14:46.915","Text":"Part c, we were asked if the sequence is bounded."},{"Start":"14:46.915 ","End":"14:49.780","Text":"The answer of course is yes,"},{"Start":"14:49.780 ","End":"14:55.570","Text":"because it has some upper bound,"},{"Start":"14:55.570 ","End":"14:58.970","Text":"it\u0027s bounded above, and it has lower bound,"},{"Start":"14:58.970 ","End":"15:00.545","Text":"so it\u0027s bounded below,"},{"Start":"15:00.545 ","End":"15:03.125","Text":"and if it\u0027s bounded above and below,"},{"Start":"15:03.125 ","End":"15:05.565","Text":"then it\u0027s just bounded."},{"Start":"15:05.565 ","End":"15:10.175","Text":"That answers that example exercise."},{"Start":"15:10.175 ","End":"15:13.775","Text":"I want to end this clip with a useful theorem,"},{"Start":"15:13.775 ","End":"15:16.865","Text":"turns out to be very useful, in fact."},{"Start":"15:16.865 ","End":"15:23.410","Text":"If you have a sequence that\u0027s monotonic and it\u0027s bounded above and below,"},{"Start":"15:23.410 ","End":"15:26.135","Text":"that means of course, then it converges,"},{"Start":"15:26.135 ","End":"15:28.530","Text":"meaning it has a limit."},{"Start":"15:28.710 ","End":"15:33.620","Text":"If a sequence is increasing or decreasing and is also bounded,"},{"Start":"15:33.620 ","End":"15:35.870","Text":"then it has a limit."},{"Start":"15:35.870 ","End":"15:39.360","Text":"I\u0027m ending this clip here."}],"ID":32378},{"Watched":false,"Name":"Exercise 1","Duration":"2m 33s","ChapterTopicVideoID":26301,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have A,"},{"Start":"00:03.450 ","End":"00:06.210","Text":"a non-empty subset of the reels."},{"Start":"00:06.210 ","End":"00:09.540","Text":"It doesn\u0027t say, but it\u0027s bounded below."},{"Start":"00:09.540 ","End":"00:12.720","Text":"We know this because we\u0027re told that it has an infimum,"},{"Start":"00:12.720 ","End":"00:15.840","Text":"a greatest lower bound, which is Alpha."},{"Start":"00:15.840 ","End":"00:21.960","Text":"Our task is to show that there is a sequence a_n where all the members come"},{"Start":"00:21.960 ","End":"00:29.775","Text":"from A and the sequence converges to Alpha."},{"Start":"00:29.775 ","End":"00:37.360","Text":"For each n, Alpha is the greatest lower bound of the sequence."},{"Start":"00:37.360 ","End":"00:42.020","Text":"Alpha plus 1 over n can\u0027t be a lower bound because if it was,"},{"Start":"00:42.020 ","End":"00:44.335","Text":"it would be greater than the greatest."},{"Start":"00:44.335 ","End":"00:47.675","Text":"Now, what does it mean that it\u0027s not a lower bound?"},{"Start":"00:47.675 ","End":"00:51.050","Text":"There is some element of a,"},{"Start":"00:51.050 ","End":"00:53.484","Text":"I call it a_n,"},{"Start":"00:53.484 ","End":"01:01.505","Text":"such that a_n is less than Alpha plus 1 over n, strictly less than."},{"Start":"01:01.505 ","End":"01:07.880","Text":"By the way, a_n is still bigger or equal to Alpha because Alpha is a lower bound."},{"Start":"01:07.880 ","End":"01:12.710","Text":"We repeat this procedure for each n and choose some a_n less"},{"Start":"01:12.710 ","End":"01:17.630","Text":"than Alpha plus 1 over n and we get the sequence that it\u0027s all members of a,"},{"Start":"01:17.630 ","End":"01:19.340","Text":"so it\u0027s a subset."},{"Start":"01:19.340 ","End":"01:26.510","Text":"All we have to do now is show that the limit as n goes to infinity of a_n is Alpha."},{"Start":"01:26.510 ","End":"01:34.005","Text":"We\u0027re going to use the Epsilon N definition of the limit."},{"Start":"01:34.005 ","End":"01:39.020","Text":"Let\u0027s take Epsilon bigger than 0 arbitrary."},{"Start":"01:39.020 ","End":"01:44.905","Text":"We can choose an integer capital N bigger than 1 over Epsilon."},{"Start":"01:44.905 ","End":"01:49.175","Text":"What we have is that for all n bigger than or equal to N,"},{"Start":"01:49.175 ","End":"01:51.800","Text":"the absolute value of a_n minus Alpha,"},{"Start":"01:51.800 ","End":"01:57.350","Text":"we can drop the absolute value because a_n is bigger or equal to Alpha."},{"Start":"01:57.350 ","End":"01:59.465","Text":"So this is a_n minus Alpha."},{"Start":"01:59.465 ","End":"02:02.060","Text":"That\u0027s less than 1 over n,"},{"Start":"02:02.060 ","End":"02:09.085","Text":"because a_n is less than Alpha plus 1 over n. Since n is bigger or equal to N,"},{"Start":"02:09.085 ","End":"02:12.350","Text":"then this reciprocal inverts the direction of"},{"Start":"02:12.350 ","End":"02:15.935","Text":"the inequality and this is less than Epsilon."},{"Start":"02:15.935 ","End":"02:19.830","Text":"In short, this absolute value of a_n minus Alpha less than"},{"Start":"02:19.830 ","End":"02:24.725","Text":"Epsilon for all N bigger or equal to n and this proves the convergence."},{"Start":"02:24.725 ","End":"02:28.850","Text":"We\u0027ve shown that for each Epsilon there is an N depending on Epsilon"},{"Start":"02:28.850 ","End":"02:34.060","Text":"such that this is less than this and so we\u0027re done."}],"ID":32379},{"Watched":false,"Name":"Exercise 2","Duration":"1m 12s","ChapterTopicVideoID":26302,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.004","Text":"I\u0027m going to rephrase this exercise in simple words."},{"Start":"00:04.004 ","End":"00:05.700","Text":"What we have to show,"},{"Start":"00:05.700 ","End":"00:08.040","Text":"is that if we have a rational number,"},{"Start":"00:08.040 ","End":"00:10.515","Text":"we can find a sequence of irrationals,"},{"Start":"00:10.515 ","End":"00:12.720","Text":"that converge to it."},{"Start":"00:12.720 ","End":"00:15.120","Text":"There are many ways of doing this."},{"Start":"00:15.120 ","End":"00:18.390","Text":"It\u0027s constructive proof."},{"Start":"00:18.390 ","End":"00:26.310","Text":"For example, you could take xn as x naught plus pi over n. For each n,"},{"Start":"00:26.310 ","End":"00:28.800","Text":"x naught is rational."},{"Start":"00:28.800 ","End":"00:33.060","Text":"Pi over N is irrational because Pi is irrational."},{"Start":"00:33.060 ","End":"00:38.070","Text":"The sum of a rational and an irrational gives us an irrational."},{"Start":"00:38.070 ","End":"00:41.325","Text":"Now we\u0027ll have to show is that,"},{"Start":"00:41.325 ","End":"00:45.160","Text":"xn converges to x naught."},{"Start":"00:46.390 ","End":"00:51.535","Text":"Limit as n goes to infinity of xn,"},{"Start":"00:51.535 ","End":"00:54.800","Text":"is, now this x naught is a constant,"},{"Start":"00:54.800 ","End":"00:57.170","Text":"so I can take a constant in front of the limit,"},{"Start":"00:57.170 ","End":"01:01.100","Text":"so it\u0027s x-naught plus the limit of Pi over N,"},{"Start":"01:01.100 ","End":"01:05.175","Text":"and Pi over n goes to 0."},{"Start":"01:05.175 ","End":"01:07.350","Text":"We end up with just x naught,"},{"Start":"01:07.350 ","End":"01:12.430","Text":"and that\u0027s this part. We\u0027re done."}],"ID":32380},{"Watched":false,"Name":"Exercise 3","Duration":"3m 10s","ChapterTopicVideoID":26303,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have a sequence x_n of"},{"Start":"00:03.570 ","End":"00:08.955","Text":"real numbers and we have to prove or disprove the following 2 statements."},{"Start":"00:08.955 ","End":"00:18.795","Text":"A says that if x_n goes to 0 and y_n is bounded then the product x_n y_n goes to 0."},{"Start":"00:18.795 ","End":"00:26.100","Text":"In part B similar this time x_n goes to infinity and also y_n is a bounded sequence."},{"Start":"00:26.100 ","End":"00:30.525","Text":"Then the question is, does x_n y_n go to infinity?"},{"Start":"00:30.525 ","End":"00:32.505","Text":"Well, we\u0027ll do them 1 at a time."},{"Start":"00:32.505 ","End":"00:34.050","Text":"First of all A,"},{"Start":"00:34.050 ","End":"00:36.060","Text":"this turns out to be true."},{"Start":"00:36.060 ","End":"00:40.250","Text":"It\u0027s actually a useful result and you should remember this,"},{"Start":"00:40.250 ","End":"00:41.869","Text":"it will be useful in future."},{"Start":"00:41.869 ","End":"00:48.620","Text":"In words, it says that a null sequence times a bounded sequence is a null sequence."},{"Start":"00:48.620 ","End":"00:50.975","Text":"Null sequence is 1 that goes to 0."},{"Start":"00:50.975 ","End":"00:52.940","Text":"Anyway, let\u0027s prove this."},{"Start":"00:52.940 ","End":"00:56.600","Text":"Let M be a bound for y_n."},{"Start":"00:56.600 ","End":"00:59.015","Text":"If it\u0027s bounded, then it has a bound."},{"Start":"00:59.015 ","End":"01:04.790","Text":"What that means is that all the y_n are bounded in absolute value by this"},{"Start":"01:04.790 ","End":"01:11.840","Text":"M. Note that 0 is less than or equal to the absolute value of x_n y_n."},{"Start":"01:11.840 ","End":"01:14.500","Text":"The absolute value of anything is bigger or equal to 0."},{"Start":"01:14.500 ","End":"01:18.060","Text":"We can break this up because absolute value of x_n is"},{"Start":"01:18.060 ","End":"01:21.990","Text":"non-negative and absolute value of y_n is less than or equal to M,"},{"Start":"01:21.990 ","End":"01:24.170","Text":"we get this inequality."},{"Start":"01:24.170 ","End":"01:29.960","Text":"That means that the limit of this part is 0,"},{"Start":"01:29.960 ","End":"01:33.365","Text":"the absolute value of x_n also goes to 0."},{"Start":"01:33.365 ","End":"01:35.420","Text":"I\u0027ll say more on this in a moment."},{"Start":"01:35.420 ","End":"01:42.710","Text":"The limit of absolute value of x_n y_n goes to 0 by the sandwich theorem,"},{"Start":"01:42.710 ","End":"01:45.170","Text":"it\u0027s sandwiched between this and 0,"},{"Start":"01:45.170 ","End":"01:48.240","Text":"which you can think of as a 0 sequence."},{"Start":"01:48.980 ","End":"01:55.555","Text":"That means that the limit of x_n y_n without the absolute value is 0."},{"Start":"01:55.555 ","End":"02:00.110","Text":"Now I said I\u0027d comment on both this and this."},{"Start":"02:00.110 ","End":"02:07.280","Text":"In general, a_n goes to 0 if and only if the absolute value of a_n goes to 0."},{"Start":"02:07.280 ","End":"02:11.700","Text":"That explains why absolute value of x_n goes to 0."},{"Start":"02:11.700 ","End":"02:15.365","Text":"It also explains the other way how we got from this to this."},{"Start":"02:15.365 ","End":"02:16.610","Text":"For null sequence is,"},{"Start":"02:16.610 ","End":"02:19.835","Text":"it doesn\u0027t matter if you put an absolute value or not,"},{"Start":"02:19.835 ","End":"02:23.395","Text":"they\u0027re both null sequences or neither 1 is."},{"Start":"02:23.395 ","End":"02:26.775","Text":"Now part B turns out to be false."},{"Start":"02:26.775 ","End":"02:33.545","Text":"I mean, it might go to infinity and it might not but in general, not."},{"Start":"02:33.545 ","End":"02:36.395","Text":"All you have to do is give 1 counterexample."},{"Start":"02:36.395 ","End":"02:43.890","Text":"Let x_n be n and y_n be 1 over n. Now,"},{"Start":"02:43.890 ","End":"02:52.355","Text":"certainly x_n goes to infinity and also the sequence y_n is bounded."},{"Start":"02:52.355 ","End":"02:56.420","Text":"We can see that the absolute value of y_n is always less than or equal to 1."},{"Start":"02:56.420 ","End":"02:59.390","Text":"But the limit of x_n,"},{"Start":"02:59.390 ","End":"03:02.420","Text":"y_n is the limit of n times 1 over n, which is the limit of 1,"},{"Start":"03:02.420 ","End":"03:06.400","Text":"which is 1 and that\u0027s not infinity."},{"Start":"03:06.500 ","End":"03:10.390","Text":"That concludes B and we\u0027re done."}],"ID":32381},{"Watched":false,"Name":"Exercise 4","Duration":"2m 39s","ChapterTopicVideoID":26304,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, x_n is a sequence of real numbers and"},{"Start":"00:04.110 ","End":"00:08.715","Text":"we have to prove or disprove a and b."},{"Start":"00:08.715 ","End":"00:16.635","Text":"A says that if the sequence x_n plus 1 over n x_n converges, then x_n converges."},{"Start":"00:16.635 ","End":"00:18.990","Text":"Turns out this 1 is true."},{"Start":"00:18.990 ","End":"00:23.460","Text":"In b, it\u0027s very similar except there\u0027s a squared here."},{"Start":"00:23.460 ","End":"00:25.710","Text":"It turns out that b is false."},{"Start":"00:25.710 ","End":"00:28.020","Text":"Let\u0027s get started with a."},{"Start":"00:28.020 ","End":"00:32.685","Text":"Let y_n be this sequence here."},{"Start":"00:32.685 ","End":"00:37.590","Text":"Notice that it is just x_n times 1 plus 1 over"},{"Start":"00:37.590 ","End":"00:42.165","Text":"n. Let z_n be"},{"Start":"00:42.165 ","End":"00:48.660","Text":"just this part here so we can write x_n as y_n over z_n."},{"Start":"00:48.660 ","End":"00:57.450","Text":"It\u0027s this, divided by just the 1 plus 1 over n. Y_n converges,"},{"Start":"00:57.450 ","End":"01:02.430","Text":"that\u0027s given, and z_n converges to 1,"},{"Start":"01:02.430 ","End":"01:04.875","Text":"and is also non-0."},{"Start":"01:04.875 ","End":"01:13.300","Text":"We can look at the quotient of 2 converging sequences and conclude that x_n converges."},{"Start":"01:13.300 ","End":"01:18.140","Text":"Whenever the numerator converges and the denominator is not 0 and converges,"},{"Start":"01:18.140 ","End":"01:21.200","Text":"then the quotient converges to the quotient."},{"Start":"01:21.200 ","End":"01:23.815","Text":"Actually, we can say more."},{"Start":"01:23.815 ","End":"01:29.130","Text":"The limit of x_n is the limit of y_n over the limit of z_n."},{"Start":"01:29.130 ","End":"01:33.285","Text":"The limit of z_n is 1,"},{"Start":"01:33.285 ","End":"01:36.440","Text":"so we\u0027re just left with the limit of y_n."},{"Start":"01:36.440 ","End":"01:43.100","Text":"It turns out that x_n converges to the same limit as what this converges to."},{"Start":"01:43.100 ","End":"01:47.090","Text":"Now in part B, it\u0027s false so we need to provide a counterexample."},{"Start":"01:47.090 ","End":"01:48.425","Text":"There\u0027s many possible."},{"Start":"01:48.425 ","End":"01:50.195","Text":"I\u0027m just giving you an example."},{"Start":"01:50.195 ","End":"01:54.340","Text":"Take x_n, which is minus 1 to the n,"},{"Start":"01:54.340 ","End":"02:00.515","Text":"the oscillating sequence that goes back and forth from minus 1 to 1, it doesn\u0027t converge."},{"Start":"02:00.515 ","End":"02:07.964","Text":"But if we take x_n squared plus 1 over n x_n, x_n squared is 1."},{"Start":"02:07.964 ","End":"02:12.065","Text":"1 over n x_n is this over n,"},{"Start":"02:12.065 ","End":"02:15.840","Text":"and this certainly converges to 0."},{"Start":"02:15.880 ","End":"02:20.000","Text":"We could use the previous exercise,"},{"Start":"02:20.000 ","End":"02:25.100","Text":"something bounded minus 1 to the n times something converging to 0,"},{"Start":"02:25.100 ","End":"02:30.540","Text":"1 over n also converges to 0 so this thing goes to 1."},{"Start":"02:30.640 ","End":"02:35.450","Text":"That\u0027s our counterexample because this converges,"},{"Start":"02:35.450 ","End":"02:39.300","Text":"but this doesn\u0027t, and we\u0027re done."}],"ID":32382},{"Watched":false,"Name":"Exercise 5","Duration":"5m 5s","ChapterTopicVideoID":26305,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"In this exercise, we have 2 numbers,"},{"Start":"00:04.875 ","End":"00:09.120","Text":"b_1 and a_1, both positive b_1 less than a_1."},{"Start":"00:09.120 ","End":"00:13.095","Text":"We\u0027re going to define a sequence recursively,"},{"Start":"00:13.095 ","End":"00:16.275","Text":"a sequence a_n, and a sequence b_n,"},{"Start":"00:16.275 ","End":"00:22.560","Text":"a_n plus 1 will be the arithmetic mean of a_n and b_n,"},{"Start":"00:22.560 ","End":"00:28.140","Text":"and b_n plus 1 will be the geometric mean of a_n and b_n."},{"Start":"00:28.140 ","End":"00:31.670","Text":"For all n belonging to the set of natural numbers,"},{"Start":"00:31.670 ","End":"00:34.354","Text":"which means that this is recursive definition."},{"Start":"00:34.354 ","End":"00:40.100","Text":"Now, our task is to show that both sequences a_n and b_n converge,"},{"Start":"00:40.100 ","End":"00:44.090","Text":"and more than that, that they converge to the same limit."},{"Start":"00:44.090 ","End":"00:49.240","Text":"As a hint, we\u0027re going to use the AM-GM inequality,"},{"Start":"00:49.240 ","End":"00:51.815","Text":"this mean, the arithmetic mean, geometric mean."},{"Start":"00:51.815 ","End":"00:57.410","Text":"It\u0027s known that the geometric mean of"},{"Start":"00:57.410 ","End":"01:03.070","Text":"2 positive numbers is less than the arithmetic mean,"},{"Start":"01:03.070 ","End":"01:05.955","Text":"provided that the 2 numbers are not equal,"},{"Start":"01:05.955 ","End":"01:09.465","Text":"otherwise, we get equality here."},{"Start":"01:09.465 ","End":"01:16.545","Text":"The first claim is that for all n 0s less than b_n and less than a_n,"},{"Start":"01:16.545 ","End":"01:20.575","Text":"and a proof by induction, abbreviated proof."},{"Start":"01:20.575 ","End":"01:24.940","Text":"N equals 1, we can see by ourselves that from here."},{"Start":"01:24.940 ","End":"01:27.175","Text":"This is just the inductive step,"},{"Start":"01:27.175 ","End":"01:30.805","Text":"assuming it\u0027s true for a_n that show it\u0027s true for n plus 1."},{"Start":"01:30.805 ","End":"01:36.205","Text":"So b_n plus 1 is the square root of a_n, b n. Now,"},{"Start":"01:36.205 ","End":"01:38.965","Text":"each of these is positive,"},{"Start":"01:38.965 ","End":"01:43.840","Text":"so the square root of this is also going to be positive."},{"Start":"01:43.840 ","End":"01:46.735","Text":"That\u0027s the bigger than 0 part."},{"Start":"01:46.735 ","End":"01:50.425","Text":"Now I need to show that b_n is less than a_n."},{"Start":"01:50.425 ","End":"01:53.020","Text":"Well, I mean, for n plus 1,"},{"Start":"01:53.020 ","End":"01:56.485","Text":"b_n plus 1 is the square root of a _n b_n."},{"Start":"01:56.485 ","End":"02:02.370","Text":"By the hint, this is less than a_n plus b_n/ 2,"},{"Start":"02:02.370 ","End":"02:04.365","Text":"and that\u0027s just a_n plus 1."},{"Start":"02:04.365 ","End":"02:07.005","Text":"So this is less than this."},{"Start":"02:07.005 ","End":"02:09.815","Text":"We\u0027ve proven this by induction."},{"Start":"02:09.815 ","End":"02:17.760","Text":"The next claim is that the sequence a_n is decreasing, again by induction."},{"Start":"02:18.100 ","End":"02:22.400","Text":"Well, a_1 is whatever it is."},{"Start":"02:22.400 ","End":"02:31.895","Text":"All I have to show is that each subsequent term is less than the previous term,"},{"Start":"02:31.895 ","End":"02:34.715","Text":"so a_n plus 1,"},{"Start":"02:34.715 ","End":"02:39.270","Text":"which is this, the arithmetic mean."},{"Start":"02:41.320 ","End":"02:46.190","Text":"We can also write less than or equal to a_n plus a_n/2."},{"Start":"02:46.190 ","End":"02:51.914","Text":"In other words, I\u0027ve replaced b_n with a_n because b_n is less than a_n,"},{"Start":"02:51.914 ","End":"02:55.460","Text":"and so We get that a_n plus 1 is less than a_n."},{"Start":"02:55.460 ","End":"02:58.700","Text":"Well, less than or equal to is all we need for decreasing."},{"Start":"02:58.700 ","End":"03:01.620","Text":"It\u0027s actually strictly decreasing."},{"Start":"03:01.850 ","End":"03:07.770","Text":"Similarly, b_n is increasing and we have to"},{"Start":"03:07.770 ","End":"03:13.290","Text":"show is that b_n plus 1 is greater than b_n."},{"Start":"03:13.290 ","End":"03:16.935","Text":"Well, b_n plus 1 is the square root of a_n, b_n."},{"Start":"03:16.935 ","End":"03:22.460","Text":"That\u0027s bigger or equal to actually bigger than square root of b_n times b_n,"},{"Start":"03:22.460 ","End":"03:26.150","Text":"since a_n is bigger than b_n and that\u0027s b_n."},{"Start":"03:26.150 ","End":"03:30.155","Text":"That\u0027s the b_n part."},{"Start":"03:30.155 ","End":"03:34.730","Text":"Now this is increasing and this is decreasing."},{"Start":"03:34.730 ","End":"03:39.960","Text":"Claim next is that both are bounded."},{"Start":"03:40.300 ","End":"03:45.930","Text":"That\u0027s easy to see because each b_n is bigger or equal"},{"Start":"03:45.930 ","End":"03:50.775","Text":"to b_ 1 and b_n is less than or equal to a_n,"},{"Start":"03:50.775 ","End":"03:53.160","Text":"and a_n is less than or equal to a_1,"},{"Start":"03:53.160 ","End":"04:03.300","Text":"so b_n and a_n are both bounded in the interval from b_1 to a_1."},{"Start":"04:03.300 ","End":"04:09.230","Text":"Now we apply the monotone convergence criterion for increasing,"},{"Start":"04:09.230 ","End":"04:14.435","Text":"unbounded or decreasing unbounded and conclude that they both converge."},{"Start":"04:14.435 ","End":"04:18.365","Text":"The last part we have to show is that the limits are equal."},{"Start":"04:18.365 ","End":"04:23.030","Text":"Let\u0027s call this limit a and this limit b,"},{"Start":"04:23.030 ","End":"04:29.680","Text":"so the limit of a_n plus 1 is the same as the limit of a_n."},{"Start":"04:29.680 ","End":"04:31.330","Text":"If you just shift by 1,"},{"Start":"04:31.330 ","End":"04:33.040","Text":"it doesn\u0027t change the limit."},{"Start":"04:33.040 ","End":"04:35.840","Text":"What we get is that a,"},{"Start":"04:35.840 ","End":"04:39.255","Text":"which is the limit of a_n plus 1,"},{"Start":"04:39.255 ","End":"04:42.835","Text":"is the limit of a_n plus 1,"},{"Start":"04:42.835 ","End":"04:46.720","Text":"which is a_n plus b_n/2,"},{"Start":"04:46.720 ","End":"04:51.025","Text":"and that\u0027s equal to a plus b/2."},{"Start":"04:51.025 ","End":"04:55.055","Text":"Now once we have that, a equals a plus b/2,"},{"Start":"04:55.055 ","End":"04:58.250","Text":"by simple algebra, we get 2 a equals a plus b,"},{"Start":"04:58.250 ","End":"05:00.425","Text":"so a equals b."},{"Start":"05:00.425 ","End":"05:03.290","Text":"That\u0027s this last part."},{"Start":"05:03.290 ","End":"05:05.670","Text":"We are done."}],"ID":32383},{"Watched":false,"Name":"Exercise 6","Duration":"3m 14s","ChapterTopicVideoID":26306,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"In this exercise, we\u0027re given a positive number a,"},{"Start":"00:03.705 ","End":"00:07.170","Text":"and we\u0027re also given some x_1 which is positive,"},{"Start":"00:07.170 ","End":"00:13.065","Text":"and from this we build a sequence x_n recursively by"},{"Start":"00:13.065 ","End":"00:19.485","Text":"x_n plus 1 is 1/2 of x_n plus a over x_n."},{"Start":"00:19.485 ","End":"00:24.840","Text":"We have to show that the sequence x_n converges to the square root of a."},{"Start":"00:24.840 ","End":"00:28.799","Text":"First, we\u0027ll show that all the x_n in the sequence are positive,"},{"Start":"00:28.799 ","End":"00:30.705","Text":"and this is by induction."},{"Start":"00:30.705 ","End":"00:36.795","Text":"X_1 is positive, so here\u0027s just the inductive step from k to k plus 1."},{"Start":"00:36.795 ","End":"00:38.580","Text":"If x_k is positive,"},{"Start":"00:38.580 ","End":"00:41.610","Text":"a over x_k is positive."},{"Start":"00:41.610 ","End":"00:45.275","Text":"Then if we take the average of x_k and a over x_k,"},{"Start":"00:45.275 ","End":"00:48.150","Text":"that will also be positive."},{"Start":"00:48.190 ","End":"00:52.430","Text":"Now we apply the AM-GM inequality,"},{"Start":"00:52.430 ","End":"00:57.500","Text":"which says that the arithmetic mean is bigger or equal to the geometric mean."},{"Start":"00:57.500 ","End":"00:59.390","Text":"So x_n plus 1,"},{"Start":"00:59.390 ","End":"01:00.995","Text":"which is equal to this,"},{"Start":"01:00.995 ","End":"01:03.860","Text":"which is the arithmetic mean of these 2,"},{"Start":"01:03.860 ","End":"01:07.715","Text":"is bigger or equal to the geometric mean of these 2,"},{"Start":"01:07.715 ","End":"01:11.430","Text":"and that\u0027s equal to square root of a."},{"Start":"01:11.900 ","End":"01:18.920","Text":"All the members of the sequence x_n are bigger or equal to the square root of a,"},{"Start":"01:18.920 ","End":"01:21.765","Text":"at least from 2 onwards."},{"Start":"01:21.765 ","End":"01:26.885","Text":"The next thing we\u0027ll show is that the sequence is monotonically decreasing,"},{"Start":"01:26.885 ","End":"01:32.555","Text":"and we\u0027ll evaluate the difference between 2 consecutive terms,"},{"Start":"01:32.555 ","End":"01:35.170","Text":"x_n plus 1 minus x_n,"},{"Start":"01:35.170 ","End":"01:37.725","Text":"but the definition of x_n plus 1,"},{"Start":"01:37.725 ","End":"01:40.395","Text":"it\u0027s equal to this minus x_n,"},{"Start":"01:40.395 ","End":"01:43.680","Text":"and a bit of simplification gives us,"},{"Start":"01:43.680 ","End":"01:48.455","Text":"this is equal to 1/2 of a minus x_n squared over x_n."},{"Start":"01:48.455 ","End":"01:53.270","Text":"Now, this is negative or at least less than or equal to"},{"Start":"01:53.270 ","End":"01:58.220","Text":"0 because x_n is bigger or equal to square root of a."},{"Start":"01:58.220 ","End":"02:01.540","Text":"So this numerator is non-positive."},{"Start":"02:01.540 ","End":"02:04.880","Text":"We\u0027ve just shown that x_n is decreasing and we"},{"Start":"02:04.880 ","End":"02:10.375","Text":"showed that it\u0027s bounded below by square root of a."},{"Start":"02:10.375 ","End":"02:13.625","Text":"By the monotone sequence theorem,"},{"Start":"02:13.625 ","End":"02:17.840","Text":"we have that x_n converges to some limit,"},{"Start":"02:17.840 ","End":"02:23.390","Text":"call it L. From this definition of x_n plus 1,"},{"Start":"02:23.390 ","End":"02:26.090","Text":"what we can get if we let n go to infinity,"},{"Start":"02:26.090 ","End":"02:29.000","Text":"we can put the limits in here and here."},{"Start":"02:29.000 ","End":"02:32.490","Text":"Notice that all the x_n is non-zero,"},{"Start":"02:32.490 ","End":"02:34.080","Text":"so we can do that."},{"Start":"02:34.080 ","End":"02:38.055","Text":"What this gives us is that L,"},{"Start":"02:38.055 ","End":"02:41.750","Text":"I mean the limit of x_n is the same as the limit of x_n plus 1,"},{"Start":"02:41.750 ","End":"02:47.270","Text":"is equal to 1/2 of L plus a over L. Now what we need is a bit of algebra."},{"Start":"02:47.270 ","End":"02:54.095","Text":"From this, we get the 2L squared is L squared plus a multiplying both sides by 2L."},{"Start":"02:54.095 ","End":"02:59.630","Text":"Then we can get that L squared equals a so that L is the square root of a."},{"Start":"02:59.630 ","End":"03:07.050","Text":"Obviously L is positive because all the x_n\u0027s are positive,"},{"Start":"03:07.050 ","End":"03:14.340","Text":"and that\u0027s what we have to show that the sequence converges to route a so we\u0027re done."}],"ID":32384},{"Watched":false,"Name":"Exercise 7","Duration":"2m 43s","ChapterTopicVideoID":26307,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.269","Text":"In this exercise, we have a sequence x_n in the open interval 0, 1."},{"Start":"00:06.269 ","End":"00:08.640","Text":"Suppose we have the inequality,"},{"Start":"00:08.640 ","End":"00:11.865","Text":"as written here for all n,"},{"Start":"00:11.865 ","End":"00:15.660","Text":"we have to show that the sequence is monotone,"},{"Start":"00:15.660 ","End":"00:18.670","Text":"and to find its limit."},{"Start":"00:19.270 ","End":"00:25.095","Text":"From this inequality, we can get that"},{"Start":"00:25.095 ","End":"00:33.570","Text":"the square root of x_n times 1 minus x_n plus 1 is bigger than,"},{"Start":"00:33.570 ","End":"00:35.774","Text":"we take the 4 over to the other side,"},{"Start":"00:35.774 ","End":"00:37.305","Text":"and then take the square root,"},{"Start":"00:37.305 ","End":"00:39.420","Text":"so that\u0027s a half,"},{"Start":"00:39.420 ","End":"00:42.380","Text":"and from the arithmetic mean,"},{"Start":"00:42.380 ","End":"00:47.120","Text":"geometric mean inequality, what we get is that"},{"Start":"00:47.120 ","End":"00:54.050","Text":"the average arithmetic of x_n and 1 minus x_n plus 1 is bigger,"},{"Start":"00:54.050 ","End":"00:56.960","Text":"or equal to the geometric mean of the same 2 quantities,"},{"Start":"00:56.960 ","End":"01:00.125","Text":"but this, as we just saw,"},{"Start":"01:00.125 ","End":"01:03.485","Text":"is bigger than 1.5."},{"Start":"01:03.485 ","End":"01:09.645","Text":"That x_n plus 1 minus x_n plus 1 is bigger than 1,"},{"Start":"01:09.645 ","End":"01:12.870","Text":"which means take away the 1 from"},{"Start":"01:12.870 ","End":"01:16.620","Text":"both sides and bring the x_n plus 1 to the other side, we get this."},{"Start":"01:16.620 ","End":"01:20.850","Text":"The sequence is monotone decreasing."},{"Start":"01:20.850 ","End":"01:25.845","Text":"It\u0027s also bounded, it\u0027s bounded in the interval from 0 to 1,"},{"Start":"01:25.845 ","End":"01:31.230","Text":"so x_n must have some limit, call it x-naught."},{"Start":"01:32.770 ","End":"01:38.610","Text":"Back to this inequality."},{"Start":"01:38.840 ","End":"01:42.345","Text":"We take the limit of both sides,"},{"Start":"01:42.345 ","End":"01:45.635","Text":"and when we take the limit of an inequality,"},{"Start":"01:45.635 ","End":"01:49.250","Text":"it could go from strict to non-strict."},{"Start":"01:49.250 ","End":"01:52.680","Text":"Otherwise, we could get a big or equal to here."},{"Start":"01:53.840 ","End":"01:56.655","Text":"This gives us that,"},{"Start":"01:56.655 ","End":"01:59.010","Text":"taking the limit, x_n goes to x-naught,"},{"Start":"01:59.010 ","End":"02:00.780","Text":"x_n plus 1 also goes to x-naught,"},{"Start":"02:00.780 ","End":"02:03.470","Text":"that we get this inequality,"},{"Start":"02:03.470 ","End":"02:05.650","Text":"which you can rewrite like this,"},{"Start":"02:05.650 ","End":"02:07.970","Text":"and we\u0027ve got a quadratic inequality,"},{"Start":"02:07.970 ","End":"02:09.800","Text":"but this is a perfect square."},{"Start":"02:09.800 ","End":"02:13.205","Text":"It\u0027s 2 x-naught minus 1 squared."},{"Start":"02:13.205 ","End":"02:17.835","Text":"When you have something squared less than or equal to 0,"},{"Start":"02:17.835 ","End":"02:20.780","Text":"the square has to be bigger or equal to 0 also,"},{"Start":"02:20.780 ","End":"02:22.970","Text":"so it has to be equal to 0,"},{"Start":"02:22.970 ","End":"02:27.165","Text":"and that\u0027s only possible when this is 0."},{"Start":"02:27.165 ","End":"02:31.475","Text":"Extracting x-naught, it gives us that x-naught is a half."},{"Start":"02:31.475 ","End":"02:36.420","Text":"X-naught is the limit of the sequence,"},{"Start":"02:36.420 ","End":"02:38.760","Text":"so that\u0027s the answer."},{"Start":"02:38.760 ","End":"02:40.995","Text":"The limit, as n goes to infinity of x_n,"},{"Start":"02:40.995 ","End":"02:44.260","Text":"is 1.5, and we\u0027re done."}],"ID":32385},{"Watched":false,"Name":"Exercise 8","Duration":"2m 50s","ChapterTopicVideoID":26308,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"In this exercise, we have a non-empty set of real numbers A,"},{"Start":"00:05.340 ","End":"00:10.230","Text":"and we have another real number, x naught."},{"Start":"00:10.230 ","End":"00:19.050","Text":"Our task is to show that there\u0027s a sequence a_ n of elements in A such"},{"Start":"00:19.050 ","End":"00:23.300","Text":"that the distance from x naught to a_ n"},{"Start":"00:23.300 ","End":"00:28.770","Text":"converges to the distance from x naught to the set A."},{"Start":"00:28.770 ","End":"00:30.040","Text":"In case you\u0027ve forgotten,"},{"Start":"00:30.040 ","End":"00:36.320","Text":"the definition of a distance from a point to a set is the infimum or"},{"Start":"00:36.320 ","End":"00:43.890","Text":"greatest lower bound of all the individual distances of x to members of A."},{"Start":"00:44.870 ","End":"00:48.090","Text":"The solution. First of all,"},{"Start":"00:48.090 ","End":"00:56.370","Text":"let\u0027s denote S as this set that\u0027s in the curly brackets,"},{"Start":"00:56.370 ","End":"01:02.000","Text":"and the set of all distances of x to a point in A so"},{"Start":"01:02.000 ","End":"01:08.910","Text":"that the infimum is what we call d of x naught,"},{"Start":"01:08.910 ","End":"01:14.885","Text":"A, and we\u0027ll call that capital D. Now we\u0027re going to build a sequence."},{"Start":"01:14.885 ","End":"01:18.110","Text":"Choose any arbitrary n,"},{"Start":"01:18.110 ","End":"01:20.255","Text":"which is a natural number."},{"Start":"01:20.255 ","End":"01:29.510","Text":"Then D plus 1 over n is not a lower bound of S because D is the greatest lower bound,"},{"Start":"01:29.510 ","End":"01:31.820","Text":"so there can\u0027t be another lower bound bigger than it."},{"Start":"01:31.820 ","End":"01:35.135","Text":"Now, what does it mean that it\u0027s not a lower bound?"},{"Start":"01:35.135 ","End":"01:45.165","Text":"It means that some element of the set is less than D plus 1 over n. Now,"},{"Start":"01:45.165 ","End":"01:50.340","Text":"s is the absolute value of x naught minus a."},{"Start":"01:51.680 ","End":"01:55.640","Text":"This particular a that we chose for this n,"},{"Start":"01:55.640 ","End":"01:57.515","Text":"we\u0027ll call it a_ n,"},{"Start":"01:57.515 ","End":"02:03.645","Text":"and so we get a sequence a_ n. What it satisfies,"},{"Start":"02:03.645 ","End":"02:07.185","Text":"just plug a_n instead of a here,"},{"Start":"02:07.185 ","End":"02:14.930","Text":"is that x naught minus a_ n is less than D plus 1 over n. But it\u0027s"},{"Start":"02:14.930 ","End":"02:23.790","Text":"also bigger or equal to D because D is the infimum of all such expressions."},{"Start":"02:24.260 ","End":"02:28.040","Text":"Now, if we let n go to infinity,"},{"Start":"02:28.040 ","End":"02:30.275","Text":"we get a sandwich here."},{"Start":"02:30.275 ","End":"02:31.850","Text":"The limit, on the 1 hand,"},{"Start":"02:31.850 ","End":"02:35.220","Text":"is bigger or equal to D,"},{"Start":"02:35.930 ","End":"02:40.010","Text":"and it\u0027s also less than or equal to the limit of this,"},{"Start":"02:40.010 ","End":"02:43.415","Text":"which is D, so the limit is exactly D,"},{"Start":"02:43.415 ","End":"02:46.220","Text":"which is d of x naught,"},{"Start":"02:46.220 ","End":"02:50.460","Text":"A as required, and we are done."}],"ID":32386},{"Watched":false,"Name":"Exercise 9","Duration":"2m 49s","ChapterTopicVideoID":26309,"CourseChapterTopicPlaylistID":254167,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we have a bounded sequence, a_k,"},{"Start":"00:04.770 ","End":"00:07.860","Text":"k goes from 1 to infinity, say,"},{"Start":"00:07.860 ","End":"00:09.855","Text":"and for every n,"},{"Start":"00:09.855 ","End":"00:12.990","Text":"we define another sequence, x_n."},{"Start":"00:12.990 ","End":"00:20.100","Text":"By x_n is the supremum of all the a_k in this sequence,"},{"Start":"00:20.100 ","End":"00:23.250","Text":"where the index k is less than n."},{"Start":"00:23.250 ","End":"00:28.215","Text":"This is the same as the maximum because it\u0027s a finite set."},{"Start":"00:28.215 ","End":"00:34.275","Text":"We have to show that the new sequence, x_n converges."},{"Start":"00:34.275 ","End":"00:38.955","Text":"Now, I\u0027m going to state something that may seem obvious."},{"Start":"00:38.955 ","End":"00:47.255","Text":"If I have 2 subsets of the real numbers and they\u0027re both bounded above,"},{"Start":"00:47.255 ","End":"00:54.500","Text":"then the supremum of A has to be less than or equal to the supremum of B."},{"Start":"00:54.500 ","End":"00:57.589","Text":"In other words, if you increase the set,"},{"Start":"00:57.589 ","End":"01:01.580","Text":"you can only increase the supremum."},{"Start":"01:01.580 ","End":"01:05.460","Text":"It seems clear because there\u0027s more terms here,"},{"Start":"01:05.460 ","End":"01:07.400","Text":"so the maximum could grow,"},{"Start":"01:07.400 ","End":"01:08.840","Text":"it could stay the same,"},{"Start":"01:08.840 ","End":"01:11.630","Text":"but it certainly can\u0027t be less."},{"Start":"01:11.630 ","End":"01:15.440","Text":"I\u0027ll give you a formal proof of that."},{"Start":"01:15.440 ","End":"01:20.035","Text":"If you have any element a in A,"},{"Start":"01:20.035 ","End":"01:21.740","Text":"then it also belongs to B."},{"Start":"01:21.740 ","End":"01:24.375","Text":"That\u0027s the definition of set containment."},{"Start":"01:24.375 ","End":"01:28.490","Text":"As a member of B, it\u0027s less than or equal to the supremum of B."},{"Start":"01:28.490 ","End":"01:33.985","Text":"Supremum of B is an upper bound for the set A."},{"Start":"01:33.985 ","End":"01:35.660","Text":"If it\u0027s an upper bound,"},{"Start":"01:35.660 ","End":"01:39.200","Text":"it has to be bigger or equal to the least upper bound."},{"Start":"01:39.200 ","End":"01:42.485","Text":"If m is less than n,"},{"Start":"01:42.485 ","End":"01:50.390","Text":"then the set of all a_k where k is less than m is contained in the set of a_k,"},{"Start":"01:50.390 ","End":"01:51.830","Text":"where k is less than n,"},{"Start":"01:51.830 ","End":"01:56.620","Text":"that means there can only be more elements here."},{"Start":"01:56.620 ","End":"01:58.564","Text":"But what we said above,"},{"Start":"01:58.564 ","End":"02:02.090","Text":"the supremum of this is less than or equal to supremum of this."},{"Start":"02:02.090 ","End":"02:05.940","Text":"In other words, x_m is less than x_n."},{"Start":"02:05.940 ","End":"02:08.460","Text":"So x_n is increasing."},{"Start":"02:08.460 ","End":"02:12.770","Text":"Actually it should be less than or equal to here, but never mind."},{"Start":"02:12.770 ","End":"02:18.790","Text":"Now clearly, if M is an upper bound for the infinite sequence a_k,"},{"Start":"02:18.790 ","End":"02:25.820","Text":"it\u0027s also an upper bound for the partial finite sequence from 1 to n."},{"Start":"02:25.820 ","End":"02:29.555","Text":"This is just x_n,"},{"Start":"02:29.555 ","End":"02:31.480","Text":"the least upper bound."},{"Start":"02:31.480 ","End":"02:36.810","Text":"So x_n is less than or equal to M."},{"Start":"02:36.810 ","End":"02:42.410","Text":"Now what we have is that the sequence x_n is increasing and bounded."},{"Start":"02:42.410 ","End":"02:45.460","Text":"Also by the theorem it must converge."},{"Start":"02:45.460 ","End":"02:50.140","Text":"That\u0027s what we have to show and so we are done."}],"ID":32387}],"Thumbnail":null,"ID":254167},{"Name":"Limit of a Sequence - Limit Arithmetic","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Limit of a Sequence - Limit Arithmetic","Duration":"11m 12s","ChapterTopicVideoID":29720,"CourseChapterTopicPlaylistID":294602,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":31333},{"Watched":false,"Name":"Exercise 1","Duration":"1m 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","End":"00:11.844","Text":"also called the squeeze theorem."},{"Start":"00:11.844 ","End":"00:15.415","Text":"I\u0027ll illustrate with an example and then you\u0027ll see"},{"Start":"00:15.415 ","End":"00:19.945","Text":"where it gets its name and we\u0027ll do a formal definition."},{"Start":"00:19.945 ","End":"00:26.815","Text":"The example I want is the limit as n goes to infinity"},{"Start":"00:26.815 ","End":"00:34.270","Text":"of the nth root of 4^n plus 10^n."},{"Start":"00:34.270 ","End":"00:37.480","Text":"Now, what seems to be the problem?"},{"Start":"00:37.480 ","End":"00:39.580","Text":"Why can\u0027t we just substitute?"},{"Start":"00:39.580 ","End":"00:42.575","Text":"Well, this is equal to,"},{"Start":"00:42.575 ","End":"00:44.960","Text":"if I put it in exponential notation,"},{"Start":"00:44.960 ","End":"00:48.125","Text":"4^n plus 10^n,"},{"Start":"00:48.125 ","End":"00:54.590","Text":"to the power of 1 over n. Now,"},{"Start":"00:54.590 ","End":"00:59.465","Text":"we have this part and this tends to infinity."},{"Start":"00:59.465 ","End":"01:02.240","Text":"This goes to infinity and this goes to infinity."},{"Start":"01:02.240 ","End":"01:08.710","Text":"Infinity plus infinity is infinity and 1 over infinity is 0."},{"Start":"01:08.710 ","End":"01:12.350","Text":"This is the limit of the form infinity to the power of 0,"},{"Start":"01:12.350 ","End":"01:15.040","Text":"which is one of the undefined forms."},{"Start":"01:15.040 ","End":"01:17.285","Text":"Here\u0027s what we\u0027ll do."},{"Start":"01:17.285 ","End":"01:22.160","Text":"I\u0027m going to try and sandwich this expression"},{"Start":"01:22.160 ","End":"01:27.190","Text":"between something that\u0027s bigger or equal and something that\u0027s less than or equal to."},{"Start":"01:27.190 ","End":"01:28.715","Text":"This is what we\u0027ll do."},{"Start":"01:28.715 ","End":"01:31.780","Text":"I\u0027m just going to work on this expression."},{"Start":"01:31.780 ","End":"01:41.180","Text":"The nth root of 4^n plus 10^n is certainly less than,"},{"Start":"01:41.180 ","End":"01:45.680","Text":"but it\u0027s enough for me less than or equal to the nth root,"},{"Start":"01:45.680 ","End":"01:47.315","Text":"if I make this bigger,"},{"Start":"01:47.315 ","End":"01:53.045","Text":"I could write 10^n plus 10^n because 10 is bigger than 4."},{"Start":"01:53.045 ","End":"02:00.140","Text":"On the other hand, this is bigger or equal to the nth root of 10^n,"},{"Start":"02:00.140 ","End":"02:02.290","Text":"I just dropped the 4^n."},{"Start":"02:02.290 ","End":"02:05.070","Text":"Obviously it can only get smaller."},{"Start":"02:05.070 ","End":"02:09.785","Text":"Now, let\u0027s see what the limit of each of these is."},{"Start":"02:09.785 ","End":"02:12.855","Text":"One of them is just 10."},{"Start":"02:12.855 ","End":"02:15.085","Text":"This is equal to 10,"},{"Start":"02:15.085 ","End":"02:19.900","Text":"and therefore the limit is 10."},{"Start":"02:19.900 ","End":"02:26.895","Text":"This equals the nth root of 2 times 10^n."},{"Start":"02:26.895 ","End":"02:32.670","Text":"They\u0027ll write this as 2 times 10^n to the power of 1 over n."},{"Start":"02:32.670 ","End":"02:39.435","Text":"This is equal to 2^1 over n times 10^n to the 1 over n,"},{"Start":"02:39.435 ","End":"02:41.955","Text":"which means times 10."},{"Start":"02:41.955 ","End":"02:47.160","Text":"This tends to also 10,"},{"Start":"02:47.160 ","End":"02:50.190","Text":"because if n goes to infinity,"},{"Start":"02:50.190 ","End":"02:56.500","Text":"1 over n goes to 0 and we have 2^0 here, which is 1."},{"Start":"02:56.990 ","End":"03:00.680","Text":"This thing is sandwiched between something"},{"Start":"03:00.680 ","End":"03:04.495","Text":"that goes to 10 and something else that goes to 10."},{"Start":"03:04.495 ","End":"03:08.410","Text":"By the sandwich theorem or squeeze theorem,"},{"Start":"03:08.410 ","End":"03:10.775","Text":"which we haven\u0027t defined precisely yet,"},{"Start":"03:10.775 ","End":"03:15.170","Text":"it makes sense that it is also goes to 10."},{"Start":"03:15.170 ","End":"03:17.390","Text":"That\u0027s the general idea for a sequence."},{"Start":"03:17.390 ","End":"03:21.275","Text":"Find something bigger and something smaller that are easier to compute,"},{"Start":"03:21.275 ","End":"03:24.410","Text":"but which go to the same limit and the thing"},{"Start":"03:24.410 ","End":"03:29.050","Text":"that\u0027s in the middle also will go to that limit."},{"Start":"03:29.270 ","End":"03:33.370","Text":"That was an informal introduction to the sandwich theorem."},{"Start":"03:33.370 ","End":"03:37.860","Text":"I\u0027d like to write it out more precisely."},{"Start":"03:37.860 ","End":"03:42.150","Text":"Let\u0027s say that we\u0027re given 3 sequences,"},{"Start":"03:42.150 ","End":"03:44.730","Text":"a_n, b_ n,"},{"Start":"03:44.730 ","End":"03:50.220","Text":"and c_ n,"},{"Start":"03:50.220 ","End":"03:55.615","Text":"such that the following conditions hold,"},{"Start":"03:55.615 ","End":"04:04.915","Text":"a_n is less than or equal to b_n but greater or equal to c_n."},{"Start":"04:04.915 ","End":"04:07.880","Text":"This is for all n,"},{"Start":"04:07.880 ","End":"04:19.200","Text":"and the limit as n goes to infinity of c_n is some finite L"},{"Start":"04:19.200 ","End":"04:29.525","Text":"and the limit as n goes to infinity of b_n is the same L. Then"},{"Start":"04:29.525 ","End":"04:36.604","Text":"what we have is that the limit as n goes to infinity of a_n is also"},{"Start":"04:36.604 ","End":"04:44.195","Text":"equal to L. It\u0027s sandwiched between L and L. Let\u0027s do another example."},{"Start":"04:44.195 ","End":"04:50.125","Text":"This example, we want to find the limit as n goes to infinity"},{"Start":"04:50.125 ","End":"04:57.695","Text":"of 1 over n^2 plus 1 plus 1 over n^2 plus 2 plus,"},{"Start":"04:57.695 ","End":"05:02.460","Text":"etc, up to 1 over n^2 plus n,"},{"Start":"05:02.460 ","End":"05:04.670","Text":"there\u0027s n terms in this sum."},{"Start":"05:04.670 ","End":"05:07.240","Text":"We want to know what this is."},{"Start":"05:07.240 ","End":"05:10.235","Text":"Of course, we\u0027re going to use the sandwich theorem."},{"Start":"05:10.235 ","End":"05:14.030","Text":"This bit here is our a_n."},{"Start":"05:14.030 ","End":"05:21.480","Text":"What we want to do is sandwich it in between 2 other sequences."},{"Start":"05:21.480 ","End":"05:27.970","Text":"What I\u0027m going to do here is take 1 over n^2 plus 1,"},{"Start":"05:29.210 ","End":"05:37.860","Text":"n times n^2 plus 1,1 over n^2 plus 1."},{"Start":"05:37.860 ","End":"05:41.355","Text":"There\u0027s n terms here, all identical."},{"Start":"05:41.355 ","End":"05:43.865","Text":"Now, why is it less than or equal to?"},{"Start":"05:43.865 ","End":"05:45.650","Text":"Reciprocal is the biggest one,"},{"Start":"05:45.650 ","End":"05:48.350","Text":"the denominator is smallest."},{"Start":"05:48.350 ","End":"05:50.270","Text":"This has the smallest denominator,"},{"Start":"05:50.270 ","End":"05:51.470","Text":"so this is the biggest."},{"Start":"05:51.470 ","End":"05:53.810","Text":"You probably guessed it on the other side,"},{"Start":"05:53.810 ","End":"05:56.150","Text":"I\u0027m going to take the smallest term,"},{"Start":"05:56.150 ","End":"05:59.255","Text":"which is this one also n times."},{"Start":"05:59.255 ","End":"06:06.980","Text":"It\u0027s 1 over n^2 plus n plus 1 over n^2 plus"},{"Start":"06:06.980 ","End":"06:15.790","Text":"n plus 1 over n^2 plus n. Also there\u0027s n terms here."},{"Start":"06:15.790 ","End":"06:19.040","Text":"Now since all the terms are identical,"},{"Start":"06:19.040 ","End":"06:22.605","Text":"this is just n times one of them."},{"Start":"06:22.605 ","End":"06:30.045","Text":"It\u0027s n over n^2 plus n less than or equal to a_n."},{"Start":"06:30.045 ","End":"06:35.565","Text":"Here we have n over n^2 plus 1."},{"Start":"06:35.565 ","End":"06:38.445","Text":"Now let\u0027s label this c_n,"},{"Start":"06:38.445 ","End":"06:41.115","Text":"and we\u0027ll label this b_n."},{"Start":"06:41.115 ","End":"06:45.245","Text":"Now let\u0027s see what are the limits of c_n and b _n."},{"Start":"06:45.245 ","End":"06:49.310","Text":"The limit as n goes to infinity of c_ n,"},{"Start":"06:49.310 ","End":"06:57.425","Text":"which is n over n^2 plus n. You just have to look at the leading coefficients."},{"Start":"06:57.425 ","End":"06:59.420","Text":"When you have a polynomial over polynomial,"},{"Start":"06:59.420 ","End":"07:03.235","Text":"this is like n over n^2,"},{"Start":"07:03.235 ","End":"07:09.225","Text":"which is 1 over n. The limit is 0."},{"Start":"07:09.225 ","End":"07:19.665","Text":"The limit is n goes to infinity of b_n is the limit of n over n^2 plus 1."},{"Start":"07:19.665 ","End":"07:25.875","Text":"Again, we just have to look at the leading terms in numerator and denominator."},{"Start":"07:25.875 ","End":"07:27.840","Text":"This is again n over n^2,"},{"Start":"07:27.840 ","End":"07:31.365","Text":"which is 1 over n. It\u0027s also 0."},{"Start":"07:31.365 ","End":"07:37.110","Text":"If c_n goes to 0 and b _n goes to 0"},{"Start":"07:37.110 ","End":"07:43.395","Text":"and we have that"},{"Start":"07:43.395 ","End":"07:49.155","Text":"a_n is sandwiched between c_ n and b _n."},{"Start":"07:49.155 ","End":"07:53.255","Text":"All these together by the sandwich theorem,"},{"Start":"07:53.255 ","End":"07:59.715","Text":"imply that a_n also tends to 0."},{"Start":"07:59.715 ","End":"08:03.610","Text":"That\u0027s the answer. The limit is 0."},{"Start":"08:03.610 ","End":"08:09.455","Text":"Next, I\u0027m going to prove a useful theorem with the help of the sandwich theorem."},{"Start":"08:09.455 ","End":"08:16.650","Text":"But before that, I need to remind you of what a bounded sequences in case you forgot."},{"Start":"08:16.850 ","End":"08:19.755","Text":"I have a sequence a_n,"},{"Start":"08:19.755 ","End":"08:27.115","Text":"then I say that it\u0027s bounded if a_n"},{"Start":"08:27.115 ","End":"08:35.210","Text":"is an absolute value less than or equal to M. To be more precise,"},{"Start":"08:35.210 ","End":"08:42.570","Text":"I could say there exists an M such that a_n is less than or equal to m,"},{"Start":"08:42.570 ","End":"08:49.355","Text":"absolute value should really say for all n. There is an alternative of this,"},{"Start":"08:49.355 ","End":"08:57.690","Text":"which instead of this condition says that a_n is between 2 numbers,"},{"Start":"08:57.690 ","End":"09:02.739","Text":"m and M in place of this."},{"Start":"09:02.739 ","End":"09:05.599","Text":"But we\u0027ll stick with this version."},{"Start":"09:05.599 ","End":"09:09.560","Text":"I\u0027m just letting you know that you might see this alternative."},{"Start":"09:09.560 ","End":"09:12.740","Text":"Now the theorem, in this theorem,"},{"Start":"09:12.740 ","End":"09:14.705","Text":"I have 2 sequences,"},{"Start":"09:14.705 ","End":"09:17.330","Text":"a_n and b_ n,"},{"Start":"09:17.330 ","End":"09:23.080","Text":"such that a_n is bounded"},{"Start":"09:23.080 ","End":"09:33.060","Text":"and b_n tends to 0."},{"Start":"09:33.060 ","End":"09:34.810","Text":"If I have this,"},{"Start":"09:34.810 ","End":"09:38.500","Text":"then it implies that a third sequence a_n,"},{"Start":"09:38.500 ","End":"09:44.000","Text":"b_n also tends to 0."},{"Start":"09:45.430 ","End":"09:51.130","Text":"That\u0027s a theorem and it turns out to be useful and I\u0027ll do an example of it in a moment."},{"Start":"09:51.130 ","End":"09:54.160","Text":"But I\u0027d like to actually prove it this time."},{"Start":"09:54.160 ","End":"09:55.705","Text":"We usually omit proofs,"},{"Start":"09:55.705 ","End":"10:00.550","Text":"but this is an easy proof and it follows from the sandwich theorem."},{"Start":"10:00.550 ","End":"10:08.860","Text":"If absolute value of a_n is less than or equal to M,"},{"Start":"10:08.860 ","End":"10:14.900","Text":"then absolute value of a_n, b_n,"},{"Start":"10:14.900 ","End":"10:18.840","Text":"which is absolute value of a_n times Epsilon absolute value of"},{"Start":"10:18.840 ","End":"10:24.230","Text":"b_n is less than or equal to M times the absolute value of b_n."},{"Start":"10:24.230 ","End":"10:33.440","Text":"Now, in general, suppose I have the absolute value of A is less than or equal to B."},{"Start":"10:33.440 ","End":"10:41.870","Text":"This implies that A has to be between B and minus B,"},{"Start":"10:41.870 ","End":"10:43.655","Text":"or the other way round."},{"Start":"10:43.655 ","End":"10:46.955","Text":"If this is true, a_n,"},{"Start":"10:46.955 ","End":"10:52.685","Text":"b_n is less than or equal to plus this thing,"},{"Start":"10:52.685 ","End":"10:54.950","Text":"which is just itself."},{"Start":"10:54.950 ","End":"10:59.270","Text":"It is already positive or non-negative and it\u0027s bigger"},{"Start":"10:59.270 ","End":"11:03.935","Text":"or equal to minus M absolute value of b_n."},{"Start":"11:03.935 ","End":"11:07.210","Text":"This is sandwiched between these 2."},{"Start":"11:07.210 ","End":"11:10.535","Text":"Because b_n tends to 0,"},{"Start":"11:10.535 ","End":"11:15.830","Text":"the absolute value of b_n tends to absolute value of 0,"},{"Start":"11:15.830 ","End":"11:17.585","Text":"which is also 0."},{"Start":"11:17.585 ","End":"11:23.565","Text":"This tends to minus M absolute value of 0,"},{"Start":"11:23.565 ","End":"11:29.470","Text":"and this tends to plus M absolute value of 0,"},{"Start":"11:29.470 ","End":"11:33.555","Text":"which is 0 in both cases."},{"Start":"11:33.555 ","End":"11:35.610","Text":"By the sandwich theorem,"},{"Start":"11:35.610 ","End":"11:39.640","Text":"if this goes to 0 and this goes to 0,"},{"Start":"11:39.640 ","End":"11:44.405","Text":"then this is forced to also go to 0."},{"Start":"11:44.405 ","End":"11:46.880","Text":"It\u0027s trapped between 0 and 0,"},{"Start":"11:46.880 ","End":"11:49.085","Text":"so it\u0027s also 0."},{"Start":"11:49.085 ","End":"11:53.375","Text":"Now I\u0027ll give an example of the usefulness of this theorem."},{"Start":"11:53.375 ","End":"11:54.985","Text":"Before the example,"},{"Start":"11:54.985 ","End":"11:59.055","Text":"remember there\u0027s a way of phrasing this that some use."},{"Start":"11:59.055 ","End":"12:00.770","Text":"When a sequence tends to 0,"},{"Start":"12:00.770 ","End":"12:02.600","Text":"it\u0027s called a null sequence."},{"Start":"12:02.600 ","End":"12:06.425","Text":"Sometimes it\u0027s summarized by saying"},{"Start":"12:06.425 ","End":"12:11.065","Text":"a bounded sequence times"},{"Start":"12:11.065 ","End":"12:18.330","Text":"a null sequence equals a null sequence."},{"Start":"12:18.330 ","End":"12:21.005","Text":"Then it\u0027s a more compact way of saying it."},{"Start":"12:21.005 ","End":"12:23.915","Text":"Now let\u0027s get to that example."},{"Start":"12:23.915 ","End":"12:31.745","Text":"We want to evaluate what is the limit as n goes to infinity"},{"Start":"12:31.745 ","End":"12:40.915","Text":"of 1 over n times sine of n^2 plus 2n plus 8."},{"Start":"12:40.915 ","End":"12:42.990","Text":"What does this equal?"},{"Start":"12:42.990 ","End":"12:47.090","Text":"Of course we\u0027re going to use what we just said here."},{"Start":"12:47.090 ","End":"12:52.620","Text":"Notice that this is a null sequence or it tends to"},{"Start":"12:52.620 ","End":"12:59.890","Text":"0 and that this sequence is bounded."},{"Start":"12:59.890 ","End":"13:02.600","Text":"Do you see why this is bounded?"},{"Start":"13:02.600 ","End":"13:05.990","Text":"Because in general, the sine function is bounded."},{"Start":"13:05.990 ","End":"13:07.220","Text":"The sine of anything,"},{"Start":"13:07.220 ","End":"13:11.740","Text":"say Theta is always less than or equal to 1."},{"Start":"13:11.740 ","End":"13:17.340","Text":"This is the M. Null sequence,"},{"Start":"13:17.340 ","End":"13:21.855","Text":"bounded sequence, equal the null sequence."},{"Start":"13:21.855 ","End":"13:26.530","Text":"The limit has to be 0."},{"Start":"13:26.600 ","End":"13:30.130","Text":"That\u0027s it for this clip."}],"ID":31170},{"Watched":false,"Name":"Exercise 1","Duration":"2m 56s","ChapterTopicVideoID":29558,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to evaluate the limit as n goes to infinity of"},{"Start":"00:05.280 ","End":"00:10.545","Text":"sine n over n. It looks familiar, but it\u0027s not."},{"Start":"00:10.545 ","End":"00:14.865","Text":"What\u0027s familiar is sine t over t when t goes to 0."},{"Start":"00:14.865 ","End":"00:16.779","Text":"But here, we\u0027re going to infinity,"},{"Start":"00:16.779 ","End":"00:18.345","Text":"so that\u0027s something else."},{"Start":"00:18.345 ","End":"00:22.140","Text":"The thing is that the denominator we know goes to infinity,"},{"Start":"00:22.140 ","End":"00:24.450","Text":"but the numerator doesn\u0027t have a limit."},{"Start":"00:24.450 ","End":"00:25.980","Text":"As n goes to infinity,"},{"Start":"00:25.980 ","End":"00:30.360","Text":"sine n just keeps oscillating."},{"Start":"00:30.360 ","End":"00:33.260","Text":"If it was function sine x,"},{"Start":"00:33.260 ","End":"00:34.575","Text":"it would just keep going from 1,"},{"Start":"00:34.575 ","End":"00:36.580","Text":"minus 1, 1, minus 1."},{"Start":"00:36.580 ","End":"00:39.220","Text":"Anyway, it doesn\u0027t have a limit."},{"Start":"00:39.220 ","End":"00:44.795","Text":"So what to do? Sine n doesn\u0027t have a limit,"},{"Start":"00:44.795 ","End":"00:47.000","Text":"but it is bounded."},{"Start":"00:47.000 ","End":"00:49.040","Text":"Now, I\u0027ll start again."},{"Start":"00:49.040 ","End":"00:50.630","Text":"I want to rephrase."},{"Start":"00:50.630 ","End":"00:55.900","Text":"I want to let our sequence a_n is sine n"},{"Start":"00:55.900 ","End":"01:03.245","Text":"over n. I want to write a_n as the product of 2 sequences,"},{"Start":"01:03.245 ","End":"01:05.915","Text":"b_n times C_n,"},{"Start":"01:05.915 ","End":"01:11.925","Text":"where b_n is 1 over n,"},{"Start":"01:11.925 ","End":"01:18.350","Text":"and c_n is sine n. Now we know"},{"Start":"01:18.350 ","End":"01:25.790","Text":"that b_n goes to 0 because n goes to infinity,"},{"Start":"01:25.790 ","End":"01:28.468","Text":"so it obviously goes to 0,"},{"Start":"01:28.468 ","End":"01:31.955","Text":"and C_n is bounded."},{"Start":"01:31.955 ","End":"01:33.575","Text":"What do I mean by bounded?"},{"Start":"01:33.575 ","End":"01:35.134","Text":"Just like with functions,"},{"Start":"01:35.134 ","End":"01:44.329","Text":"it\u0027s got an upper limit and lower limit because C_n is between 1 and minus 1."},{"Start":"01:44.329 ","End":"01:50.624","Text":"The sine of anything is always between minus 1 and 1."},{"Start":"01:50.624 ","End":"01:58.895","Text":"There\u0027s a theorem that something goes to 0 times something is bounded,"},{"Start":"01:58.895 ","End":"02:01.610","Text":"then it also goes to 0."},{"Start":"02:01.610 ","End":"02:06.565","Text":"I\u0027ll just write that. It\u0027s a proposition."},{"Start":"02:06.565 ","End":"02:11.570","Text":"If b_n goes to 0 as the sequence b_n,"},{"Start":"02:11.570 ","End":"02:17.550","Text":"and C_n is bounded,"},{"Start":"02:19.660 ","End":"02:26.014","Text":"then the sequence b_n times C_n also goes to 0,"},{"Start":"02:26.014 ","End":"02:29.970","Text":"which in our case is a_n."},{"Start":"02:29.970 ","End":"02:33.305","Text":"So the answer is 0."},{"Start":"02:33.305 ","End":"02:40.370","Text":"So we can say that the limit n goes to infinity"},{"Start":"02:40.370 ","End":"02:48.950","Text":"of sine n over n is equal to 0 by the theorem."},{"Start":"02:48.950 ","End":"02:56.850","Text":"Tending to 0 times bounded also tends to 0. We\u0027re done."}],"ID":31171},{"Watched":false,"Name":"Exercise 2","Duration":"2m 27s","ChapterTopicVideoID":29559,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, we have to evaluate this limit and goes"},{"Start":"00:03.990 ","End":"00:08.910","Text":"to infinity of cosine 2n plus 1."},{"Start":"00:08.910 ","End":"00:13.935","Text":"Now, if I break it up into cosine and then times 1,"},{"Start":"00:13.935 ","End":"00:16.769","Text":"the 1 part goes to 0,"},{"Start":"00:16.769 ","End":"00:19.080","Text":"but the cosine of 2n plus 1."},{"Start":"00:19.080 ","End":"00:20.745","Text":"What does that go to?"},{"Start":"00:20.745 ","End":"00:22.335","Text":"It doesn\u0027t have a limit,"},{"Start":"00:22.335 ","End":"00:24.315","Text":"but let\u0027s just say we don\u0027t know."},{"Start":"00:24.315 ","End":"00:28.005","Text":"What I\u0027m about to say is good even if we don\u0027t know."},{"Start":"00:28.005 ","End":"00:30.660","Text":"Let\u0027s just get some notation."},{"Start":"00:30.660 ","End":"00:35.690","Text":"I\u0027ll write our original sequence as A_n,"},{"Start":"00:35.690 ","End":"00:40.220","Text":"so A_n is cosine of 2n plus 1."},{"Start":"00:40.220 ","End":"00:45.835","Text":"I mentioned writing it as a product so we\u0027ll take B_n to be"},{"Start":"00:45.835 ","End":"00:54.165","Text":"1 and C_n to be cosine of 2n plus 1."},{"Start":"00:54.165 ","End":"01:00.430","Text":"Then our A_n will be equal to B_n times C_n."},{"Start":"01:00.430 ","End":"01:04.130","Text":"Now there\u0027s a theorem when we have a product,"},{"Start":"01:04.130 ","End":"01:05.795","Text":"as we do here,"},{"Start":"01:05.795 ","End":"01:09.995","Text":"and this 1 goes to 0,"},{"Start":"01:09.995 ","End":"01:13.105","Text":"and this 1 is bounded."},{"Start":"01:13.105 ","End":"01:18.395","Text":"Then A_n also goes to 0."},{"Start":"01:18.395 ","End":"01:20.360","Text":"If you don\u0027t remember what bounded is,"},{"Start":"01:20.360 ","End":"01:26.345","Text":"it just means that it\u0027s got an upper and lower limit."},{"Start":"01:26.345 ","End":"01:36.170","Text":"In this case, we can say that C_n is between minus 1 and 1 because it\u0027s a cosine."},{"Start":"01:36.170 ","End":"01:39.590","Text":"A cosine is always between minus 1 and 1."},{"Start":"01:39.590 ","End":"01:46.110","Text":"We have tends to 0 times bounded a equals tends to 0,"},{"Start":"01:46.110 ","End":"01:48.435","Text":"and so A_n,"},{"Start":"01:48.435 ","End":"01:51.755","Text":"tends to 0 also."},{"Start":"01:51.755 ","End":"02:02.735","Text":"In other words, the limit as n goes to infinity of cosine of 2n plus 1 is 0."},{"Start":"02:02.735 ","End":"02:06.905","Text":"Just remember that I\u0027ll just even write it symbolically."},{"Start":"02:06.905 ","End":"02:15.380","Text":"If we have something tends to 0 times some other sequence that\u0027s bounded,"},{"Start":"02:15.380 ","End":"02:21.995","Text":"then this also is a tends to 0 sequence."},{"Start":"02:21.995 ","End":"02:23.795","Text":"Useful to remember."},{"Start":"02:23.795 ","End":"02:27.690","Text":"Anyway, we are done. That\u0027s the answer."}],"ID":31172},{"Watched":false,"Name":"Exercise 3","Duration":"1m 42s","ChapterTopicVideoID":29560,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.360","Text":"In this exercise, we have to evaluate the limit as n goes to infinity of this expression."},{"Start":"00:06.360 ","End":"00:09.255","Text":"It\u0027s not clear what\u0027s happening here."},{"Start":"00:09.255 ","End":"00:11.685","Text":"The 3n goes to infinity,"},{"Start":"00:11.685 ","End":"00:14.040","Text":"sine n doesn\u0027t have a limit,"},{"Start":"00:14.040 ","End":"00:16.680","Text":"similarly, infinity doesn\u0027t have a limit,"},{"Start":"00:16.680 ","End":"00:19.035","Text":"not clear what\u0027s going on here."},{"Start":"00:19.035 ","End":"00:22.335","Text":"But there is a way of simplifying it,"},{"Start":"00:22.335 ","End":"00:27.240","Text":"we divide everything by n then we get the limit."},{"Start":"00:27.240 ","End":"00:29.385","Text":"As n goes to infinity,"},{"Start":"00:29.385 ","End":"00:31.250","Text":"we take the top and divide it by n,"},{"Start":"00:31.250 ","End":"00:36.450","Text":"it will give us 3 plus sine n over n,"},{"Start":"00:36.450 ","End":"00:41.790","Text":"and on the bottom 4 plus cosine n over"},{"Start":"00:41.790 ","End":"00:48.920","Text":"n. Now each of these sine n over n and cosine n over n,"},{"Start":"00:48.920 ","End":"00:51.865","Text":"they both go to 0."},{"Start":"00:51.865 ","End":"00:55.170","Text":"We\u0027ve seen this before, for example,"},{"Start":"00:55.170 ","End":"01:03.405","Text":"sine n over n is 1 over n times sine n,"},{"Start":"01:03.405 ","End":"01:07.265","Text":"and this tends to 0."},{"Start":"01:07.265 ","End":"01:10.280","Text":"This other 1 is bounded,"},{"Start":"01:10.280 ","End":"01:14.815","Text":"sine and cosine are both bounded between minus 1 and 1,"},{"Start":"01:14.815 ","End":"01:21.270","Text":"and there\u0027s a theorem that tends to 0 times bounded also tends to 0."},{"Start":"01:21.270 ","End":"01:25.885","Text":"Similarly, with the other 1 with the cosine n over n,"},{"Start":"01:25.885 ","End":"01:29.415","Text":"exactly this similarly goes to 0."},{"Start":"01:29.415 ","End":"01:36.885","Text":"What we\u0027re left with here is 3 plus 0 over 4 plus 0,"},{"Start":"01:36.885 ","End":"01:41.890","Text":"which is 3/4 and that\u0027s the answer."}],"ID":31173},{"Watched":false,"Name":"Exercise 4","Duration":"2m 13s","ChapterTopicVideoID":29561,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.270","Text":"In this exercise, we have to evaluate the limit as n goes to infinity of,"},{"Start":"00:06.270 ","End":"00:07.525","Text":"well, it\u0027s a mixture here,"},{"Start":"00:07.525 ","End":"00:08.610","Text":"it\u0027s a bit of polynomial,"},{"Start":"00:08.610 ","End":"00:10.260","Text":"a bit of trigonometric,"},{"Start":"00:10.260 ","End":"00:14.700","Text":"basically infinity plus doesn\u0027t have a limit over the same,"},{"Start":"00:14.700 ","End":"00:16.440","Text":"and not clear what\u0027s going on."},{"Start":"00:16.440 ","End":"00:18.300","Text":"But we can simplify everything."},{"Start":"00:18.300 ","End":"00:21.810","Text":"If we just divide top and bottom by n^2,"},{"Start":"00:21.810 ","End":"00:26.100","Text":"then we will get a limit which is easier to understand."},{"Start":"00:26.100 ","End":"00:27.840","Text":"Divide the top by n^2."},{"Start":"00:27.840 ","End":"00:36.855","Text":"We have 3 plus 1 plus Sin 2n over n^2,"},{"Start":"00:36.855 ","End":"00:47.110","Text":"over 1 plus Cosine 3n over n^2."},{"Start":"00:47.110 ","End":"00:52.170","Text":"Now each of these pieces has a limit."},{"Start":"00:52.430 ","End":"00:56.175","Text":"This is just 3."},{"Start":"00:56.175 ","End":"00:58.195","Text":"This is just 1."},{"Start":"00:58.195 ","End":"01:01.670","Text":"This goes to 0."},{"Start":"01:01.670 ","End":"01:03.635","Text":"What about these 2?"},{"Start":"01:03.635 ","End":"01:10.354","Text":"These also go to 0 because they are of the form"},{"Start":"01:10.354 ","End":"01:17.835","Text":"something that tends to 0 times something bounded,"},{"Start":"01:17.835 ","End":"01:20.510","Text":"I\u0027m just writing this very shorthand and symbolically,"},{"Start":"01:20.510 ","End":"01:23.735","Text":"equal something that tends to 0."},{"Start":"01:23.735 ","End":"01:25.430","Text":"Now where is the product?"},{"Start":"01:25.430 ","End":"01:27.500","Text":"This is Sin 2n."},{"Start":"01:27.500 ","End":"01:31.420","Text":"For example, no the other way around."},{"Start":"01:31.420 ","End":"01:34.315","Text":"This is the 1^2,"},{"Start":"01:34.315 ","End":"01:37.095","Text":"and this is the Sin 2n."},{"Start":"01:37.095 ","End":"01:44.005","Text":"This is bounded between minus 1 and 1."},{"Start":"01:44.005 ","End":"01:45.845","Text":"This goes to 0,"},{"Start":"01:45.845 ","End":"01:48.335","Text":"so this product goes to 0."},{"Start":"01:48.335 ","End":"01:53.569","Text":"Similarly, if we had Cosine 3n, it\u0027s all a cosine,"},{"Start":"01:53.569 ","End":"01:56.083","Text":"it\u0027s still bounded between minus 1 and 1,"},{"Start":"01:56.083 ","End":"01:58.310","Text":"and also 1^2,"},{"Start":"01:58.310 ","End":"02:02.340","Text":"so it similarly goes to 0."},{"Start":"02:02.340 ","End":"02:06.300","Text":"After all this we get 3 plus 0 plus 0,"},{"Start":"02:06.300 ","End":"02:08.820","Text":"over 1 plus 0,"},{"Start":"02:08.820 ","End":"02:10.283","Text":"which is equal to 3."},{"Start":"02:10.283 ","End":"02:13.120","Text":"That\u0027s the answer."}],"ID":31174},{"Watched":false,"Name":"Exercise 5","Duration":"2m 32s","ChapterTopicVideoID":29562,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.425","Text":"Here we have to compute the limit as n goes to infinity of all this."},{"Start":"00:04.425 ","End":"00:06.420","Text":"Not clear what\u0027s going on."},{"Start":"00:06.420 ","End":"00:08.745","Text":"The 3n goes to infinity."},{"Start":"00:08.745 ","End":"00:12.360","Text":"Not sure about what happens with the arctangent."},{"Start":"00:12.360 ","End":"00:13.950","Text":"Actually I do,"},{"Start":"00:13.950 ","End":"00:16.425","Text":"as n goes to infinity so does this."},{"Start":"00:16.425 ","End":"00:20.770","Text":"An arctangent of infinity is Pi over 2."},{"Start":"00:20.770 ","End":"00:27.390","Text":"But here, I\u0027m not sure what the limit of n minus natural log of n is."},{"Start":"00:27.390 ","End":"00:29.040","Text":"Well, I know it\u0027s infinity,"},{"Start":"00:29.040 ","End":"00:31.410","Text":"but you don\u0027t maybe so we just"},{"Start":"00:31.410 ","End":"00:34.020","Text":"say that it\u0027s infinity plus something, infinity plus something."},{"Start":"00:34.020 ","End":"00:35.295","Text":"We don\u0027t know what\u0027s going on."},{"Start":"00:35.295 ","End":"00:37.640","Text":"We have to do some simplification."},{"Start":"00:37.640 ","End":"00:42.125","Text":"I suggest dividing top and bottom by n. We get"},{"Start":"00:42.125 ","End":"00:47.720","Text":"the limit as n goes to infinity of 3 plus 1 over"},{"Start":"00:47.720 ","End":"00:57.555","Text":"n arctangent of 2n minus 3 over 4 plus"},{"Start":"00:57.555 ","End":"01:04.655","Text":"1 over n arctangent of n minus natural log of"},{"Start":"01:04.655 ","End":"01:08.090","Text":"n. Now let\u0027s say I can show that this actually has"},{"Start":"01:08.090 ","End":"01:12.680","Text":"a limit and that both of the arctangents tend to Pi over 2, but I don\u0027t need it."},{"Start":"01:12.680 ","End":"01:15.155","Text":"If you don\u0027t know that we can get around it,"},{"Start":"01:15.155 ","End":"01:19.225","Text":"all I need to know is that the arctangent is bounded."},{"Start":"01:19.225 ","End":"01:26.610","Text":"Arctangent of x is strictly between,"},{"Start":"01:26.610 ","End":"01:28.620","Text":"can\u0027t even equal,"},{"Start":"01:28.620 ","End":"01:32.475","Text":"Pi over 2 and minus Pi over 2."},{"Start":"01:32.475 ","End":"01:35.285","Text":"Look what we have here. These two are constants."},{"Start":"01:35.285 ","End":"01:37.970","Text":"Now this goes to 0,"},{"Start":"01:37.970 ","End":"01:40.205","Text":"and this goes to 0."},{"Start":"01:40.205 ","End":"01:47.510","Text":"This part is bounded and this arctangent is bounded."},{"Start":"01:47.510 ","End":"01:52.320","Text":"They\u0027re both between minus Pi over 2 and Pi over 2."},{"Start":"01:52.330 ","End":"01:59.320","Text":"As you recall, something that goes to 0 times something bounded also goes to 0."},{"Start":"01:59.320 ","End":"02:02.510","Text":"I\u0027ll write it again, something that tends to 0,"},{"Start":"02:02.510 ","End":"02:05.070","Text":"I\u0027ll write it in shorthand,"},{"Start":"02:05.070 ","End":"02:15.120","Text":"times something that\u0027s bounded also is something that goes to 0."},{"Start":"02:15.420 ","End":"02:25.320","Text":"What we have here is 3 plus 0 over 4 plus 0,"},{"Start":"02:25.320 ","End":"02:31.120","Text":"which is 3/4. We\u0027re done."}],"ID":31175},{"Watched":false,"Name":"Exercise 6","Duration":"4m 18s","ChapterTopicVideoID":29563,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have to compute the limit as n goes to"},{"Start":"00:03.690 ","End":"00:07.935","Text":"infinity of this strange sequence."},{"Start":"00:07.935 ","End":"00:10.230","Text":"2^n plus 3^n plus 4^n,"},{"Start":"00:10.230 ","End":"00:13.935","Text":"all to the power of 1 over n. What do we do here?"},{"Start":"00:13.935 ","End":"00:16.875","Text":"Now I\u0027m actually going to show you 2 ways of solving this."},{"Start":"00:16.875 ","End":"00:21.000","Text":"The first way is just we do a bit of algebra first."},{"Start":"00:21.000 ","End":"00:24.030","Text":"We can say that this is the limit."},{"Start":"00:24.030 ","End":"00:34.020","Text":"If I take 4^n outside the brackets then I\u0027ll have 2 over 4^n"},{"Start":"00:34.020 ","End":"00:38.820","Text":"plus 3 over 4^n plus 4 over"},{"Start":"00:38.820 ","End":"00:45.820","Text":"4^n and all this,"},{"Start":"00:46.100 ","End":"00:51.000","Text":"the whole thing to the power of 1 over"},{"Start":"00:51.000 ","End":"00:57.905","Text":"n. Now n goes to infinity."},{"Start":"00:57.905 ","End":"01:00.875","Text":"I can take the power of 1 over n separately."},{"Start":"01:00.875 ","End":"01:03.980","Text":"I can say that this is equal to."},{"Start":"01:03.980 ","End":"01:09.650","Text":"Well, 4^n to the 1 over n is just 4, it\u0027s a constant."},{"Start":"01:09.650 ","End":"01:11.600","Text":"I could bring it in front of the limit or leave it here."},{"Start":"01:11.600 ","End":"01:16.920","Text":"Meanwhile, 4 times whatever is here."},{"Start":"01:18.350 ","End":"01:25.185","Text":"I copied this here and this is to the power of 1 over n and the 4 is separate."},{"Start":"01:25.185 ","End":"01:27.540","Text":"Now everything has a limit."},{"Start":"01:27.540 ","End":"01:29.630","Text":"2 over 4 is a 1/2,"},{"Start":"01:29.630 ","End":"01:33.290","Text":"something smaller than 1^n goes to 0."},{"Start":"01:33.290 ","End":"01:35.300","Text":"This goes to 0."},{"Start":"01:35.300 ","End":"01:45.240","Text":"This thing is just equal to 1 so it tends to 1 and this goes to 0."},{"Start":"01:45.240 ","End":"01:47.250","Text":"What do we get all together?"},{"Start":"01:47.250 ","End":"01:56.055","Text":"We get 4 times 0 plus 0 plus 1^0."},{"Start":"01:56.055 ","End":"02:02.710","Text":"Well, this is 1^0 is 1 and so the answer is 4."},{"Start":"02:02.710 ","End":"02:08.180","Text":"Now I want to show you an alternative way using the sandwich theorem,"},{"Start":"02:08.180 ","End":"02:10.830","Text":"also known as the squeeze theorem."},{"Start":"02:10.830 ","End":"02:16.800","Text":"I can take the expression 2^n plus 3^n plus 4^n and"},{"Start":"02:16.800 ","End":"02:23.815","Text":"sandwich it between two things."},{"Start":"02:23.815 ","End":"02:30.875","Text":"I can say that this is bigger or equal to 4^n certainly there\u0027s less terms."},{"Start":"02:30.875 ","End":"02:37.970","Text":"But I can also say it\u0027s less than or equal to 4^n plus 4^n plus 4^n."},{"Start":"02:37.970 ","End":"02:41.490","Text":"In other words, 3 times 4^n."},{"Start":"02:42.490 ","End":"02:47.320","Text":"If I take everything to a positive power 1 over"},{"Start":"02:47.320 ","End":"02:54.630","Text":"n then it still preserves the inequality."},{"Start":"02:54.910 ","End":"03:04.940","Text":"Now, I can say that my limit as n goes to"},{"Start":"03:04.940 ","End":"03:10.710","Text":"infinity of 2^n plus 3^n plus"},{"Start":"03:10.710 ","End":"03:13.350","Text":"4^n to the 1 over"},{"Start":"03:13.350 ","End":"03:18.425","Text":"n is going to be sandwiched between the limit of this and the limit of this."},{"Start":"03:18.425 ","End":"03:24.940","Text":"Now, this thing equals 4 so the limit is 4."},{"Start":"03:28.160 ","End":"03:33.060","Text":"The limit of this is, well,"},{"Start":"03:33.060 ","End":"03:38.980","Text":"it\u0027s equal to 3^1 over n times 4."},{"Start":"03:39.590 ","End":"03:47.415","Text":"This thing tends to 3^1 over n is 3^0 is 1."},{"Start":"03:47.415 ","End":"03:49.725","Text":"This also goes to 4."},{"Start":"03:49.725 ","End":"03:53.430","Text":"If this is sandwiched between two things that go to 4,"},{"Start":"03:53.430 ","End":"03:57.160","Text":"this thing also has to go to 4."},{"Start":"03:58.190 ","End":"04:02.030","Text":"I didn\u0027t write it too technically, you follow me."},{"Start":"04:02.030 ","End":"04:03.470","Text":"It\u0027s sandwiched between two things."},{"Start":"04:03.470 ","End":"04:04.700","Text":"The limit here is 4,"},{"Start":"04:04.700 ","End":"04:06.335","Text":"the limit here is 4."},{"Start":"04:06.335 ","End":"04:13.459","Text":"This is necessarily with limit 4 also as n goes to infinity."},{"Start":"04:13.459 ","End":"04:18.870","Text":"We\u0027ve got the same answer and that\u0027s encouraging and that\u0027s it."}],"ID":31176},{"Watched":false,"Name":"Exercise 7","Duration":"3m 17s","ChapterTopicVideoID":29564,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we have to evaluate the limit as n goes"},{"Start":"00:04.470 ","End":"00:10.635","Text":"to infinity of n factorial over n^n."},{"Start":"00:10.635 ","End":"00:14.820","Text":"The strategy here will be to use the squeeze theorem,"},{"Start":"00:14.820 ","End":"00:17.385","Text":"also known as the sandwich theorem."},{"Start":"00:17.385 ","End":"00:19.740","Text":"What I\u0027m going to do is squeeze"},{"Start":"00:19.740 ","End":"00:24.120","Text":"the general term between 2 things that go to the same limit."},{"Start":"00:24.120 ","End":"00:28.875","Text":"What I can say is that n factorial is"},{"Start":"00:28.875 ","End":"00:36.135","Text":"n times n minus 1 n minus 2 and so on,"},{"Start":"00:36.135 ","End":"00:40.305","Text":"down to 3, 2,"},{"Start":"00:40.305 ","End":"00:50.000","Text":"1. n to the power of n has the same number of factors that just all n,"},{"Start":"00:50.000 ","End":"00:53.945","Text":"n times n times n times, etc,"},{"Start":"00:53.945 ","End":"00:59.435","Text":"n times n times n. Now,"},{"Start":"00:59.435 ","End":"01:06.100","Text":"this general term, I\u0027m going to say is less than and bigger than,"},{"Start":"01:06.100 ","End":"01:10.005","Text":"it could be less than or equal to that will work also."},{"Start":"01:10.005 ","End":"01:13.590","Text":"This thing is bigger than 0."},{"Start":"01:13.590 ","End":"01:17.230","Text":"It\u0027s also less than,"},{"Start":"01:17.690 ","End":"01:22.045","Text":"I\u0027ll put less than or equal to because n could be 1."},{"Start":"01:22.045 ","End":"01:27.710","Text":"It\u0027s less than or equal to what I get if I take the same denominator,"},{"Start":"01:27.710 ","End":"01:29.645","Text":"let me write that n,"},{"Start":"01:29.645 ","End":"01:33.469","Text":"n, n, n,"},{"Start":"01:33.469 ","End":"01:38.030","Text":"n, n and ellipsis."},{"Start":"01:38.030 ","End":"01:43.490","Text":"Now, instead of all these factors,"},{"Start":"01:43.490 ","End":"01:49.550","Text":"I\u0027ll take n so that except for the first 1,"},{"Start":"01:49.550 ","End":"01:53.520","Text":"which I\u0027ll make 1."},{"Start":"01:53.750 ","End":"01:56.550","Text":"Sorry, I\u0027ll do that again."},{"Start":"01:56.550 ","End":"02:00.645","Text":"Except for the last 1 which will be a 1,"},{"Start":"02:00.645 ","End":"02:05.200","Text":"I\u0027ll make all of this n on the top."},{"Start":"02:05.560 ","End":"02:10.325","Text":"Now, all of them are less than or equal to n. If I make them n,"},{"Start":"02:10.325 ","End":"02:12.860","Text":"I can only get larger."},{"Start":"02:12.860 ","End":"02:16.330","Text":"You can probably see what\u0027s happening now."},{"Start":"02:16.330 ","End":"02:18.660","Text":"All these ns,"},{"Start":"02:18.660 ","End":"02:23.490","Text":"n minus 1 factors cancel with all these ns,"},{"Start":"02:23.490 ","End":"02:29.005","Text":"and what we\u0027re left with is on this side is just"},{"Start":"02:29.005 ","End":"02:36.140","Text":"1 over n. Now we\u0027ll let n go to infinity."},{"Start":"02:36.140 ","End":"02:42.260","Text":"Now, this doesn\u0027t depend on n so when n goes to infinity, this goes to 0."},{"Start":"02:42.260 ","End":"02:50.705","Text":"This when n goes to infinity, goes to 0."},{"Start":"02:50.705 ","End":"02:59.345","Text":"This is sandwiched in between 0 and 0 so therefore this has to also go to 0."},{"Start":"02:59.345 ","End":"03:05.140","Text":"I\u0027ll just write that by the sandwich theorem,"},{"Start":"03:05.140 ","End":"03:08.315","Text":"also known as the squeeze theorem."},{"Start":"03:08.315 ","End":"03:12.755","Text":"There is a squeeze between 0 and 0, it has to be 0."},{"Start":"03:12.755 ","End":"03:16.740","Text":"That\u0027s the answer and we\u0027re done."}],"ID":31177},{"Watched":false,"Name":"Exercise 8","Duration":"5m 23s","ChapterTopicVideoID":29565,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we need to evaluate this limit,"},{"Start":"00:04.500 ","End":"00:05.850","Text":"n goes to infinity,"},{"Start":"00:05.850 ","End":"00:09.045","Text":"the nth root of this expression."},{"Start":"00:09.045 ","End":"00:13.695","Text":"We\u0027ll do it by use of the squeeze theorem,"},{"Start":"00:13.695 ","End":"00:16.810","Text":"also known as the sandwich theorem."},{"Start":"00:16.940 ","End":"00:22.100","Text":"We need to sandwich the general term between 2 things,"},{"Start":"00:22.100 ","End":"00:26.580","Text":"1 above and 1 below and both of those go to the same limit"},{"Start":"00:26.580 ","End":"00:31.830","Text":"and this one gets caught in the middle and also goes to that limit, so to speak."},{"Start":"00:31.830 ","End":"00:36.335","Text":"First of all, I\u0027ll take just the inside without the nth root."},{"Start":"00:36.335 ","End":"00:42.740","Text":"1 plus 2^4n plus 1 over"},{"Start":"00:42.740 ","End":"00:53.010","Text":"n. I\u0027m claiming that this is certainly bigger than 2^4n."},{"Start":"00:53.210 ","End":"00:55.470","Text":"There\u0027s 2 reasons. First of all,"},{"Start":"00:55.470 ","End":"00:59.640","Text":"I subtracted the 1 from here that made it smaller and then I"},{"Start":"00:59.640 ","End":"01:04.790","Text":"reduce this 1 over n from the exponent and that made it even smaller."},{"Start":"01:04.790 ","End":"01:06.605","Text":"This is bigger than this."},{"Start":"01:06.605 ","End":"01:10.525","Text":"On the other hand, they want to get this to be smaller than something,"},{"Start":"01:10.525 ","End":"01:17.180","Text":"1 plus 2^4n plus 1 over n is going to be less than."},{"Start":"01:17.180 ","End":"01:19.415","Text":"Now, here\u0027s what I\u0027m going to do."},{"Start":"01:19.415 ","End":"01:25.560","Text":"I\u0027m going to replace the 1 by 2^4n."},{"Start":"01:26.900 ","End":"01:30.165","Text":"This is certainly bigger than this."},{"Start":"01:30.165 ","End":"01:41.055","Text":"This I\u0027m going to replace by 2^4n plus 1."},{"Start":"01:41.055 ","End":"01:46.215","Text":"This is certainly less than this,"},{"Start":"01:46.215 ","End":"01:49.050","Text":"unless n happens to be 1 and then they\u0027re equal,"},{"Start":"01:49.050 ","End":"01:51.510","Text":"but anything from n above 1,"},{"Start":"01:51.510 ","End":"01:54.105","Text":"1 over n is less than 1."},{"Start":"01:54.105 ","End":"01:57.900","Text":"The limit to mean this is less than this."},{"Start":"01:57.900 ","End":"02:01.590","Text":"In any event, this is less than this."},{"Start":"02:01.900 ","End":"02:06.935","Text":"I\u0027m going to work on this a little bit till we get it in a better shape."},{"Start":"02:06.935 ","End":"02:10.025","Text":"This is equal to,"},{"Start":"02:10.025 ","End":"02:19.820","Text":"this is 2^4n plus and using the rules of exponents is 2^4n times 2^1."},{"Start":"02:19.820 ","End":"02:23.160","Text":"It\u0027s twice 2^4n,"},{"Start":"02:24.010 ","End":"02:32.670","Text":"so altogether, 3 times 2^4n."},{"Start":"02:32.930 ","End":"02:38.520","Text":"What we have now is that 2^4n is"},{"Start":"02:38.520 ","End":"02:44.850","Text":"less than 1 plus 2^4n plus 1 over n from here."},{"Start":"02:44.850 ","End":"02:53.170","Text":"From this one, we get less than 3 times 2^4n."},{"Start":"02:53.890 ","End":"02:56.915","Text":"Now, let\u0027s take the nth root."},{"Start":"02:56.915 ","End":"03:04.505","Text":"But taking the nth root of something is the same as taking the power of 1 over n,"},{"Start":"03:04.505 ","End":"03:12.940","Text":"so (2^4n)^1 is less than."},{"Start":"03:13.130 ","End":"03:17.390","Text":"The inequality is preserved if you raise to a positive power,"},{"Start":"03:17.390 ","End":"03:22.370","Text":"that\u0027s okay, to the nth root, and this time,"},{"Start":"03:22.370 ","End":"03:25.265","Text":"I\u0027ll write it as the root not as an exponent,"},{"Start":"03:25.265 ","End":"03:27.620","Text":"because this is what we have originally,"},{"Start":"03:27.620 ","End":"03:34.625","Text":"4n plus 1 over n. This is less than,"},{"Start":"03:34.625 ","End":"03:38.000","Text":"I\u0027ll take this^1 over n. I can do it separately,"},{"Start":"03:38.000 ","End":"03:40.100","Text":"3 to the 1 over n,"},{"Start":"03:40.100 ","End":"03:47.705","Text":"2^4n to the 1 over n. Now,"},{"Start":"03:47.705 ","End":"03:53.665","Text":"this is our general an."},{"Start":"03:53.665 ","End":"03:59.820","Text":"What we have now is that an is"},{"Start":"03:59.820 ","End":"04:09.724","Text":"less than 3^1 over n. This by rules of exponents is just 2 to the fourth of 16,"},{"Start":"04:09.724 ","End":"04:17.380","Text":"I\u0027ll leave it as 2 to the fourth and is greater than 2 to the 4th."},{"Start":"04:20.290 ","End":"04:27.610","Text":"Now, let\u0027s take the limit as n goes to infinity for each of these 3."},{"Start":"04:27.610 ","End":"04:29.620","Text":"This is a constant."},{"Start":"04:29.620 ","End":"04:32.060","Text":"This just stays what it is."},{"Start":"04:32.060 ","End":"04:34.030","Text":"I\u0027ll write it as 16."},{"Start":"04:34.030 ","End":"04:40.595","Text":"This goes to,"},{"Start":"04:40.595 ","End":"04:43.880","Text":"it is 16, so it goes to 16,"},{"Start":"04:43.880 ","End":"04:50.890","Text":"3 to the 1 over n goes to 1 because this goes to 0."},{"Start":"04:50.890 ","End":"04:54.630","Text":"This goes to 3 to the 0 which is 1."},{"Start":"04:54.630 ","End":"05:04.200","Text":"What we have is that an is sandwiched between 16 and 1 times 16 is 16 and so"},{"Start":"05:04.200 ","End":"05:14.270","Text":"it also must go to 16 by the squeeze theorem, the sandwich theorem."},{"Start":"05:14.270 ","End":"05:16.430","Text":"That is the answer."},{"Start":"05:16.430 ","End":"05:22.530","Text":"The limit is equal to 16, and we\u0027re done."}],"ID":31178},{"Watched":false,"Name":"Exercise 9","Duration":"7m 46s","ChapterTopicVideoID":29566,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.289","Text":"In this exercise, we have to evaluate this limit."},{"Start":"00:04.289 ","End":"00:07.500","Text":"Let\u0027s say this is a_n."},{"Start":"00:07.500 ","End":"00:09.045","Text":"So members out."},{"Start":"00:09.045 ","End":"00:11.490","Text":"A_1, n is 1."},{"Start":"00:11.490 ","End":"00:13.110","Text":"So if n is 1,"},{"Start":"00:13.110 ","End":"00:17.064","Text":"we go up to twice 1 minus 1 is just 1,"},{"Start":"00:17.064 ","End":"00:20.400","Text":"and here we go from 2 up to 2,"},{"Start":"00:20.400 ","End":"00:26.100","Text":"so it\u0027s just 1.5. a_2 twice 2 minus 1 is 3."},{"Start":"00:26.100 ","End":"00:33.620","Text":"So it\u0027s 1 times 3 over 2 times 4. a_3, actually,"},{"Start":"00:33.620 ","End":"00:36.200","Text":"the 3 is just the number of factors,"},{"Start":"00:36.200 ","End":"00:38.165","Text":"1 times 3 times 5,"},{"Start":"00:38.165 ","End":"00:40.475","Text":"because twice 3 minus 1 is 5,"},{"Start":"00:40.475 ","End":"00:41.960","Text":"because all the odd numbers,"},{"Start":"00:41.960 ","End":"00:44.945","Text":"and here\u0027s even numbers up to twice 3,"},{"Start":"00:44.945 ","End":"00:48.720","Text":"2 times 4 times 6, and so on."},{"Start":"00:48.720 ","End":"00:51.965","Text":"The question is, what does this go to in the limit?"},{"Start":"00:51.965 ","End":"00:53.525","Text":"We\u0027re given a hint."},{"Start":"00:53.525 ","End":"00:55.880","Text":"I\u0027ll do this at the end."},{"Start":"00:55.880 ","End":"00:58.895","Text":"So let\u0027s assume that this is true."},{"Start":"00:58.895 ","End":"01:00.770","Text":"How do we use this?"},{"Start":"01:00.770 ","End":"01:02.924","Text":"The answer is, the sandwich theorem,"},{"Start":"01:02.924 ","End":"01:08.215","Text":"the squeeze theorem is what we have is that a_n,"},{"Start":"01:08.215 ","End":"01:10.620","Text":"This is a_n,"},{"Start":"01:10.620 ","End":"01:17.268","Text":"is less than 1 over the square root of 2 n plus 1,"},{"Start":"01:17.268 ","End":"01:23.125","Text":"and it\u0027s also clear that a_n is a positive number."},{"Start":"01:23.125 ","End":"01:25.390","Text":"All these numbers are positive,"},{"Start":"01:25.390 ","End":"01:28.135","Text":"so it\u0027s bigger than 0."},{"Start":"01:28.135 ","End":"01:32.830","Text":"Each of these is a series in n. Well,"},{"Start":"01:32.830 ","End":"01:34.300","Text":"this one\u0027s a constant series,"},{"Start":"01:34.300 ","End":"01:37.405","Text":"but these two, the one on the left and the right have a limit."},{"Start":"01:37.405 ","End":"01:42.055","Text":"This thing, as n goes to infinity,"},{"Start":"01:42.055 ","End":"01:47.440","Text":"it goes to 0 because 2n plus 1 goes to infinity."},{"Start":"01:47.440 ","End":"01:49.390","Text":"Square root of infinity is infinity."},{"Start":"01:49.390 ","End":"01:51.385","Text":"1 over infinity is 0."},{"Start":"01:51.385 ","End":"01:53.425","Text":"0 is a constant,"},{"Start":"01:53.425 ","End":"01:55.780","Text":"so its limit is 0."},{"Start":"01:55.780 ","End":"02:00.363","Text":"So the limit of a_n is trapped,"},{"Start":"02:00.363 ","End":"02:03.540","Text":"squeezed between 0 and 0,"},{"Start":"02:03.540 ","End":"02:08.740","Text":"so it also goes to 0 as n goes to infinity."},{"Start":"02:08.740 ","End":"02:10.655","Text":"So this is the answer."},{"Start":"02:10.655 ","End":"02:16.520","Text":"But I can\u0027t say we\u0027re done because we still have this induction to prove."},{"Start":"02:16.520 ","End":"02:19.820","Text":"I just copied it down there."},{"Start":"02:19.820 ","End":"02:21.500","Text":"Let\u0027s go about proving it."},{"Start":"02:21.500 ","End":"02:24.980","Text":"Start off with n=1."},{"Start":"02:24.980 ","End":"02:28.205","Text":"Well, we have a_1, it\u0027s 1/2."},{"Start":"02:28.205 ","End":"02:30.530","Text":"It says here less than,"},{"Start":"02:30.530 ","End":"02:32.180","Text":"but I have to put a question mark."},{"Start":"02:32.180 ","End":"02:38.900","Text":"We have to show 1 over square root of twice 1 plus 1."},{"Start":"02:38.900 ","End":"02:45.200","Text":"It\u0027s 1 over the square root of 2 times 1 is 2,"},{"Start":"02:45.200 ","End":"02:46.455","Text":"plus 1 is 3,"},{"Start":"02:46.455 ","End":"02:50.034","Text":"is a half less than 1 over square root of 3,"},{"Start":"02:50.034 ","End":"02:53.360","Text":"and the answer is yes."},{"Start":"02:53.360 ","End":"03:00.666","Text":"The reason it\u0027s true is because 2 is bigger than square root of 3,"},{"Start":"03:00.666 ","End":"03:04.160","Text":"and we invert the inequality."},{"Start":"03:04.160 ","End":"03:07.190","Text":"You might ask, why is 2 bigger than the square root of 3?"},{"Start":"03:07.190 ","End":"03:11.305","Text":"Well, you could square both sides and say 4 is bigger than 3,"},{"Start":"03:11.305 ","End":"03:13.280","Text":"2^2 is bigger than this."},{"Start":"03:13.280 ","End":"03:18.095","Text":"Or you could just evaluate the square root of 3 is 1.7 something."},{"Start":"03:18.095 ","End":"03:20.720","Text":"So it\u0027s less than 2."},{"Start":"03:20.720 ","End":"03:23.255","Text":"Okay, that\u0027s n=1."},{"Start":"03:23.255 ","End":"03:26.375","Text":"Now, we do the induction step."},{"Start":"03:26.375 ","End":"03:30.530","Text":"Assume this is true for a certain n,"},{"Start":"03:30.530 ","End":"03:36.355","Text":"let\u0027s show that it\u0027s true for the next n. So let\u0027s say we know this,"},{"Start":"03:36.355 ","End":"03:38.853","Text":"we have to show that,"},{"Start":"03:38.853 ","End":"03:40.370","Text":"or let me just copy this."},{"Start":"03:40.370 ","End":"03:49.295","Text":"I\u0027ll have to extend this because this is a_n and I now have to write out a_n plus 1,"},{"Start":"03:49.295 ","End":"03:53.030","Text":"which means that on the denominator,"},{"Start":"03:53.030 ","End":"03:58.645","Text":"we stop at twice n plus 1, 2n plus 2."},{"Start":"03:58.645 ","End":"04:04.280","Text":"Here, I replace n by n plus 1 and I get 2n plus 1."},{"Start":"04:04.280 ","End":"04:06.365","Text":"It\u0027s 2n plus 2 minus 1."},{"Start":"04:06.365 ","End":"04:12.920","Text":"We have to show that this is less than 1 over the square root."},{"Start":"04:12.920 ","End":"04:15.335","Text":"If I put n plus 1,"},{"Start":"04:15.335 ","End":"04:19.705","Text":"then I get 2n plus 3."},{"Start":"04:19.705 ","End":"04:23.585","Text":"We\u0027re taking for granted now that this is true,"},{"Start":"04:23.585 ","End":"04:24.920","Text":"and with the help of this,"},{"Start":"04:24.920 ","End":"04:28.460","Text":"we have to show that this is true."},{"Start":"04:28.460 ","End":"04:32.150","Text":"I copied this here."},{"Start":"04:32.150 ","End":"04:36.410","Text":"Now I start with this expression and I\u0027ll show that this is less than this."},{"Start":"04:36.410 ","End":"04:42.710","Text":"First of all, I can use the induction hypothesis to say that this part is"},{"Start":"04:42.710 ","End":"04:50.330","Text":"less than 1 over root 2n plus 1."},{"Start":"04:50.330 ","End":"04:53.000","Text":"This over, this is a positive number."},{"Start":"04:53.000 ","End":"04:59.034","Text":"So you can multiply both sides of a_n inequality by a positive number,"},{"Start":"04:59.034 ","End":"05:02.568","Text":"2n plus 1 over 2n plus 2,"},{"Start":"05:02.568 ","End":"05:09.049","Text":"and now I have to show that this is less than this."},{"Start":"05:09.049 ","End":"05:12.350","Text":"This, by the way, can be simplified a bit."},{"Start":"05:12.350 ","End":"05:17.460","Text":"The square root of 2n plus 1 goes into 2n plus 1,"},{"Start":"05:17.460 ","End":"05:20.300","Text":"the square root of 2n plus 1 times."},{"Start":"05:20.300 ","End":"05:23.705","Text":"So I can write it like this."},{"Start":"05:23.705 ","End":"05:30.634","Text":"Now, I have to show that this is less than this."},{"Start":"05:30.634 ","End":"05:38.054","Text":"We still have to show that root 2n plus 1 over 2n plus 2"},{"Start":"05:38.054 ","End":"05:46.485","Text":"is less than 1 over root 2n plus 3."},{"Start":"05:46.485 ","End":"05:49.280","Text":"We\u0027ve reduced our problem to this."},{"Start":"05:49.280 ","End":"05:52.080","Text":"If we have 4 positive numbers,"},{"Start":"05:52.080 ","End":"05:53.585","Text":"a, b, c and d,"},{"Start":"05:53.585 ","End":"05:59.210","Text":"a over b is less than c over d. If and only"},{"Start":"05:59.210 ","End":"06:05.420","Text":"if this diagonal is less than this diagonal,"},{"Start":"06:05.420 ","End":"06:09.725","Text":"i.e., ad is less than bc."},{"Start":"06:09.725 ","End":"06:13.925","Text":"So we want to apply this to our case,"},{"Start":"06:13.925 ","End":"06:19.495","Text":"and see, is this times this less than this times this?"},{"Start":"06:19.495 ","End":"06:27.735","Text":"Root 2n plus 1 root 2n plus 3,"},{"Start":"06:27.735 ","End":"06:32.770","Text":"is it less than 2n plus 2?"},{"Start":"06:32.770 ","End":"06:34.040","Text":"This is equivalent to this,"},{"Start":"06:34.040 ","End":"06:35.840","Text":"so we\u0027re reduced to showing this."},{"Start":"06:35.840 ","End":"06:38.315","Text":"Now, when both sides are positive,"},{"Start":"06:38.315 ","End":"06:41.390","Text":"this is if and only if the squares are equal."},{"Start":"06:41.390 ","End":"06:46.625","Text":"So this is if and only if 2n plus 1,"},{"Start":"06:46.625 ","End":"06:49.400","Text":"2n plus 3,"},{"Start":"06:49.400 ","End":"06:56.495","Text":"less than 2n plus 2^2,"},{"Start":"06:56.495 ","End":"06:57.785","Text":"if you multiply this out,"},{"Start":"06:57.785 ","End":"07:02.300","Text":"we get 4n^2 plus"},{"Start":"07:02.300 ","End":"07:09.515","Text":"2n plus 6n is plus 8n plus 3."},{"Start":"07:09.515 ","End":"07:17.905","Text":"This is 4n^2 plus 8n plus 4."},{"Start":"07:17.905 ","End":"07:23.965","Text":"This will be true if and only if 3 is less than 4,"},{"Start":"07:23.965 ","End":"07:28.295","Text":"because this part cancels with this part."},{"Start":"07:28.295 ","End":"07:29.930","Text":"Now, this is true,"},{"Start":"07:29.930 ","End":"07:31.175","Text":"so this is true,"},{"Start":"07:31.175 ","End":"07:32.593","Text":"so this is true,"},{"Start":"07:32.593 ","End":"07:34.224","Text":"so this is true,"},{"Start":"07:34.224 ","End":"07:35.300","Text":"so this is true,"},{"Start":"07:35.300 ","End":"07:42.110","Text":"and that means that this is less than this,"},{"Start":"07:42.110 ","End":"07:46.260","Text":"and so we are done."}],"ID":31179},{"Watched":false,"Name":"Exercise 10","Duration":"5m 56s","ChapterTopicVideoID":29567,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we have to evaluate the limit as n goes to"},{"Start":"00:03.840 ","End":"00:11.010","Text":"infinity of this sequence and each term is a sum."},{"Start":"00:11.010 ","End":"00:16.365","Text":"Let\u0027s call this general term an."},{"Start":"00:16.365 ","End":"00:19.880","Text":"We\u0027re going to solve it using the squeeze theorem,"},{"Start":"00:19.880 ","End":"00:21.680","Text":"also known as the sandwich theorem,"},{"Start":"00:21.680 ","End":"00:26.160","Text":"which says that if I can find 2 other theories,"},{"Start":"00:26.160 ","End":"00:34.626","Text":"let\u0027s say bn and cn such that an is squeezed in between both of these and"},{"Start":"00:34.626 ","End":"00:39.312","Text":"if bn tends to limit L as n goes to infinity"},{"Start":"00:39.312 ","End":"00:44.240","Text":"and cn also turns to L as n goes to infinity,"},{"Start":"00:44.240 ","End":"00:48.710","Text":"then an is squeezed,"},{"Start":"00:48.710 ","End":"00:55.500","Text":"it\u0027s forced to also go to the limit L as n goes to infinity."},{"Start":"00:55.780 ","End":"00:58.675","Text":"Let\u0027s squeeze an."},{"Start":"00:58.675 ","End":"01:07.280","Text":"an is going to be less than or equal to something and greater or equal to something else."},{"Start":"01:07.280 ","End":"01:11.725","Text":"If we look at the terms in this sum,"},{"Start":"01:11.725 ","End":"01:15.390","Text":"which one is the biggest and which one is the smallest?"},{"Start":"01:15.390 ","End":"01:18.035","Text":"Now, if it\u0027s 1 over something,"},{"Start":"01:18.035 ","End":"01:23.010","Text":"the smallest denominator is the biggest number."},{"Start":"01:23.010 ","End":"01:26.730","Text":"This is the smallest denominator,"},{"Start":"01:26.730 ","End":"01:32.840","Text":"so this is the greatest of these."},{"Start":"01:32.840 ","End":"01:35.915","Text":"How many are there? There are n of them."},{"Start":"01:35.915 ","End":"01:41.560","Text":"This is the least, the smallest."},{"Start":"01:41.720 ","End":"01:47.130","Text":"If I take n times the greatest,"},{"Start":"01:47.130 ","End":"01:49.603","Text":"that should be bigger or equal to,"},{"Start":"01:49.603 ","End":"01:56.160","Text":"so it\u0027s n over the square root of n squared plus 1."},{"Start":"01:56.500 ","End":"01:58.940","Text":"This is the smallest,"},{"Start":"01:58.940 ","End":"02:08.600","Text":"so n times 1 over the square root of n^2 plus n. Now,"},{"Start":"02:08.600 ","End":"02:12.610","Text":"I\u0027m going to show that each of these turns to 1."},{"Start":"02:12.610 ","End":"02:17.845","Text":"Let\u0027s start with this one."},{"Start":"02:17.845 ","End":"02:23.535","Text":"Then we have that the limit as n goes to infinity of"},{"Start":"02:23.535 ","End":"02:29.615","Text":"n over square root of n^2 plus 1 is equal to."},{"Start":"02:29.615 ","End":"02:33.320","Text":"What I\u0027m going to do is factorize."},{"Start":"02:33.320 ","End":"02:40.798","Text":"In the denominator, I\u0027m going to take n^2 out of what\u0027s under the square root."},{"Start":"02:40.798 ","End":"02:46.800","Text":"I\u0027ll just work on this side."},{"Start":"02:47.710 ","End":"02:56.255","Text":"N over the square root of n^2 plus 1 is equal to n over the square root."},{"Start":"02:56.255 ","End":"03:02.840","Text":"I\u0027ll take n^2 out the brackets and I have 1 plus 1 over n^2 here."},{"Start":"03:02.840 ","End":"03:08.450","Text":"Now, this is equal to n over,"},{"Start":"03:08.450 ","End":"03:11.975","Text":"the square root I can break up into 2 square roots,"},{"Start":"03:11.975 ","End":"03:19.370","Text":"the square root of n^2 and then the square root of 1 plus 1 over n^2."},{"Start":"03:19.370 ","End":"03:28.610","Text":"Notice that this is equal to n. Normally it could be plus or minus n,"},{"Start":"03:28.610 ","End":"03:30.140","Text":"it would be the absolute value of n,"},{"Start":"03:30.140 ","End":"03:31.760","Text":"but n is positive."},{"Start":"03:31.760 ","End":"03:37.910","Text":"Now, this n cancels with this n and that leaves us 1 here."},{"Start":"03:37.910 ","End":"03:41.170","Text":"I\u0027ll take this and I\u0027ll go back here."},{"Start":"03:41.170 ","End":"03:46.565","Text":"We have the limit as n goes to infinity of 1"},{"Start":"03:46.565 ","End":"03:54.645","Text":"over the square root of 1 plus 1 over n^2."},{"Start":"03:54.645 ","End":"03:59.960","Text":"Now when n^2 goes to infinity,"},{"Start":"03:59.960 ","End":"04:03.935","Text":"this thing goes to 0."},{"Start":"04:03.935 ","End":"04:08.490","Text":"The limit is 1 and that\u0027s this 1 here,"},{"Start":"04:08.490 ","End":"04:11.090","Text":"so I\u0027ll shade it in the same color."},{"Start":"04:11.090 ","End":"04:17.280","Text":"Now, we want to work on this one."},{"Start":"04:18.070 ","End":"04:23.180","Text":"I\u0027ll do at the side, I\u0027ll simplify this."},{"Start":"04:23.180 ","End":"04:29.480","Text":"n over the square root of n^2 plus n. It\u0027s almost the same as this,"},{"Start":"04:29.480 ","End":"04:30.770","Text":"I could almost copy it."},{"Start":"04:30.770 ","End":"04:36.170","Text":"It\u0027s n over the square root of n^2 times"},{"Start":"04:36.170 ","End":"04:41.510","Text":"1 plus 1 over n. That\u0027s really the only difference,"},{"Start":"04:41.510 ","End":"04:44.960","Text":"instead of n^2 here I have n. This is equal to."},{"Start":"04:44.960 ","End":"04:48.995","Text":"I can say that this is n over"},{"Start":"04:48.995 ","End":"04:58.680","Text":"n square root of 1 plus 1 over n and the n cancels and leaves a 1 here."},{"Start":"04:58.680 ","End":"05:05.945","Text":"Back here, we have the limit as n goes to infinity,"},{"Start":"05:05.945 ","End":"05:11.690","Text":"n over the square root of n^2 plus n,"},{"Start":"05:11.690 ","End":"05:13.265","Text":"that\u0027s the only difference."},{"Start":"05:13.265 ","End":"05:17.465","Text":"This will be the limit as n goes to infinity"},{"Start":"05:17.465 ","End":"05:22.820","Text":"of 1 over the square root of 1 plus and I\u0027m copying from here,"},{"Start":"05:22.820 ","End":"05:25.985","Text":"1 over n and just as above,"},{"Start":"05:25.985 ","End":"05:27.440","Text":"it\u0027s almost the same thing,"},{"Start":"05:27.440 ","End":"05:29.615","Text":"this also goes to 0."},{"Start":"05:29.615 ","End":"05:35.405","Text":"This is equal to 1 and I\u0027ll shade that also."},{"Start":"05:35.405 ","End":"05:37.520","Text":"We have the sandwich,"},{"Start":"05:37.520 ","End":"05:39.930","Text":"this yellow turns to 1,"},{"Start":"05:39.930 ","End":"05:43.275","Text":"this green goes to 1."},{"Start":"05:43.275 ","End":"05:46.295","Text":"Hence by the squeeze theorem,"},{"Start":"05:46.295 ","End":"05:49.130","Text":"as n goes to infinity, r, a, n,"},{"Start":"05:49.130 ","End":"05:51.020","Text":"which is between the two of them,"},{"Start":"05:51.020 ","End":"05:55.920","Text":"also goes to 1 and that\u0027s it."}],"ID":31180},{"Watched":false,"Name":"Exercise 11","Duration":"4m 59s","ChapterTopicVideoID":29568,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this exercise, there are 2 parts and each part is sequence and we have to"},{"Start":"00:05.580 ","End":"00:11.205","Text":"prove that the sequence goes to 0 using the Sandwich Theorem."},{"Start":"00:11.205 ","End":"00:16.020","Text":"In part a, sequence is called a_n and in part b it\u0027s called b_n."},{"Start":"00:16.020 ","End":"00:21.390","Text":"Part a is a product of factors starting with"},{"Start":"00:21.390 ","End":"00:27.000","Text":"the cube root of 2 and going up to the 2n plus 1th root of 2."},{"Start":"00:27.000 ","End":"00:29.130","Text":"Notice that if n is 1,"},{"Start":"00:29.130 ","End":"00:31.515","Text":"we get 3 twice 1 plus 1,"},{"Start":"00:31.515 ","End":"00:33.795","Text":"then twice 2 plus 1 is 5."},{"Start":"00:33.795 ","End":"00:36.720","Text":"This corresponds to n equals 1,"},{"Start":"00:36.720 ","End":"00:38.250","Text":"n equals 2,"},{"Start":"00:38.250 ","End":"00:42.635","Text":"and the general n and there are n factors."},{"Start":"00:42.635 ","End":"00:48.830","Text":"We could write this product using the Pi notation as the product k equals"},{"Start":"00:48.830 ","End":"00:56.555","Text":"1-n of the square root of 2 minus the 2k plus 1th root of 2."},{"Start":"00:56.555 ","End":"00:59.629","Text":"Like so. Now I\u0027m going to estimate the typical"},{"Start":"00:59.629 ","End":"01:03.430","Text":"factor in the product to be between something and something."},{"Start":"01:03.430 ","End":"01:05.625","Text":"Let\u0027s start with the root,"},{"Start":"01:05.625 ","End":"01:11.430","Text":"the 2k plus first root of 2 is less than the square root of 2,"},{"Start":"01:11.430 ","End":"01:13.620","Text":"like the cube root of 2 is less than the square root."},{"Start":"01:13.620 ","End":"01:16.670","Text":"The 5th root is less than the square root, and so on."},{"Start":"01:16.670 ","End":"01:21.220","Text":"Also, the 2k plus first root of 2 is bigger than 1."},{"Start":"01:21.220 ","End":"01:23.285","Text":"Now from this, we can conclude this."},{"Start":"01:23.285 ","End":"01:29.370","Text":"The square root of 2 is bigger than the 2k plus first root of 2,"},{"Start":"01:29.370 ","End":"01:34.520","Text":"so this minus this is bigger than 0 and also because this is bigger than this,"},{"Start":"01:34.520 ","End":"01:37.720","Text":"when you subtract each 1 from 2,"},{"Start":"01:37.720 ","End":"01:39.994","Text":"it\u0027ll be the other way round."},{"Start":"01:39.994 ","End":"01:44.540","Text":"Square root of 2 minus this will be less than square root of 2 minus 1,"},{"Start":"01:44.540 ","End":"01:46.850","Text":"so we have this inequality,"},{"Start":"01:46.850 ","End":"01:50.340","Text":"and this is true for k equals 1,"},{"Start":"01:50.340 ","End":"01:52.220","Text":"2 and so on up to n,"},{"Start":"01:52.220 ","End":"01:56.945","Text":"but we can multiply them all together and get 0 times 0 times"},{"Start":"01:56.945 ","End":"02:01.610","Text":"0 and factors is less than this times this times this."},{"Start":"02:01.610 ","End":"02:03.560","Text":"Here where k is 1, here k is 2,"},{"Start":"02:03.560 ","End":"02:08.280","Text":"here k is n and is less than the square root of 2 minus"},{"Start":"02:08.280 ","End":"02:13.570","Text":"1 to the power of n. Now this here is a_n,"},{"Start":"02:13.570 ","End":"02:19.260","Text":"so we have 0 less than a_n less than root 2 minus 1 to the power"},{"Start":"02:19.260 ","End":"02:25.290","Text":"of n. The square root of 2 minus 1 is between 0 and 1."},{"Start":"02:25.290 ","End":"02:28.995","Text":"Then to the power of n tends to 0."},{"Start":"02:28.995 ","End":"02:32.460","Text":"It\u0027s roughly equal to 0.4 something."},{"Start":"02:32.460 ","End":"02:34.230","Text":"Could just say it was a half,"},{"Start":"02:34.230 ","End":"02:38.015","Text":"a half times a half times a half keeps getting smaller and smaller and goes to 0."},{"Start":"02:38.015 ","End":"02:43.060","Text":"Something between 0 and 1 to the power of n goes to 0 as n goes to infinity."},{"Start":"02:43.060 ","End":"02:48.500","Text":"If a_n is sandwiched between 0 and something that tends to 0,"},{"Start":"02:48.500 ","End":"02:50.255","Text":"by the Sandwich Theorem,"},{"Start":"02:50.255 ","End":"02:54.450","Text":"it tends to 0 and that\u0027s part a."},{"Start":"02:54.450 ","End":"02:57.840","Text":"Now part b. As a reminder,"},{"Start":"02:57.840 ","End":"03:00.360","Text":"b_n is equal to n plus 1 to"},{"Start":"03:00.360 ","End":"03:05.640","Text":"the Alpha minus n^Alpha and Alpha is some constant between 0 and 1."},{"Start":"03:05.640 ","End":"03:09.990","Text":"We have to show that b_n goes to 0 using the Sandwich Theorem."},{"Start":"03:09.990 ","End":"03:14.990","Text":"Note that n plus 1 to the Alpha is bigger than n^Alpha,"},{"Start":"03:14.990 ","End":"03:17.060","Text":"so that b_n is bigger than 0."},{"Start":"03:17.060 ","End":"03:19.030","Text":"We\u0027ll need that later."},{"Start":"03:19.030 ","End":"03:22.010","Text":"Now, b_n is equal to,"},{"Start":"03:22.010 ","End":"03:24.590","Text":"if you take n^Alpha outside the brackets,"},{"Start":"03:24.590 ","End":"03:28.849","Text":"we get the following and then using the rules of exponents,"},{"Start":"03:28.849 ","End":"03:32.900","Text":"we can take n plus 1 over n all to the power of Alpha."},{"Start":"03:32.900 ","End":"03:43.060","Text":"N plus 1 over n is 1 plus 1 over n and because Alpha is between 0 and 1,"},{"Start":"03:43.060 ","End":"03:46.625","Text":"this is less than this."},{"Start":"03:46.625 ","End":"03:51.570","Text":"Something positive increases as Alpha increases,"},{"Start":"03:51.570 ","End":"03:54.800","Text":"Alpha is less than 1 and this is less than this, 20 subtract 1,"},{"Start":"03:54.800 ","End":"03:58.645","Text":"it\u0027s still less and multiply by a positive number, it\u0027s still less."},{"Start":"03:58.645 ","End":"04:01.320","Text":"We can cancel the 1 minus 1,"},{"Start":"04:01.320 ","End":"04:05.540","Text":"so this is just 1 over n. N^Alpha times 1 over n,"},{"Start":"04:05.540 ","End":"04:07.700","Text":"if we put this in the denominator,"},{"Start":"04:07.700 ","End":"04:11.380","Text":"we have 1 over n to the 1 minus Alpha."},{"Start":"04:11.380 ","End":"04:15.600","Text":"Now 1 minus Alpha is bigger than 0,"},{"Start":"04:15.600 ","End":"04:18.240","Text":"because 1 is bigger than Alpha."},{"Start":"04:18.240 ","End":"04:27.030","Text":"We\u0027ve just shown that b_n is less than 1 over n to the power of 1 minus Alpha,"},{"Start":"04:27.030 ","End":"04:30.480","Text":"but also b_n is bigger than 0,"},{"Start":"04:30.480 ","End":"04:36.400","Text":"so b_n is between 0 and 1 over n to the 1 minus Alpha."},{"Start":"04:36.400 ","End":"04:38.360","Text":"Now, when n goes to infinity,"},{"Start":"04:38.360 ","End":"04:40.715","Text":"because 1 minus Alpha is positive,"},{"Start":"04:40.715 ","End":"04:42.530","Text":"the denominator goes to infinity,"},{"Start":"04:42.530 ","End":"04:44.635","Text":"so the fraction goes to 0,"},{"Start":"04:44.635 ","End":"04:49.205","Text":"so b_n is sandwiched between 0 and something that tends to 0,"},{"Start":"04:49.205 ","End":"04:54.715","Text":"and therefore, it also goes to 0 by the Sandwich Theorem."},{"Start":"04:54.715 ","End":"04:59.460","Text":"We\u0027ve proven what we had to prove and that concludes this exercise."}],"ID":31181},{"Watched":false,"Name":"Exercise 12","Duration":"3m 10s","ChapterTopicVideoID":29569,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.915","Text":"In this exercise, x is a fixed positive real number, arbitrary but fixed."},{"Start":"00:06.915 ","End":"00:10.980","Text":"We consider the sequence a_n, where n varies,"},{"Start":"00:10.980 ","End":"00:12.030","Text":"1, 2, 3, 4, 5,"},{"Start":"00:12.030 ","End":"00:15.165","Text":"etc, a_n equals this."},{"Start":"00:15.165 ","End":"00:18.705","Text":"This notation is the floor function."},{"Start":"00:18.705 ","End":"00:21.180","Text":"Here\u0027s a graph of the floor function."},{"Start":"00:21.180 ","End":"00:24.360","Text":"It\u0027s the integer closest to the number,"},{"Start":"00:24.360 ","End":"00:25.860","Text":"but on the left of it."},{"Start":"00:25.860 ","End":"00:28.050","Text":"If a is already a whole number,"},{"Start":"00:28.050 ","End":"00:29.940","Text":"then it\u0027s just equal to a."},{"Start":"00:29.940 ","End":"00:33.330","Text":"We have to prove that the limit as n goes to infinity of"},{"Start":"00:33.330 ","End":"00:37.100","Text":"the sequence a_n is bigger than 2."},{"Start":"00:37.100 ","End":"00:41.490","Text":"Whenever you see sequence problem with the floor function in it,"},{"Start":"00:41.490 ","End":"00:44.720","Text":"you can bet that it\u0027s going to involve the sandwich theorem."},{"Start":"00:44.720 ","End":"00:49.055","Text":"Now, the floor function of a is always less than or equal to a."},{"Start":"00:49.055 ","End":"00:51.110","Text":"We only go to the left."},{"Start":"00:51.110 ","End":"00:53.930","Text":"But it\u0027s always bigger than a minus 1."},{"Start":"00:53.930 ","End":"00:56.720","Text":"Because if a minus 1 was a whole number,"},{"Start":"00:56.720 ","End":"00:58.543","Text":"then a would be a whole number."},{"Start":"00:58.543 ","End":"00:59.990","Text":"It can\u0027t be equal here."},{"Start":"00:59.990 ","End":"01:02.615","Text":"Here less than or equal to here, less than."},{"Start":"01:02.615 ","End":"01:04.400","Text":"Now from this inequality,"},{"Start":"01:04.400 ","End":"01:07.040","Text":"we can get an inequality for a_n."},{"Start":"01:07.040 ","End":"01:08.915","Text":"If you look at the coloring,"},{"Start":"01:08.915 ","End":"01:15.440","Text":"we can just replace this by x^2 n^2 and get a less than or equal to,"},{"Start":"01:15.440 ","End":"01:18.940","Text":"and replace it by x^2 n^2 minus 1 and get it less than."},{"Start":"01:18.940 ","End":"01:22.970","Text":"The 6n plus doesn\u0027t change the direction of the inequality."},{"Start":"01:22.970 ","End":"01:25.770","Text":"Dividing by a positive number also doesn\u0027t change,"},{"Start":"01:25.770 ","End":"01:27.710","Text":"so this is the inequality."},{"Start":"01:27.710 ","End":"01:29.300","Text":"Now let\u0027s label them."},{"Start":"01:29.300 ","End":"01:32.100","Text":"This is our original a_n,"},{"Start":"01:32.100 ","End":"01:34.510","Text":"we\u0027ll call this b_n and this c_n."},{"Start":"01:34.510 ","End":"01:36.320","Text":"But the square root of x^2,"},{"Start":"01:36.320 ","End":"01:40.295","Text":"n^2 is xn as x is positive."},{"Start":"01:40.295 ","End":"01:42.950","Text":"Now the idea is to use the sandwich theorem on a_n,"},{"Start":"01:42.950 ","End":"01:47.005","Text":"so let\u0027s figure out the limit of b_n and the limit of c_n."},{"Start":"01:47.005 ","End":"01:49.505","Text":"Hopefully they\u0027ll give the same result."},{"Start":"01:49.505 ","End":"01:51.935","Text":"Then we can deduce the limit of a_n."},{"Start":"01:51.935 ","End":"01:53.390","Text":"Let\u0027s start with c_n."},{"Start":"01:53.390 ","End":"01:56.240","Text":"As n goes to infinity,"},{"Start":"01:56.240 ","End":"01:59.895","Text":"we can divide top and bottom by"},{"Start":"01:59.895 ","End":"02:02.900","Text":"n. The only thing"},{"Start":"02:02.900 ","End":"02:06.500","Text":"involving n now is the square root of 2 over n. When n goes to infinity,"},{"Start":"02:06.500 ","End":"02:08.345","Text":"this goes to 0."},{"Start":"02:08.345 ","End":"02:11.555","Text":"We\u0027re left with 6 plus x over 3."},{"Start":"02:11.555 ","End":"02:13.370","Text":"Now for b_n,"},{"Start":"02:13.370 ","End":"02:15.980","Text":"if we divide top and bottom by n,"},{"Start":"02:15.980 ","End":"02:18.275","Text":"the difference is that we have,"},{"Start":"02:18.275 ","End":"02:21.260","Text":"instead of just x, we have this expression,"},{"Start":"02:21.260 ","End":"02:26.375","Text":"square root of x^2 minus 1 over n^2."},{"Start":"02:26.375 ","End":"02:28.685","Text":"When n goes to infinity,"},{"Start":"02:28.685 ","End":"02:33.665","Text":"this part goes to 0 and the 1 over n^2 goes to 0."},{"Start":"02:33.665 ","End":"02:39.910","Text":"We\u0027re left with 6 plus square root of x^2 is x and over 3. The same."},{"Start":"02:39.910 ","End":"02:41.750","Text":"Now that we have the same,"},{"Start":"02:41.750 ","End":"02:47.030","Text":"we can substitute these 2 limits in this inequality that b_n is less than a_n,"},{"Start":"02:47.030 ","End":"02:48.605","Text":"less than or equal to c_n,"},{"Start":"02:48.605 ","End":"02:51.335","Text":"b_n goes to 6 plus x over 3,"},{"Start":"02:51.335 ","End":"02:55.090","Text":"c_n goes to 6 plus x over 3."},{"Start":"02:55.090 ","End":"02:57.495","Text":"By the sandwich rule,"},{"Start":"02:57.495 ","End":"03:01.245","Text":"a_n also goes to 6 plus x over 3."},{"Start":"03:01.245 ","End":"03:04.350","Text":"This is equal to 2 plus x over 3,"},{"Start":"03:04.350 ","End":"03:05.580","Text":"which is bigger than 2."},{"Start":"03:05.580 ","End":"03:08.180","Text":"That\u0027s what we have to show that the limit is bigger than 2."},{"Start":"03:08.180 ","End":"03:10.980","Text":"That concludes this exercise."}],"ID":31182},{"Watched":false,"Name":"Exercise 13","Duration":"3m 40s","ChapterTopicVideoID":29570,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we have to compute the following limit,"},{"Start":"00:04.065 ","End":"00:07.710","Text":"which is different from what we used to."},{"Start":"00:07.710 ","End":"00:10.725","Text":"We\u0027ve have had things with roots,"},{"Start":"00:10.725 ","End":"00:15.030","Text":"but usually inside there are the same exponent."},{"Start":"00:15.030 ","End":"00:18.780","Text":"Here the exponents are all different and it\u0027s hard to tell"},{"Start":"00:18.780 ","End":"00:22.755","Text":"which of these 4 terms goes to infinity faster."},{"Start":"00:22.755 ","End":"00:25.349","Text":"But that usually determines how we proceed."},{"Start":"00:25.349 ","End":"00:27.460","Text":"Let\u0027s see what we can do anyway."},{"Start":"00:27.460 ","End":"00:30.660","Text":"One of the things we can do is separate these exponents,"},{"Start":"00:30.660 ","End":"00:36.105","Text":"take the constant part aside using the product rule for exponents."},{"Start":"00:36.105 ","End":"00:41.535","Text":"Here we can say 2 to the 3n^2 is 2 to the 3 to the n^2."},{"Start":"00:41.535 ","End":"00:46.155","Text":"Similarly, 3 to 2 to the n^2 is 3 to the 2n^2."},{"Start":"00:46.155 ","End":"00:49.580","Text":"We can do that here, here and here."},{"Start":"00:49.580 ","End":"00:52.775","Text":"2^3 is 8,"},{"Start":"00:52.775 ","End":"00:54.915","Text":"3^2 is 9,"},{"Start":"00:54.915 ","End":"00:58.590","Text":"4^1.5 is 8,"},{"Start":"00:58.590 ","End":"01:02.670","Text":"4^1.5 is 2^2 to the 1.5,"},{"Start":"01:02.670 ","End":"01:07.225","Text":"2 times 1.5 is 3, 2^3 is 8."},{"Start":"01:07.225 ","End":"01:12.005","Text":"Now it\u0027s a bit easier to see which of these goes to infinity fastest."},{"Start":"01:12.005 ","End":"01:13.620","Text":"Here we have 8, 9,"},{"Start":"01:13.620 ","End":"01:16.500","Text":"and 8, and 9 is the biggest."},{"Start":"01:16.500 ","End":"01:19.880","Text":"As for the 100, it\u0027s to the power of n,"},{"Start":"01:19.880 ","End":"01:23.560","Text":"which is of lower order than n^2."},{"Start":"01:23.560 ","End":"01:25.955","Text":"Here\u0027s the inequality that we can get."},{"Start":"01:25.955 ","End":"01:30.770","Text":"What\u0027s under the radical is certainly bigger than 9n^2."},{"Start":"01:30.770 ","End":"01:34.880","Text":"I claim it\u0027s also less than, the following expression."},{"Start":"01:34.880 ","End":"01:36.180","Text":"Instead of 8,"},{"Start":"01:36.180 ","End":"01:39.360","Text":"we can write 9 and that just increases it."},{"Start":"01:39.360 ","End":"01:40.935","Text":"Instead of the 8 here,"},{"Start":"01:40.935 ","End":"01:43.260","Text":"we can write 9 here."},{"Start":"01:43.260 ","End":"01:46.395","Text":"The 9 here and the 9 here are the same,"},{"Start":"01:46.395 ","End":"01:53.685","Text":"and I claim that 100 to the n is less than 9 to the n^2."},{"Start":"01:53.685 ","End":"01:56.880","Text":"This here is 9 to the n^n."},{"Start":"01:56.880 ","End":"02:00.300","Text":"We\u0027re really comparing 9 to the n with a 100."},{"Start":"02:00.300 ","End":"02:02.235","Text":"When n is bigger than 2,"},{"Start":"02:02.235 ","End":"02:07.445","Text":"9^n is bigger than 100 and we\u0027re letting n go to infinity."},{"Start":"02:07.445 ","End":"02:11.000","Text":"Here I just wrote out what I just said,"},{"Start":"02:11.000 ","End":"02:13.095","Text":"that when n is bigger than 2,"},{"Start":"02:13.095 ","End":"02:15.600","Text":"9 to the n^2 bigger than 100^n,"},{"Start":"02:15.600 ","End":"02:19.680","Text":"because 9^n is bigger than 100 then raise both sides to the power of"},{"Start":"02:19.680 ","End":"02:25.755","Text":"n. Let\u0027s simplify this right-hand part here,"},{"Start":"02:25.755 ","End":"02:29.535","Text":"we\u0027ll take 9 to the n^2 out, and we have,"},{"Start":"02:29.535 ","End":"02:31.710","Text":"this is less than 1,"},{"Start":"02:31.710 ","End":"02:34.335","Text":"this is equal to 3."},{"Start":"02:34.335 ","End":"02:38.040","Text":"This is equal to 1,024,"},{"Start":"02:38.040 ","End":"02:41.505","Text":"and this coefficient is 1."},{"Start":"02:41.505 ","End":"02:43.950","Text":"This is less than this."},{"Start":"02:43.950 ","End":"02:46.170","Text":"Now, put back the radical,"},{"Start":"02:46.170 ","End":"02:52.620","Text":"the n^2 root of each of the 3 pieces of the inequality."},{"Start":"02:52.620 ","End":"02:56.030","Text":"I\u0027ve colored the 2 pieces on the end."},{"Start":"02:56.030 ","End":"03:00.980","Text":"I\u0027m going to show that each of these goes to the same thing as n goes to infinity."},{"Start":"03:00.980 ","End":"03:02.510","Text":"Let\u0027s start with this one."},{"Start":"03:02.510 ","End":"03:05.450","Text":"The nth root of 9 to the n^2 is just 9,"},{"Start":"03:05.450 ","End":"03:07.295","Text":"so the limit is 9."},{"Start":"03:07.295 ","End":"03:08.960","Text":"As for the other one,"},{"Start":"03:08.960 ","End":"03:11.765","Text":"the limit of this is the limit of,"},{"Start":"03:11.765 ","End":"03:18.680","Text":"we can separate the root applied to 1,029 separately and 9 to the n^2 separately."},{"Start":"03:18.680 ","End":"03:22.765","Text":"The limit as n goes to infinity of this is 1,"},{"Start":"03:22.765 ","End":"03:25.500","Text":"and this is just 9,"},{"Start":"03:25.500 ","End":"03:27.525","Text":"so the limit is 9."},{"Start":"03:27.525 ","End":"03:29.974","Text":"It\u0027s the same here and here."},{"Start":"03:29.974 ","End":"03:35.370","Text":"Which means that the bit in the middle also goes to 9 by the Sandwich rule,"},{"Start":"03:35.370 ","End":"03:37.680","Text":"and this is what we had to compute,"},{"Start":"03:37.680 ","End":"03:40.900","Text":"the answer is 9 and we\u0027re done."}],"ID":31183},{"Watched":false,"Name":"Exercise 14","Duration":"2m 7s","ChapterTopicVideoID":29571,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.805","Text":"In this exercise, we\u0027re asked to compute the following limit as n goes to infinity."},{"Start":"00:06.805 ","End":"00:09.760","Text":"We\u0027re going to be using the sandwich rule."},{"Start":"00:09.760 ","End":"00:14.740","Text":"Now, this expression, if we write it out without sigma,"},{"Start":"00:14.740 ","End":"00:16.345","Text":"is equal to,"},{"Start":"00:16.345 ","End":"00:21.760","Text":"just replace k by 1 and replace k by 2 and so on until finally"},{"Start":"00:21.760 ","End":"00:27.250","Text":"we replace k by n. Note that from these n terms,"},{"Start":"00:27.250 ","End":"00:30.535","Text":"the biggest is the first one,"},{"Start":"00:30.535 ","End":"00:33.405","Text":"because the denominator is the smallest."},{"Start":"00:33.405 ","End":"00:35.425","Text":"Then the second and so on."},{"Start":"00:35.425 ","End":"00:38.025","Text":"The last one is the smallest,"},{"Start":"00:38.025 ","End":"00:40.005","Text":"and there are n of these."},{"Start":"00:40.005 ","End":"00:44.725","Text":"What we can do is say that this is bigger than"},{"Start":"00:44.725 ","End":"00:49.780","Text":"n times the smallest and less than n times the biggest."},{"Start":"00:49.780 ","End":"00:54.340","Text":"Next, we\u0027ll compute the limit of the smallest as n"},{"Start":"00:54.340 ","End":"00:59.350","Text":"goes to infinity and also of the biggest and show that it\u0027s actually the same."},{"Start":"00:59.350 ","End":"01:01.735","Text":"Then we can apply the sandwich rule."},{"Start":"01:01.735 ","End":"01:04.155","Text":"Let\u0027s start with this one."},{"Start":"01:04.155 ","End":"01:10.380","Text":"This is equal to 1 over the square root of n^2 plus 3,"},{"Start":"01:10.380 ","End":"01:14.420","Text":"and we can throw n under the square root as n^2."},{"Start":"01:14.420 ","End":"01:17.950","Text":"Here, divide top and bottom of the fraction by n^2,"},{"Start":"01:17.950 ","End":"01:19.405","Text":"and this is what we get."},{"Start":"01:19.405 ","End":"01:21.130","Text":"Well, then n goes to infinity,"},{"Start":"01:21.130 ","End":"01:23.740","Text":"3 over n^2 goes to 0."},{"Start":"01:23.740 ","End":"01:26.345","Text":"We have the square root of 1 over 1,"},{"Start":"01:26.345 ","End":"01:29.060","Text":"which is 1. That\u0027s this one."},{"Start":"01:29.060 ","End":"01:33.080","Text":"Now, this one, the limit of this, once again,"},{"Start":"01:33.080 ","End":"01:37.250","Text":"we\u0027ll put n under the square root as n^2,"},{"Start":"01:37.250 ","End":"01:39.725","Text":"divide top and bottom by n^2,"},{"Start":"01:39.725 ","End":"01:40.910","Text":"and we get this."},{"Start":"01:40.910 ","End":"01:44.360","Text":"Here we have 3 over n square root of n,"},{"Start":"01:44.360 ","End":"01:47.495","Text":"which is the square root of n over n^2."},{"Start":"01:47.495 ","End":"01:49.850","Text":"Again, this goes to infinity,"},{"Start":"01:49.850 ","End":"01:51.920","Text":"3 over this goes to 0."},{"Start":"01:51.920 ","End":"01:54.830","Text":"We have the square root of 1, which is 1."},{"Start":"01:54.830 ","End":"01:57.565","Text":"We have the same thing here and here."},{"Start":"01:57.565 ","End":"02:04.220","Text":"We can apply the sandwich rule that the bit in the middle also goes to 1."},{"Start":"02:04.220 ","End":"02:08.220","Text":"That\u0027s the answer, and we\u0027re done."}],"ID":31184},{"Watched":false,"Name":"Exercise 15","Duration":"3m 8s","ChapterTopicVideoID":29556,"CourseChapterTopicPlaylistID":294564,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.205","Text":"This is a more theoretical exercise and actually a useful result in itself."},{"Start":"00:05.205 ","End":"00:08.385","Text":"If we have a positive sequence, a_n,"},{"Start":"00:08.385 ","End":"00:14.535","Text":"which satisfies a_n plus 1 over a_n is less than or equal to q for all n,"},{"Start":"00:14.535 ","End":"00:18.420","Text":"where q for quotient is a constant less than"},{"Start":"00:18.420 ","End":"00:23.805","Text":"1 then the limit as n goes to infinity of a_n is 0."},{"Start":"00:23.805 ","End":"00:25.395","Text":"Then there\u0027s a question,"},{"Start":"00:25.395 ","End":"00:28.620","Text":"is it possible to conclude this directly from the ratio test?"},{"Start":"00:28.620 ","End":"00:30.420","Text":"First, let\u0027s prove this."},{"Start":"00:30.420 ","End":"00:32.670","Text":"We\u0027ll start by converting this to another form."},{"Start":"00:32.670 ","End":"00:34.920","Text":"Just multiply both sides by a_n."},{"Start":"00:34.920 ","End":"00:40.175","Text":"For each n, we have that a_n plus 1 is less than or equal to a_n times q."},{"Start":"00:40.175 ","End":"00:43.195","Text":"Now if we substitute different values of a_n,"},{"Start":"00:43.195 ","End":"00:44.940","Text":"let\u0027s say we\u0027ll do the first 4,"},{"Start":"00:44.940 ","End":"00:46.620","Text":"we get that n is 1,"},{"Start":"00:46.620 ","End":"00:50.670","Text":"a_n plus 1 is a_2 less than or equal to a_1q."},{"Start":"00:50.670 ","End":"00:52.830","Text":"When n equals 2,"},{"Start":"00:52.830 ","End":"00:56.600","Text":"we get that a_3 is less than or equal to a_2^3."},{"Start":"00:56.600 ","End":"00:59.450","Text":"We already have an estimate for a_2 from here."},{"Start":"00:59.450 ","End":"01:04.610","Text":"We can replace it and get a_1q times q, which is a_1q^2."},{"Start":"01:04.610 ","End":"01:05.700","Text":"When n is 3,"},{"Start":"01:05.700 ","End":"01:08.640","Text":"we get that a_4 is less than or equal to a_3q."},{"Start":"01:08.640 ","End":"01:14.285","Text":"Again, we can replace it by what it is here and get a_1q^3."},{"Start":"01:14.285 ","End":"01:20.160","Text":"When n is 4, we get the a_5 is less than or equal to a_4q,"},{"Start":"01:20.160 ","End":"01:23.920","Text":"which is less than or equal to a_1q^4."},{"Start":"01:23.920 ","End":"01:29.805","Text":"Now we can see the pattern when n is 4 here we have a 4 plus 1,"},{"Start":"01:29.805 ","End":"01:31.470","Text":"and here we have a_1q^4."},{"Start":"01:31.470 ","End":"01:38.160","Text":"In general, a_n is less than or equal to a_1q^n minus 1."},{"Start":"01:38.160 ","End":"01:40.820","Text":"We could prove this by induction,"},{"Start":"01:40.820 ","End":"01:44.885","Text":"but it\u0027s enough that we can see the pattern continues."},{"Start":"01:44.885 ","End":"01:48.455","Text":"What we have now is that a_n,"},{"Start":"01:48.455 ","End":"01:50.180","Text":"which is bigger than 0,"},{"Start":"01:50.180 ","End":"01:52.775","Text":"because we were given that a_n is a positive sequence,"},{"Start":"01:52.775 ","End":"01:58.195","Text":"is sandwiched between 0 and a_1q^n minus 1."},{"Start":"01:58.195 ","End":"01:59.510","Text":"Now when n goes to infinity,"},{"Start":"01:59.510 ","End":"02:03.200","Text":"this goes to 0 because q is a positive number"},{"Start":"02:03.200 ","End":"02:09.780","Text":"between 0 and 1 so q^n keeps getting smaller and smaller till it goes to 0 at infinity."},{"Start":"02:09.780 ","End":"02:13.565","Text":"If a_n is sandwiched between 0 and something that tends to 0,"},{"Start":"02:13.565 ","End":"02:17.315","Text":"then a_n tends to 0 by the sandwich rule."},{"Start":"02:17.315 ","End":"02:20.270","Text":"That\u0027s the first part of the exercise."},{"Start":"02:20.270 ","End":"02:25.605","Text":"The second part, we\u0027re asked whether we can deduce this from the ratio test."},{"Start":"02:25.605 ","End":"02:27.665","Text":"The answer is no, we couldn\u0027t,"},{"Start":"02:27.665 ","End":"02:35.240","Text":"because the ratio test assumes that there exists the limit of a_n plus 1 over a_n."},{"Start":"02:35.240 ","End":"02:37.490","Text":"Here we don\u0027t know that there is such a limit."},{"Start":"02:37.490 ","End":"02:41.655","Text":"Well, it assumes that either we have a limit or it\u0027s equal to infinity."},{"Start":"02:41.655 ","End":"02:46.564","Text":"Just to show you 1 phrasing of the ratio test,"},{"Start":"02:46.564 ","End":"02:49.025","Text":"which I borrowed from the Internet,"},{"Start":"02:49.025 ","End":"02:53.690","Text":"it assumes that the limit of this is L,"},{"Start":"02:53.690 ","End":"02:58.925","Text":"which is either a number less than 1 or greater than 1 or infinity."},{"Start":"02:58.925 ","End":"03:00.920","Text":"But in each case we deduce something."},{"Start":"03:00.920 ","End":"03:04.985","Text":"In our case, we don\u0027t have the existence of the limit."},{"Start":"03:04.985 ","End":"03:08.940","Text":"That concludes this exercise."}],"ID":31185}],"Thumbnail":null,"ID":294564},{"Name":"Euler Limit","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Euler Limit","Duration":"7m 34s","ChapterTopicVideoID":29666,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/29666.jpeg","UploadDate":"2022-08-01T15:50:34.0630000","DurationForVideoObject":"PT7M34S","Description":null,"MetaTitle":"Euler Limit: Video + Workbook | Proprep","MetaDescription":"Sequences - Euler Limit. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/sequences/euler-limit/vid31271","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.150","Text":"In this clip, we\u0027ll talk about a special limit due to the mathematician Euler."},{"Start":"00:07.280 ","End":"00:11.880","Text":"This is Euler\u0027s limit, it\u0027s a sequence,"},{"Start":"00:11.880 ","End":"00:18.045","Text":"1 plus 1 over n^n and we take the limit of this sequence as n goes to infinity."},{"Start":"00:18.045 ","End":"00:21.870","Text":"It turns out that this is the special number e,"},{"Start":"00:21.870 ","End":"00:27.030","Text":"the famous e which is 2.718 something."},{"Start":"00:27.030 ","End":"00:30.150","Text":"We just take this as given,"},{"Start":"00:30.150 ","End":"00:37.170","Text":"we\u0027re not going to prove this and would use this to solve certain limit problems."},{"Start":"00:37.540 ","End":"00:40.475","Text":"I want to generalize this a little bit."},{"Start":"00:40.475 ","End":"00:42.890","Text":"The letter n is just any letter,"},{"Start":"00:42.890 ","End":"00:44.855","Text":"of course, we could use another letter,"},{"Start":"00:44.855 ","End":"00:47.000","Text":"x or m or anything,"},{"Start":"00:47.000 ","End":"00:51.490","Text":"but we can also replace n by an expression."},{"Start":"00:51.490 ","End":"00:55.729","Text":"In general, we\u0027re going to have something like,"},{"Start":"00:55.729 ","End":"01:00.950","Text":"let\u0027s use a smiley to represent that expression and we\u0027ll see some examples."},{"Start":"01:00.950 ","End":"01:03.215","Text":"Something goes to infinity,"},{"Start":"01:03.215 ","End":"01:07.585","Text":"1 plus 1 over that something to the power of that same something."},{"Start":"01:07.585 ","End":"01:09.645","Text":"As an example,"},{"Start":"01:09.645 ","End":"01:14.375","Text":"if I had 2n in place of n or in place of smiley,"},{"Start":"01:14.375 ","End":"01:17.105","Text":"then I would get this."},{"Start":"01:17.105 ","End":"01:20.250","Text":"But you might say,"},{"Start":"01:20.750 ","End":"01:24.680","Text":"here, it must go to infinity and it\u0027s a smiley."},{"Start":"01:24.680 ","End":"01:31.370","Text":"So yes, technically, we should put here that 2n goes to infinity."},{"Start":"01:31.370 ","End":"01:33.170","Text":"But when n goes to infinity,"},{"Start":"01:33.170 ","End":"01:36.680","Text":"2n goes to infinity so we don\u0027t bother,"},{"Start":"01:36.680 ","End":"01:39.200","Text":"we leave it like that even though technically this"},{"Start":"01:39.200 ","End":"01:43.050","Text":"should be 2n if it\u0027s going to fit this template."},{"Start":"01:43.460 ","End":"01:48.170","Text":"Another example, instead of the smiley,"},{"Start":"01:48.170 ","End":"01:54.240","Text":"I now have n^2 plus 1 and n^2 plus 1 and when n goes to infinity,"},{"Start":"01:54.240 ","End":"01:57.980","Text":"n^2 plus 1 certainly goes to infinity, so we\u0027re okay."},{"Start":"01:57.980 ","End":"02:01.670","Text":"We have to mentally check that when n goes to infinity,"},{"Start":"02:01.670 ","End":"02:03.530","Text":"what we put here does go to infinity,"},{"Start":"02:03.530 ","End":"02:12.165","Text":"then we just leave it here as n. Another example 4n plus 10 here,"},{"Start":"02:12.165 ","End":"02:13.770","Text":"4n plus 10 here,"},{"Start":"02:13.770 ","End":"02:15.150","Text":"and when n goes to infinity,"},{"Start":"02:15.150 ","End":"02:17.955","Text":"4n plus 10 goes to infinity."},{"Start":"02:17.955 ","End":"02:20.850","Text":"If it was minus 4n,"},{"Start":"02:20.850 ","End":"02:24.695","Text":"then it wouldn\u0027t be true because then if n goes to infinity,"},{"Start":"02:24.695 ","End":"02:28.850","Text":"minus 4n will go to minus infinity and you couldn\u0027t do this."},{"Start":"02:28.850 ","End":"02:32.195","Text":"So just make sure that when n goes to infinity,"},{"Start":"02:32.195 ","End":"02:33.920","Text":"this goes to infinity,"},{"Start":"02:33.920 ","End":"02:36.030","Text":"what\u0027s here and here."},{"Start":"02:36.250 ","End":"02:44.255","Text":"Now, notice that this is of the form 1 to the power of infinity."},{"Start":"02:44.255 ","End":"02:47.675","Text":"Because if n goes to infinity,"},{"Start":"02:47.675 ","End":"02:53.239","Text":"then 1 over n is using the arithmetic of infinity,"},{"Start":"02:53.239 ","End":"02:57.065","Text":"1 over infinity, which is 0."},{"Start":"02:57.065 ","End":"03:03.240","Text":"So what we get is 1 plus 0 to"},{"Start":"03:03.240 ","End":"03:09.780","Text":"the power of infinity and that\u0027s 1 to the infinity,"},{"Start":"03:09.780 ","End":"03:13.160","Text":"which is one of those undefined cases."},{"Start":"03:13.160 ","End":"03:14.930","Text":"But in this particular case,"},{"Start":"03:14.930 ","End":"03:17.810","Text":"it is defined that its e. Now,"},{"Start":"03:17.810 ","End":"03:25.115","Text":"I\u0027m mentioning this because some of the exercises won\u0027t look like this template."},{"Start":"03:25.115 ","End":"03:28.370","Text":"But one strong indication,"},{"Start":"03:28.370 ","End":"03:30.320","Text":"at least to try Euler\u0027s limit,"},{"Start":"03:30.320 ","End":"03:35.640","Text":"is when you get something of the form 1 to the power of infinity."},{"Start":"03:35.660 ","End":"03:41.820","Text":"For example, if we have this,"},{"Start":"03:41.820 ","End":"03:45.595","Text":"then we see that when n goes to infinity,"},{"Start":"03:45.595 ","End":"03:49.335","Text":"1 over 2n is 0,"},{"Start":"03:49.335 ","End":"03:52.530","Text":"it\u0027s also 1 to the infinity."},{"Start":"03:52.530 ","End":"03:56.540","Text":"So the idea is to somehow use Euler\u0027s limit."},{"Start":"03:56.540 ","End":"04:02.134","Text":"Of course, we can\u0027t straight off say that this is e because it doesn\u0027t fit the template,"},{"Start":"04:02.134 ","End":"04:04.085","Text":"but it\u0027s a starting point."},{"Start":"04:04.085 ","End":"04:10.320","Text":"We then do an algebraic manipulation and we\u0027ll see examples."},{"Start":"04:11.540 ","End":"04:14.220","Text":"This was 1 to the infinity,"},{"Start":"04:14.220 ","End":"04:16.245","Text":"here\u0027s another example,"},{"Start":"04:16.245 ","End":"04:19.380","Text":"1 over 2n goes to 0,"},{"Start":"04:19.380 ","End":"04:21.315","Text":"so this goes to 1,"},{"Start":"04:21.315 ","End":"04:23.970","Text":"here n goes to infinity,"},{"Start":"04:23.970 ","End":"04:26.535","Text":"then n^2 plus 1 goes to infinity."},{"Start":"04:26.535 ","End":"04:29.900","Text":"Again, we have 1 to the infinity and we"},{"Start":"04:29.900 ","End":"04:34.100","Text":"would somehow use Euler\u0027s limit to help us solve this."},{"Start":"04:34.100 ","End":"04:37.490","Text":"In this example, it\u0027s not immediately obvious,"},{"Start":"04:37.490 ","End":"04:41.240","Text":"but let\u0027s look what\u0027s inside the brackets here."},{"Start":"04:41.240 ","End":"04:45.210","Text":"This is a polynomial over a polynomial"},{"Start":"04:45.210 ","End":"04:49.010","Text":"and we learned that if the highest powers are equal,"},{"Start":"04:49.010 ","End":"04:50.825","Text":"this is n^2, this is n^2,"},{"Start":"04:50.825 ","End":"04:54.800","Text":"we just divide the leading coefficient."},{"Start":"04:54.800 ","End":"04:56.210","Text":"You don\u0027t see them,"},{"Start":"04:56.210 ","End":"05:01.485","Text":"but it\u0027s like I had 1n^2 here and 1n^2 here."},{"Start":"05:01.485 ","End":"05:06.420","Text":"So this is 1 over 1,"},{"Start":"05:06.420 ","End":"05:08.055","Text":"which is 1,"},{"Start":"05:08.055 ","End":"05:10.350","Text":"and 2n plus 1 goes to infinity,"},{"Start":"05:10.350 ","End":"05:14.100","Text":"so again, we have 1 to the infinity."},{"Start":"05:14.100 ","End":"05:16.950","Text":"I\u0027m not going to solve these now,"},{"Start":"05:16.950 ","End":"05:19.320","Text":"we\u0027ll do one simple example in a moment,"},{"Start":"05:19.320 ","End":"05:25.355","Text":"but I wanted to mention that there are many solved examples of Euler\u0027s limit,"},{"Start":"05:25.355 ","End":"05:28.280","Text":"including this example,"},{"Start":"05:28.280 ","End":"05:30.590","Text":"and even more complicated."},{"Start":"05:30.590 ","End":"05:37.535","Text":"The idea is to use algebra to bring it to the form like this,"},{"Start":"05:37.535 ","End":"05:39.755","Text":"plus maybe some extras."},{"Start":"05:39.755 ","End":"05:41.540","Text":"Let\u0027s do an example."},{"Start":"05:41.540 ","End":"05:44.060","Text":"Let\u0027s take this limit,"},{"Start":"05:44.060 ","End":"05:46.750","Text":"we need to do some algebra,"},{"Start":"05:46.750 ","End":"05:51.945","Text":"and the first step would be to say,"},{"Start":"05:51.945 ","End":"05:54.510","Text":"4n here and n here,"},{"Start":"05:54.510 ","End":"05:58.710","Text":"I want 4n here so we just write,"},{"Start":"05:58.710 ","End":"06:03.080","Text":"for the moment, a 4 here to get it to be the same."},{"Start":"06:03.080 ","End":"06:04.490","Text":"Now, of course,"},{"Start":"06:04.490 ","End":"06:07.670","Text":"I can\u0027t do this because I\u0027ve changed the expression,"},{"Start":"06:07.670 ","End":"06:10.880","Text":"what I\u0027m going to do is compensate for this 4"},{"Start":"06:10.880 ","End":"06:15.365","Text":"here and I\u0027m going to use the rule of exponents."},{"Start":"06:15.365 ","End":"06:20.810","Text":"So what I\u0027m going to do is to take all this to"},{"Start":"06:20.810 ","End":"06:26.300","Text":"the power of a quarter and then if I multiply 4n times a quarter and back to n again,"},{"Start":"06:26.300 ","End":"06:28.405","Text":"so I haven\u0027t changed anything,"},{"Start":"06:28.405 ","End":"06:36.390","Text":"then I notice that what\u0027s inside the square brackets is Euler\u0027s limit,"},{"Start":"06:36.390 ","End":"06:40.430","Text":"and I could think of this as 4n goes to infinity."},{"Start":"06:40.430 ","End":"06:44.555","Text":"When n goes to infinity 4n goes to infinity so that\u0027s okay."},{"Start":"06:44.555 ","End":"06:49.020","Text":"I can put the limit inside, and like I said,"},{"Start":"06:49.020 ","End":"06:53.735","Text":"the 4n and the n doesn\u0027t matter."},{"Start":"06:53.735 ","End":"06:57.920","Text":"I wouldn\u0027t normally write 4n goes to infinity, I leave it like that."},{"Start":"06:57.920 ","End":"07:02.040","Text":"But this is Euler\u0027s limit inside the square brackets."}],"ID":31271},{"Watched":false,"Name":"Exercise 1","Duration":"2m 57s","ChapterTopicVideoID":29667,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.493","Text":"In this exercise, we have to evaluate this limit."},{"Start":"00:04.493 ","End":"00:07.110","Text":"If we try to substitute n,"},{"Start":"00:07.110 ","End":"00:12.321","Text":"this goes to 0 and this goes to infinity,"},{"Start":"00:12.321 ","End":"00:16.710","Text":"so we get an expression of the form 1 to the power of infinity."},{"Start":"00:16.710 ","End":"00:22.310","Text":"This is one of the undefined indeterminate forms."},{"Start":"00:22.310 ","End":"00:24.455","Text":"It could be anything."},{"Start":"00:24.455 ","End":"00:26.360","Text":"The result could be 1,"},{"Start":"00:26.360 ","End":"00:29.870","Text":"could be 17, could even be infinity."},{"Start":"00:29.870 ","End":"00:31.505","Text":"You just can\u0027t say."},{"Start":"00:31.505 ","End":"00:36.425","Text":"But this does look very much like a famous limit."},{"Start":"00:36.425 ","End":"00:40.110","Text":"Here it is. It\u0027s one of Euler\u0027s limits."},{"Start":"00:40.110 ","End":"00:42.800","Text":"It\u0027s usually written with x or n,"},{"Start":"00:42.800 ","End":"00:45.935","Text":"but I want to keep it generic so I put a smiley."},{"Start":"00:45.935 ","End":"00:49.820","Text":"The limit as something goes to infinity of 1 plus 1"},{"Start":"00:49.820 ","End":"00:54.020","Text":"over that something to the power of that something is e,"},{"Start":"00:54.020 ","End":"00:56.990","Text":"the famous transcendental number,"},{"Start":"00:56.990 ","End":"01:04.460","Text":"e. But here we don\u0027t quite have this situation because we have n,"},{"Start":"01:04.460 ","End":"01:06.590","Text":"but here we have 2n."},{"Start":"01:06.590 ","End":"01:08.240","Text":"We need to tweak this,"},{"Start":"01:08.240 ","End":"01:11.410","Text":"adjust it so we can use this limit."},{"Start":"01:11.410 ","End":"01:15.350","Text":"What we can do is write it as the limit."},{"Start":"01:15.350 ","End":"01:20.834","Text":"I want to get everything in terms of 2n."},{"Start":"01:20.834 ","End":"01:26.315","Text":"When n goes to infinity,"},{"Start":"01:26.315 ","End":"01:27.890","Text":"2n goes to infinity,"},{"Start":"01:27.890 ","End":"01:29.420","Text":"so I don\u0027t have to bother with that 2,"},{"Start":"01:29.420 ","End":"01:31.565","Text":"but you could put it there."},{"Start":"01:31.565 ","End":"01:35.840","Text":"Then we have 1 plus 1 over 2n."},{"Start":"01:35.840 ","End":"01:38.730","Text":"I want 2n to be the smiley,"},{"Start":"01:38.730 ","End":"01:40.571","Text":"so I\u0027ll put 2n here."},{"Start":"01:40.571 ","End":"01:44.130","Text":"But now I\u0027ve gone and changed the problem."},{"Start":"01:44.130 ","End":"01:45.690","Text":"I have to fix it."},{"Start":"01:45.690 ","End":"01:50.120","Text":"What I\u0027ll do is use the rules of exponents to take this to"},{"Start":"01:50.120 ","End":"01:54.635","Text":"the power of 1/2 and then the exponent will be right."},{"Start":"01:54.635 ","End":"02:01.805","Text":"I mean, you remember your basic algebra that (a^b)^c is a^bc."},{"Start":"02:01.805 ","End":"02:06.350","Text":"Here, 2n times 1/2 is n. Now we haven\u0027t changed anything."},{"Start":"02:06.350 ","End":"02:11.645","Text":"Now what we can do is put the limit inside the exponent."},{"Start":"02:11.645 ","End":"02:13.790","Text":"This is actually like the square root."},{"Start":"02:13.790 ","End":"02:20.640","Text":"We can get the limit as n"},{"Start":"02:20.640 ","End":"02:27.710","Text":"goes to infinity of 1 plus 1/2n^2n."},{"Start":"02:27.710 ","End":"02:31.070","Text":"All this to the power of 1/2."},{"Start":"02:31.070 ","End":"02:33.050","Text":"Because the square root function is continuous,"},{"Start":"02:33.050 ","End":"02:35.030","Text":"you can pretty much do it with all the basic functions."},{"Start":"02:35.030 ","End":"02:38.010","Text":"You can put the limit inside."},{"Start":"02:38.030 ","End":"02:44.690","Text":"This now becomes the limit with the smiley and that becomes"},{"Start":"02:44.690 ","End":"02:51.570","Text":"e. This equals e^1/2,"},{"Start":"02:51.570 ","End":"02:54.530","Text":"or if you prefer, square root of e,"},{"Start":"02:54.530 ","End":"02:56.940","Text":"and that\u0027s the answer."}],"ID":31272},{"Watched":false,"Name":"Exercise 2","Duration":"2m 7s","ChapterTopicVideoID":29668,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this exercise, we have to evaluate this limit as n goes to"},{"Start":"00:03.810 ","End":"00:07.950","Text":"infinity of this expression."},{"Start":"00:07.950 ","End":"00:16.380","Text":"The part in brackets goes to 1 as n goes to infinity because this goes to infinity,"},{"Start":"00:16.380 ","End":"00:18.195","Text":"1 over infinity is 0,"},{"Start":"00:18.195 ","End":"00:20.340","Text":"and this goes to infinity."},{"Start":"00:20.340 ","End":"00:23.100","Text":"When we have 1 to the infinity,"},{"Start":"00:23.100 ","End":"00:25.740","Text":"it\u0027s one of those indeterminate forms."},{"Start":"00:25.740 ","End":"00:27.240","Text":"You can\u0027t say what the limit is,"},{"Start":"00:27.240 ","End":"00:28.575","Text":"it could be anything."},{"Start":"00:28.575 ","End":"00:32.205","Text":"But it does look very much like the Euler limit,"},{"Start":"00:32.205 ","End":"00:38.325","Text":"which is this, though it\u0027s usually written with x or n not smiley."},{"Start":"00:38.325 ","End":"00:40.590","Text":"Now, this looks like this."},{"Start":"00:40.590 ","End":"00:44.680","Text":"I want n^2 to be smiley,"},{"Start":"00:44.680 ","End":"00:47.960","Text":"but it have to do some algebraic adjustments first because"},{"Start":"00:47.960 ","End":"00:51.200","Text":"this has to be the same as this and it isn\u0027t at the moment."},{"Start":"00:51.200 ","End":"00:58.480","Text":"We\u0027ll write it as the limit of 1 plus 1 over n^2."},{"Start":"00:58.480 ","End":"01:02.185","Text":"Now here I want there to be n^2."},{"Start":"01:02.185 ","End":"01:04.850","Text":"Of course, I can\u0027t just change the problem."},{"Start":"01:04.850 ","End":"01:09.590","Text":"I have to compensate and we\u0027ll do that by raising this to"},{"Start":"01:09.590 ","End":"01:14.715","Text":"the appropriate power but the rules of exponents,"},{"Start":"01:14.715 ","End":"01:22.400","Text":"n^2 times this question mark has to be equal to n. Though clearly the question mark is"},{"Start":"01:22.400 ","End":"01:30.225","Text":"1 over n. I go back here and write 1 over n. Also the limit,"},{"Start":"01:30.225 ","End":"01:33.660","Text":"I don\u0027t have to write n^2 goes to infinity,"},{"Start":"01:33.660 ","End":"01:36.470","Text":"although I could put 2 here because when n goes to infinity,"},{"Start":"01:36.470 ","End":"01:37.925","Text":"n^2 goes to infinity."},{"Start":"01:37.925 ","End":"01:42.885","Text":"This part\u0027s okay. Now what do we do?"},{"Start":"01:42.885 ","End":"01:48.395","Text":"Well, we have the limit of something to the power of something."},{"Start":"01:48.395 ","End":"01:52.880","Text":"Now this part here, by Euler\u0027s limit,"},{"Start":"01:52.880 ","End":"01:55.700","Text":"this goes to e,"},{"Start":"01:55.700 ","End":"02:00.570","Text":"and this part here goes to 0."},{"Start":"02:00.570 ","End":"02:04.505","Text":"This limit is e^0,"},{"Start":"02:04.505 ","End":"02:07.740","Text":"which is 1 and that\u0027s it."}],"ID":31273},{"Watched":false,"Name":"Exercise 3","Duration":"2m 54s","ChapterTopicVideoID":29669,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"This exercise, we have to evaluate this limit as n goes to Infinity."},{"Start":"00:04.635 ","End":"00:08.940","Text":"If we try to put n equals Infinity,"},{"Start":"00:08.940 ","End":"00:12.690","Text":"well, it\u0027s not clear what this becomes."},{"Start":"00:12.690 ","End":"00:15.045","Text":"This goes to Infinity but what about this?"},{"Start":"00:15.045 ","End":"00:16.980","Text":"Well, I claim this goes to 1."},{"Start":"00:16.980 ","End":"00:18.765","Text":"There\u0027s many ways of seeing this,"},{"Start":"00:18.765 ","End":"00:22.733","Text":"but I\u0027d like to just rewrite this as,"},{"Start":"00:22.733 ","End":"00:24.960","Text":"well, I\u0027ll do it here,"},{"Start":"00:24.960 ","End":"00:28.230","Text":"limit as n goes to Infinity,"},{"Start":"00:28.230 ","End":"00:32.870","Text":"1 plus (2 over n)^n,"},{"Start":"00:32.870 ","End":"00:36.410","Text":"now we can see that this goes to 1 because as n goes to Infinity,"},{"Start":"00:36.410 ","End":"00:37.970","Text":"2 over n goes to 0."},{"Start":"00:37.970 ","End":"00:45.330","Text":"This is the limit of the form 1 to the Infinity and that\u0027s an indeterminate form,"},{"Start":"00:45.330 ","End":"00:47.000","Text":"the answer could be anything."},{"Start":"00:47.000 ","End":"00:52.250","Text":"But this is similar to a famous limit due to Euler,"},{"Start":"00:52.250 ","End":"00:56.405","Text":"and I like to write it with a smiley instead of an n or an x."},{"Start":"00:56.405 ","End":"01:00.680","Text":"Now, this limit looks quite a lot like this limit if we"},{"Start":"01:00.680 ","End":"01:05.115","Text":"take the smiley as n. But there\u0027s a 2 here."},{"Start":"01:05.115 ","End":"01:06.915","Text":"What to do about that?"},{"Start":"01:06.915 ","End":"01:11.775","Text":"Maybe we should get it to be 1 over something and then see."},{"Start":"01:11.775 ","End":"01:18.975","Text":"Well, 2 over n is the same as 1 over n over 2."},{"Start":"01:18.975 ","End":"01:21.075","Text":"Because n over 2 is the reciprocal,"},{"Start":"01:21.075 ","End":"01:22.740","Text":"so I\u0027m going to write it this way,"},{"Start":"01:22.740 ","End":"01:26.200","Text":"1 plus 1 over n over 2."},{"Start":"01:26.200 ","End":"01:30.200","Text":"Now, here doesn\u0027t really need any adjusting."},{"Start":"01:30.200 ","End":"01:36.500","Text":"You could write n over 2 here but it\u0027s not necessary because when n goes to Infinity,"},{"Start":"01:36.500 ","End":"01:39.335","Text":"n over 2 goes to Infinity, whatever."},{"Start":"01:39.335 ","End":"01:45.235","Text":"The thing is here if we\u0027re going to let the smiley be n over 2,"},{"Start":"01:45.235 ","End":"01:47.925","Text":"we have to have n over 2 here,"},{"Start":"01:47.925 ","End":"01:50.960","Text":"but here we have n so we\u0027ve changed the problem,"},{"Start":"01:50.960 ","End":"01:52.640","Text":"so we have to compensate,"},{"Start":"01:52.640 ","End":"01:57.695","Text":"fix it, and the way to fix it is to take this to the power of something."},{"Start":"01:57.695 ","End":"01:59.480","Text":"What is that something?"},{"Start":"01:59.480 ","End":"02:07.100","Text":"n over 2 times this has to equal n because here we had n. Well,"},{"Start":"02:07.100 ","End":"02:09.380","Text":"it\u0027s clear that this must be 2,"},{"Start":"02:09.380 ","End":"02:12.170","Text":"so we\u0027ll write a 2 here."},{"Start":"02:12.170 ","End":"02:15.730","Text":"Now, we can process this."},{"Start":"02:15.730 ","End":"02:19.400","Text":"First of all, if I square each term,"},{"Start":"02:19.400 ","End":"02:21.680","Text":"it\u0027s like taking the limit and then squaring it."},{"Start":"02:21.680 ","End":"02:27.020","Text":"I can say this is limit as n goes to Infinity,"},{"Start":"02:27.020 ","End":"02:34.270","Text":"1 plus 1 over n over 2^n over 2,"},{"Start":"02:34.270 ","End":"02:36.975","Text":"and all this squared."},{"Start":"02:36.975 ","End":"02:42.280","Text":"Now, this is like this with the smiley being n over 2."},{"Start":"02:45.140 ","End":"02:49.080","Text":"This thing goes to e,"},{"Start":"02:49.080 ","End":"02:50.610","Text":"and the 2 stays here,"},{"Start":"02:50.610 ","End":"02:54.490","Text":"so the answer is e^2. That\u0027s it."}],"ID":31274},{"Watched":false,"Name":"Exercise 4","Duration":"2m 48s","ChapterTopicVideoID":29670,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.070","Text":"In this exercise, we have to compute this limit."},{"Start":"00:04.070 ","End":"00:09.130","Text":"If we just try naively to put n equals infinity,"},{"Start":"00:09.130 ","End":"00:14.950","Text":"then this part is 1 because 1 over infinity is"},{"Start":"00:14.950 ","End":"00:20.470","Text":"0 and this part is infinity."},{"Start":"00:20.470 ","End":"00:24.520","Text":"When we have a 1 to the power of infinity situation,"},{"Start":"00:24.520 ","End":"00:29.170","Text":"it\u0027s indeterminate, we can\u0027t say what the limit will be."},{"Start":"00:29.170 ","End":"00:35.120","Text":"But this does look a lot like Euler\u0027s limit."},{"Start":"00:35.900 ","End":"00:38.745","Text":"This is not your usual 1,"},{"Start":"00:38.745 ","End":"00:42.625","Text":"it\u0027s a variant because we have a minus here."},{"Start":"00:42.625 ","End":"00:45.124","Text":"There\u0027s a variant that when you have a minus,"},{"Start":"00:45.124 ","End":"00:46.760","Text":"the limit is 1 over e,"},{"Start":"00:46.760 ","End":"00:47.960","Text":"whereas if it was a plus,"},{"Start":"00:47.960 ","End":"00:56.085","Text":"it would just be e. Now I want my smiley to be n^2."},{"Start":"00:56.085 ","End":"00:57.600","Text":"We have the limit."},{"Start":"00:57.600 ","End":"01:00.525","Text":"Now n goes to infinity,"},{"Start":"01:00.525 ","End":"01:02.682","Text":"it\u0027s the same as n^2 goes to infinity,"},{"Start":"01:02.682 ","End":"01:04.600","Text":"you don\u0027t even have to write that in."},{"Start":"01:04.600 ","End":"01:09.605","Text":"Then we have 1 minus 1 over n^2."},{"Start":"01:09.605 ","End":"01:13.250","Text":"Now to use this, I would have to have n^2 here."},{"Start":"01:13.250 ","End":"01:15.460","Text":"But I can\u0027t change the problem,"},{"Start":"01:15.460 ","End":"01:18.380","Text":"I have to make it so that it is n^2 minus 1."},{"Start":"01:18.380 ","End":"01:20.930","Text":"I\u0027ll compensate by raising it to"},{"Start":"01:20.930 ","End":"01:26.390","Text":"the appropriate power so that this times this will be n^2 minus 1."},{"Start":"01:26.390 ","End":"01:29.245","Text":"We have n^2 times question mark."},{"Start":"01:29.245 ","End":"01:31.460","Text":"Of course, we are using the rules of exponents,"},{"Start":"01:31.460 ","End":"01:34.880","Text":"I should have said, it\u0027s obvious."},{"Start":"01:34.880 ","End":"01:37.805","Text":"a to the b to the c and a to the power of bc."},{"Start":"01:37.805 ","End":"01:44.805","Text":"This times this has got to equal n^2 minus 1."},{"Start":"01:44.805 ","End":"01:49.630","Text":"Question mark is n^2 minus 1 over n^2."},{"Start":"01:52.940 ","End":"02:00.590","Text":"I find it more convenient to write as 1 minus 1 over n^2."},{"Start":"02:00.590 ","End":"02:06.650","Text":"It\u0027ll be easier when we take the limit as n goes to infinity late, soon."},{"Start":"02:06.650 ","End":"02:14.430","Text":"This is 1 minus 1 over n^2."},{"Start":"02:14.430 ","End":"02:18.560","Text":"Now, no problem really,"},{"Start":"02:18.560 ","End":"02:21.095","Text":"the limit of something to the power of something."},{"Start":"02:21.095 ","End":"02:30.105","Text":"This part here goes to 1 over e by using this rule with the smiley being n^2."},{"Start":"02:30.105 ","End":"02:36.095","Text":"This part here goes to 1 because 1 over n^2 goes to 0."},{"Start":"02:36.095 ","End":"02:42.770","Text":"What we get is just 1 over e to the power of 1,"},{"Start":"02:42.770 ","End":"02:47.160","Text":"which is 1 over e. That\u0027s the answer."}],"ID":31275},{"Watched":false,"Name":"Exercise 5","Duration":"4m 3s","ChapterTopicVideoID":29671,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this exercise, we have to evaluate the limit as n goes to infinity of"},{"Start":"00:04.680 ","End":"00:09.900","Text":"this rational expression to the power of n. Now,"},{"Start":"00:09.900 ","End":"00:15.855","Text":"the bit in brackets goes to 1 as n goes to infinity."},{"Start":"00:15.855 ","End":"00:18.225","Text":"You could, for example,"},{"Start":"00:18.225 ","End":"00:21.270","Text":"divide top and bottom by 2n,"},{"Start":"00:21.270 ","End":"00:23.280","Text":"and then you\u0027d get 1 plus something,"},{"Start":"00:23.280 ","End":"00:25.530","Text":"1 minus something that go to 0,"},{"Start":"00:25.530 ","End":"00:27.840","Text":"or you could say that with"},{"Start":"00:27.840 ","End":"00:32.400","Text":"rational expression when the polynomials on top and bottom have the same degree,"},{"Start":"00:32.400 ","End":"00:34.800","Text":"then we just look at the leading coefficients,"},{"Start":"00:34.800 ","End":"00:36.625","Text":"2 over 2 is 1."},{"Start":"00:36.625 ","End":"00:39.090","Text":"An n goes to infinity."},{"Start":"00:39.090 ","End":"00:40.815","Text":"This is a limit of the form,"},{"Start":"00:40.815 ","End":"00:42.605","Text":"1 goes to infinity,"},{"Start":"00:42.605 ","End":"00:48.745","Text":"which is 1 of the indeterminate undefined forms could be anything."},{"Start":"00:48.745 ","End":"01:00.040","Text":"What we\u0027re going to do is use Euler\u0027s Limit, which is this."},{"Start":"01:00.410 ","End":"01:04.960","Text":"This doesn\u0027t look very much like this."},{"Start":"01:05.140 ","End":"01:09.455","Text":"We have to do some algebraic manipulation."},{"Start":"01:09.455 ","End":"01:12.515","Text":"I\u0027ll just work on the bit in brackets,"},{"Start":"01:12.515 ","End":"01:20.390","Text":"2n plus 3 over 2n minus 3 is equal to,"},{"Start":"01:20.390 ","End":"01:22.850","Text":"now, if I wanted to get a 1,"},{"Start":"01:22.850 ","End":"01:27.880","Text":"I could write this as 2n minus 3."},{"Start":"01:27.880 ","End":"01:29.925","Text":"Then to compensate,"},{"Start":"01:29.925 ","End":"01:34.510","Text":"add 6 over 2n minus 3."},{"Start":"01:34.510 ","End":"01:37.580","Text":"I\u0027ll even put this in brackets to make it very clear."},{"Start":"01:37.580 ","End":"01:40.430","Text":"This will equal this over this is 1."},{"Start":"01:40.430 ","End":"01:46.780","Text":"Here, 6 over 2n minus 3."},{"Start":"01:48.620 ","End":"01:53.570","Text":"We\u0027re still not there, because here I have a 1 and here I have a 6."},{"Start":"01:53.570 ","End":"01:56.390","Text":"If I just divide top and bottom by 6,"},{"Start":"01:56.390 ","End":"01:59.905","Text":"I\u0027ll get 1 plus 1 over,"},{"Start":"01:59.905 ","End":"02:04.890","Text":"and then 2n minus 3 over 6."},{"Start":"02:04.890 ","End":"02:07.750","Text":"Now back to the limit,"},{"Start":"02:07.750 ","End":"02:10.850","Text":"which we\u0027re going to write in this form,"},{"Start":"02:10.850 ","End":"02:18.360","Text":"1 plus 1 over 2n minus 3 over 6."},{"Start":"02:18.360 ","End":"02:23.765","Text":"We have to rewrite the exponent because to use this formula,"},{"Start":"02:23.765 ","End":"02:30.910","Text":"we also have to have 2n minus 3 over 6 here."},{"Start":"02:30.910 ","End":"02:35.105","Text":"We don\u0027t have to worry about the limit because when n goes to infinity,"},{"Start":"02:35.105 ","End":"02:37.505","Text":"2n minus 3 over 6 goes to infinity."},{"Start":"02:37.505 ","End":"02:40.470","Text":"We don\u0027t need to replace this."},{"Start":"02:40.660 ","End":"02:44.585","Text":"Now, I can\u0027t just go ahead and change the problem."},{"Start":"02:44.585 ","End":"02:47.880","Text":"The original coefficient was n,"},{"Start":"02:47.880 ","End":"02:52.005","Text":"and I have 2n minus 3 over 6."},{"Start":"02:52.005 ","End":"02:54.180","Text":"I have to compensate."},{"Start":"02:54.180 ","End":"03:02.915","Text":"The way I do that is by taking this and raising it to some power to make it right,"},{"Start":"03:02.915 ","End":"03:04.820","Text":"that this is n. In other words,"},{"Start":"03:04.820 ","End":"03:06.620","Text":"because of the rules of exponents,"},{"Start":"03:06.620 ","End":"03:14.955","Text":"I want 2n minus 3 over 6 times question mark to equal n. Question mark equals,"},{"Start":"03:14.955 ","End":"03:16.830","Text":"see I bring it over to the other side,"},{"Start":"03:16.830 ","End":"03:20.205","Text":"6n over 2n minus 3."},{"Start":"03:20.205 ","End":"03:27.165","Text":"6n over 2n plus 3."},{"Start":"03:27.165 ","End":"03:29.410","Text":"Now it\u0027s okay."},{"Start":"03:29.410 ","End":"03:33.755","Text":"Now, if we look at this, we have the part in the brackets and we have the exponent."},{"Start":"03:33.755 ","End":"03:37.700","Text":"This time, we\u0027re not going to get 1 to the infinity."},{"Start":"03:37.700 ","End":"03:43.140","Text":"This thing is going to go to E by Euler\u0027s limit."},{"Start":"03:43.140 ","End":"03:46.810","Text":"This is going to go to 3."},{"Start":"03:46.810 ","End":"03:50.375","Text":"Also rational expression leading coefficients,"},{"Start":"03:50.375 ","End":"03:55.210","Text":"6 over 2 is 3."},{"Start":"03:55.850 ","End":"04:03.840","Text":"The answer comes out to be e to the power of 3. That\u0027s it."}],"ID":31276},{"Watched":false,"Name":"Exercise 6","Duration":"4m 58s","ChapterTopicVideoID":29672,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have to evaluate the limit as n goes to"},{"Start":"00:03.570 ","End":"00:09.060","Text":"infinity of this rational expression to the power of 4n^2."},{"Start":"00:09.060 ","End":"00:16.635","Text":"Now, if we just try to do it by taking the limit of this and the limit of this,"},{"Start":"00:16.635 ","End":"00:20.415","Text":"you\u0027ll see that this tends to 1,"},{"Start":"00:20.415 ","End":"00:26.070","Text":"because it\u0027s a rational expression with the same degree on top and on bottom."},{"Start":"00:26.070 ","End":"00:31.930","Text":"We just divide the leading coefficients,1 over 1 is 1,"},{"Start":"00:33.220 ","End":"00:37.100","Text":"and this goes to infinity."},{"Start":"00:37.100 ","End":"00:42.365","Text":"We have a 1 to the power of infinity form and this is indeterminate."},{"Start":"00:42.365 ","End":"00:45.035","Text":"You can\u0027t say what the limit could be."},{"Start":"00:45.035 ","End":"00:47.150","Text":"We have to use some tricks,"},{"Start":"00:47.150 ","End":"00:52.240","Text":"and I\u0027m going to reduce this to the Euler limit."},{"Start":"00:52.240 ","End":"00:55.570","Text":"This is the usual one,"},{"Start":"00:55.570 ","End":"01:01.145","Text":"but there\u0027s a variant with a minus instead of a plus, and then the limit,"},{"Start":"01:01.145 ","End":"01:05.450","Text":"instead of coming out e comes out 1 over e. Also, as you\u0027ve noted,"},{"Start":"01:05.450 ","End":"01:09.240","Text":"I use a smiley instead of a letter x or n,"},{"Start":"01:09.240 ","End":"01:13.860","Text":"and then I can substitute any expression instead of smiley."},{"Start":"01:15.010 ","End":"01:22.295","Text":"We have to do a lot or some algebraic manipulation to get this to look like this."},{"Start":"01:22.295 ","End":"01:24.470","Text":"One thing we can do,"},{"Start":"01:24.470 ","End":"01:27.485","Text":"we want it to be 1 plus 1 over something."},{"Start":"01:27.485 ","End":"01:31.115","Text":"Now notice that the numerator and denominator are almost the same."},{"Start":"01:31.115 ","End":"01:33.845","Text":"If I just take this expression,"},{"Start":"01:33.845 ","End":"01:41.230","Text":"n^2 plus n plus 1 over n^2 plus n plus 4."},{"Start":"01:41.230 ","End":"01:44.980","Text":"What I can do is instead of the plus 1,"},{"Start":"01:44.980 ","End":"01:47.190","Text":"I can plus 4,"},{"Start":"01:47.190 ","End":"01:50.105","Text":"but then minus 3 to compensate,"},{"Start":"01:50.105 ","End":"01:52.640","Text":"I\u0027ll put this in the bracket for emphasis."},{"Start":"01:52.640 ","End":"01:55.040","Text":"Now you see that this over, this is 1,"},{"Start":"01:55.040 ","End":"02:02.880","Text":"this is 1 minus 3 over n^2 plus n plus 4."},{"Start":"02:02.880 ","End":"02:05.945","Text":"Getting closer, but we want 1 over,"},{"Start":"02:05.945 ","End":"02:09.000","Text":"and it\u0027s clear that we\u0027re going to use this one."},{"Start":"02:09.320 ","End":"02:12.975","Text":"We just have to divide top and bottom by 3,"},{"Start":"02:12.975 ","End":"02:23.730","Text":"1 minus 1 over n^2 plus n plus 4 over 3."},{"Start":"02:23.730 ","End":"02:30.010","Text":"Now this is going to be the smiley here."},{"Start":"02:30.020 ","End":"02:36.330","Text":"I can rewrite this limit as the limit of 1"},{"Start":"02:36.330 ","End":"02:44.185","Text":"minus 1 over n^2 plus n plus 4 over 3,"},{"Start":"02:44.185 ","End":"02:49.290","Text":"and then here I have 4n^2."},{"Start":"02:49.290 ","End":"02:52.850","Text":"Now I also want the same expression here."},{"Start":"02:52.850 ","End":"03:00.805","Text":"I\u0027ll start by writing out n^2 plus n plus 4 over 3."},{"Start":"03:00.805 ","End":"03:03.620","Text":"Now obviously that\u0027s not legal to just change it,"},{"Start":"03:03.620 ","End":"03:07.955","Text":"but I\u0027ll compensate by taking this,"},{"Start":"03:07.955 ","End":"03:14.400","Text":"and raising it to some power that will make it go back to 4n^2."},{"Start":"03:14.420 ","End":"03:20.420","Text":"In a second, I\u0027ll just note that we can still keep the n goes to infinity,"},{"Start":"03:20.420 ","End":"03:22.010","Text":"because when n goes to infinity,"},{"Start":"03:22.010 ","End":"03:24.985","Text":"this expression also goes to infinity."},{"Start":"03:24.985 ","End":"03:27.450","Text":"Now, the matter of this."},{"Start":"03:27.450 ","End":"03:31.010","Text":"Well, the power of a power you multiply the exponents."},{"Start":"03:31.010 ","End":"03:41.500","Text":"We have n^2 plus n plus 4 over 3 times question mark equals 4n^2."},{"Start":"03:41.900 ","End":"03:44.400","Text":"The question mark equals,"},{"Start":"03:44.400 ","End":"03:47.655","Text":"I\u0027ll bring this fraction to the other side and invert it."},{"Start":"03:47.655 ","End":"03:55.905","Text":"We will get 12n^2 over n^2 plus n plus 4."},{"Start":"03:55.905 ","End":"03:59.050","Text":"Now I write it here,"},{"Start":"03:59.720 ","End":"04:07.245","Text":"(12n^2)^2 over n^2 plus n plus 4."},{"Start":"04:07.245 ","End":"04:11.045","Text":"Now we have the limit of something to the power of something."},{"Start":"04:11.045 ","End":"04:17.160","Text":"This part here, this thing goes to 1 over e,"},{"Start":"04:18.620 ","End":"04:23.175","Text":"because this is this limit,"},{"Start":"04:23.175 ","End":"04:29.875","Text":"and this is a rational expression with the same degree on top and along the bottom,"},{"Start":"04:29.875 ","End":"04:32.970","Text":"it goes to 12 over 1."},{"Start":"04:32.970 ","End":"04:34.530","Text":"You divide the leading coefficients,"},{"Start":"04:34.530 ","End":"04:38.950","Text":"this is n^2, which is 12."},{"Start":"04:39.850 ","End":"04:48.535","Text":"Our limit becomes 1 over e^12,"},{"Start":"04:48.535 ","End":"04:53.360","Text":"and now write it alternatively as e^minus 12."},{"Start":"04:53.360 ","End":"04:55.475","Text":"1 over e is e^minus 1,"},{"Start":"04:55.475 ","End":"04:58.199","Text":"and that\u0027s the answer."}],"ID":31277},{"Watched":false,"Name":"Exercise 7","Duration":"5m 51s","ChapterTopicVideoID":29673,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have to compute a limit as n goes to"},{"Start":"00:03.060 ","End":"00:07.860","Text":"infinity of this expression to the power of this."},{"Start":"00:07.860 ","End":"00:14.010","Text":"Now, if we take the limit of this part separately,"},{"Start":"00:14.010 ","End":"00:20.520","Text":"then it comes out that this goes to 1 and this goes to infinity."},{"Start":"00:20.520 ","End":"00:22.500","Text":"The 10n goes to infinity is clear."},{"Start":"00:22.500 ","End":"00:24.540","Text":"Why does this go to 1?"},{"Start":"00:24.540 ","End":"00:26.970","Text":"Because it\u0027s a rational expression of the same degree."},{"Start":"00:26.970 ","End":"00:29.399","Text":"They\u0027re both quadratic and in that case,"},{"Start":"00:29.399 ","End":"00:31.470","Text":"we take the leading coefficients,"},{"Start":"00:31.470 ","End":"00:33.870","Text":"1 over 1 is 1."},{"Start":"00:33.870 ","End":"00:39.990","Text":"Now, 1 to the infinity is one of those undefined indeterminate forms, it\u0027s useless."},{"Start":"00:39.990 ","End":"00:42.585","Text":"We have to do something else."},{"Start":"00:42.585 ","End":"00:48.474","Text":"What I\u0027m going to do here is use Euler\u0027s limit."},{"Start":"00:48.474 ","End":"00:50.815","Text":"This is Euler\u0027s limit."},{"Start":"00:50.815 ","End":"00:55.390","Text":"I like to use a smiley instead of n or x because usually,"},{"Start":"00:55.390 ","End":"00:57.370","Text":"a smiley is an expression."},{"Start":"00:57.370 ","End":"01:00.010","Text":"Now, this doesn\u0027t look very much like this,"},{"Start":"01:00.010 ","End":"01:03.670","Text":"so we need to do a bit of algebra to bring it to this form."},{"Start":"01:03.670 ","End":"01:07.540","Text":"I\u0027ll start with the rational expression first,"},{"Start":"01:07.540 ","End":"01:15.350","Text":"the n squared plus 4n plus 1 over n squared plus n plus 2."},{"Start":"01:15.350 ","End":"01:20.470","Text":"What I\u0027m going to do is equivalent to a long division with remainder,"},{"Start":"01:20.470 ","End":"01:22.570","Text":"but I\u0027m not going to do it in that form."},{"Start":"01:22.570 ","End":"01:25.130","Text":"I\u0027m going to say, okay,"},{"Start":"01:25.130 ","End":"01:29.140","Text":"on the denominator, I have n squared plus n plus 2."},{"Start":"01:29.140 ","End":"01:30.895","Text":"On the numerator,"},{"Start":"01:30.895 ","End":"01:36.840","Text":"I\u0027d like to have n squared plus n plus 2 also then this over, this is 1."},{"Start":"01:36.840 ","End":"01:40.525","Text":"But obviously, I can\u0027t just change it so what I do then is compensate."},{"Start":"01:40.525 ","End":"01:45.220","Text":"I have here plus n, so I have to add another 3n to make it 4n."},{"Start":"01:45.220 ","End":"01:46.630","Text":"Here I have plus 2,"},{"Start":"01:46.630 ","End":"01:48.395","Text":"so I have to subtract 1,"},{"Start":"01:48.395 ","End":"01:51.270","Text":"now we put brackets here and here."},{"Start":"01:51.270 ","End":"01:53.820","Text":"Now, this breaks up, this over,"},{"Start":"01:53.820 ","End":"01:59.220","Text":"this is 1 and now we have 3n minus"},{"Start":"01:59.220 ","End":"02:06.310","Text":"1 over n squared plus n plus 2."},{"Start":"02:06.830 ","End":"02:11.440","Text":"I\u0027d like to remark, this may look as complicated as this."},{"Start":"02:11.440 ","End":"02:14.785","Text":"There\u0027s a big difference besides being in this form."},{"Start":"02:14.785 ","End":"02:17.365","Text":"This thing doesn\u0027t go to 0,"},{"Start":"02:17.365 ","End":"02:23.155","Text":"but this does go to 0 because the degree on top is lower than the degree on the bottom."},{"Start":"02:23.155 ","End":"02:26.665","Text":"Anyway, we\u0027re still not in this form."},{"Start":"02:26.665 ","End":"02:30.535","Text":"I need a 1 on top also, so no problem."},{"Start":"02:30.535 ","End":"02:34.105","Text":"We\u0027ll just put this as,"},{"Start":"02:34.105 ","End":"02:35.695","Text":"if I flip it,"},{"Start":"02:35.695 ","End":"02:40.700","Text":"I can make it 1 over the upside down fraction,"},{"Start":"02:40.700 ","End":"02:50.930","Text":"n squared plus n plus 2 over 3n minus 1."},{"Start":"02:53.240 ","End":"02:56.125","Text":"Now, getting back to this,"},{"Start":"02:56.125 ","End":"02:58.330","Text":"we want the limit."},{"Start":"02:58.330 ","End":"03:01.840","Text":"As n goes to infinity,"},{"Start":"03:01.840 ","End":"03:07.960","Text":"I should remark that this thing does go to infinity when n goes to infinity."},{"Start":"03:07.960 ","End":"03:10.315","Text":"I can leave it as n goes to infinity."},{"Start":"03:10.315 ","End":"03:17.095","Text":"This goes to infinity because it behaves like n squared over 3n,"},{"Start":"03:17.095 ","End":"03:20.620","Text":"which is like 1/3n."},{"Start":"03:20.620 ","End":"03:22.780","Text":"The degree here is higher,"},{"Start":"03:22.780 ","End":"03:25.750","Text":"so it goes to plus or minus infinity,"},{"Start":"03:25.750 ","End":"03:28.160","Text":"in this case, is plus infinity."},{"Start":"03:28.250 ","End":"03:31.330","Text":"It\u0027s basically like I said,"},{"Start":"03:31.330 ","End":"03:36.055","Text":"it\u0027s n squared over 3n and that does go to infinity."},{"Start":"03:36.055 ","End":"03:40.610","Text":"Now, I also want to put the smiley up here,"},{"Start":"03:40.610 ","End":"03:42.199","Text":"which is this expression,"},{"Start":"03:42.199 ","End":"03:50.370","Text":"so I\u0027m going to write n squared plus n plus 2 over 3n minus 1."},{"Start":"03:51.200 ","End":"04:02.340","Text":"Now, I\u0027ve changed the problem because originally I had 10n and now I have this mess."},{"Start":"04:02.380 ","End":"04:07.310","Text":"I\u0027m getting ahead of myself. Let me copy that down here. Yeah, here we are."},{"Start":"04:07.310 ","End":"04:16.430","Text":"I was saying compensate by raising this to some power that will make it the same as 10n."},{"Start":"04:16.430 ","End":"04:18.785","Text":"Now an exponent of an exponent,"},{"Start":"04:18.785 ","End":"04:21.215","Text":"you multiply the exponents."},{"Start":"04:21.215 ","End":"04:31.790","Text":"What we want is n squared plus n plus 2 over 3n minus 1 times what is equal to 10n."},{"Start":"04:31.790 ","End":"04:36.380","Text":"That what is equal to 10n divided by this,"},{"Start":"04:36.380 ","End":"04:39.410","Text":"which means multiply by the flip of that."},{"Start":"04:39.410 ","End":"04:41.210","Text":"I have 10n,"},{"Start":"04:41.210 ","End":"04:48.870","Text":"3n minus 1 over n squared plus n plus 2."},{"Start":"04:50.150 ","End":"04:52.745","Text":"I\u0027ll write that here."},{"Start":"04:52.745 ","End":"04:54.365","Text":"I\u0027ll open the brackets."},{"Start":"04:54.365 ","End":"04:56.555","Text":"It\u0027s not 10,"},{"Start":"04:56.555 ","End":"05:01.800","Text":"30n squared minus 10n"},{"Start":"05:02.030 ","End":"05:07.450","Text":"over n squared plus n plus 2."},{"Start":"05:07.450 ","End":"05:12.065","Text":"We\u0027re in good shape now because"},{"Start":"05:12.065 ","End":"05:18.750","Text":"the limit of this part is just scrolled off."},{"Start":"05:21.230 ","End":"05:23.970","Text":"Well, remember you can see it there,"},{"Start":"05:23.970 ","End":"05:27.615","Text":"it\u0027s e. This limit,"},{"Start":"05:27.615 ","End":"05:33.050","Text":"because again, it\u0027s a rational expression with the same degree quadratic over quadratic,"},{"Start":"05:33.050 ","End":"05:35.515","Text":"so it\u0027s 30 over 1."},{"Start":"05:35.515 ","End":"05:39.600","Text":"This thing goes to 30."},{"Start":"05:39.600 ","End":"05:44.145","Text":"Goes to e, goes to 30."},{"Start":"05:44.145 ","End":"05:51.810","Text":"Our answer is e to the power of 30 and we\u0027re done."}],"ID":31278},{"Watched":false,"Name":"Exercise 8","Duration":"7m 26s","ChapterTopicVideoID":29674,"CourseChapterTopicPlaylistID":294583,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.180","Text":"Here we have to evaluate the limit as n goes to infinity of 1 plus tangent of 1 over"},{"Start":"00:06.180 ","End":"00:11.520","Text":"n to the power of n. If you just put n equals infinity,"},{"Start":"00:11.520 ","End":"00:13.500","Text":"1 over n is 0,"},{"Start":"00:13.500 ","End":"00:15.615","Text":"tangent of 0 is 0,"},{"Start":"00:15.615 ","End":"00:21.300","Text":"and so this is of the form 1 to the power of infinity which is no good to us."},{"Start":"00:21.300 ","End":"00:23.730","Text":"It\u0027s indeterminate, could be anything,"},{"Start":"00:23.730 ","End":"00:27.765","Text":"but this looks very much like Euler\u0027s limit."},{"Start":"00:27.765 ","End":"00:33.420","Text":"This is my Euler\u0027s limit and I like to use a smiley instead of a letter."},{"Start":"00:33.420 ","End":"00:37.350","Text":"It looks very much like this and if we didn\u0027t have the tangent here,"},{"Start":"00:37.350 ","End":"00:41.780","Text":"we would just say n is the smiley and the answer would be e,"},{"Start":"00:41.780 ","End":"00:46.340","Text":"but there\u0027s a tangent there so we\u0027re going to have to use some tricks."},{"Start":"00:46.340 ","End":"00:54.275","Text":"What we\u0027ll do is we\u0027ll get this to be in this form as follows."},{"Start":"00:54.275 ","End":"01:01.940","Text":"1 plus tangent of 1 over n is equal to 1"},{"Start":"01:01.940 ","End":"01:11.420","Text":"plus 1 over cotangent of 1 over n. The reason is,"},{"Start":"01:11.420 ","End":"01:20.190","Text":"is that the inverse"},{"Start":"01:20.190 ","End":"01:24.615","Text":"of cotangent is 1 over cotangent of Alpha."},{"Start":"01:24.615 ","End":"01:27.241","Text":"More familiar, the other way round,"},{"Start":"01:27.241 ","End":"01:30.560","Text":"cotangent Alpha is 1 over tangent Alpha,"},{"Start":"01:30.560 ","End":"01:32.650","Text":"but of course, you can reverse it."},{"Start":"01:32.650 ","End":"01:38.345","Text":"If we put tangent is 1 over cotangent then it looks very much like this."},{"Start":"01:38.345 ","End":"01:40.985","Text":"Now, back to the limit."},{"Start":"01:40.985 ","End":"01:51.000","Text":"We have the limit as n goes to infinity of 1 plus"},{"Start":"01:51.000 ","End":"01:55.560","Text":"1 over cotangent 1 over"},{"Start":"01:55.560 ","End":"02:03.870","Text":"n. Before we put the exponent,"},{"Start":"02:03.870 ","End":"02:08.555","Text":"let\u0027s just see if this thing goes to infinity or not."},{"Start":"02:08.555 ","End":"02:12.305","Text":"Cotangent 1 over n has to go to infinity,"},{"Start":"02:12.305 ","End":"02:21.345","Text":"but if I substitute t equals 1 over n,"},{"Start":"02:21.345 ","End":"02:24.755","Text":"then when n goes to infinity,"},{"Start":"02:24.755 ","End":"02:31.470","Text":"t goes to 0 from above because 1 over n is positive,"},{"Start":"02:31.470 ","End":"02:37.580","Text":"and so we get cotangent 1 over n,"},{"Start":"02:37.580 ","End":"02:40.690","Text":"the limit as n goes to infinity, I\u0027ll write it out,"},{"Start":"02:40.690 ","End":"02:49.100","Text":"limit as n goes to infinity of cotangent 1 over n is equal to the limit as t"},{"Start":"02:49.100 ","End":"02:58.820","Text":"goes to 0 from the right of cotangent t. Now,"},{"Start":"02:58.820 ","End":"03:04.910","Text":"this limit is infinity because, well,"},{"Start":"03:04.910 ","End":"03:12.800","Text":"we could say it\u0027s 1 over the limit as t goes to"},{"Start":"03:12.800 ","End":"03:21.020","Text":"0 from above of tangent t and tangent t when t goes to 0 also goes to 0,"},{"Start":"03:21.020 ","End":"03:24.240","Text":"but if t is positive, this is positive."},{"Start":"03:25.100 ","End":"03:32.740","Text":"This is 1 over 0 plus which is infinity."},{"Start":"03:32.870 ","End":"03:35.270","Text":"That was important,"},{"Start":"03:35.270 ","End":"03:36.770","Text":"that when n goes to infinity,"},{"Start":"03:36.770 ","End":"03:38.210","Text":"this also goes to infinity,"},{"Start":"03:38.210 ","End":"03:43.645","Text":"otherwise we\u0027d have to start messing around with the limit, so that\u0027s fine."},{"Start":"03:43.645 ","End":"03:50.985","Text":"Now here, we want smiley which is cotangent of 1 over n,"},{"Start":"03:50.985 ","End":"03:56.750","Text":"but now have I altered the problem because originally here it was n,"},{"Start":"03:56.750 ","End":"04:01.760","Text":"so we have to compensate to fix it and I\u0027ll fix it by raising it"},{"Start":"04:01.760 ","End":"04:07.620","Text":"to another power and we\u0027ll determine what this question mark is."},{"Start":"04:08.510 ","End":"04:11.015","Text":"I\u0027ll do this at the side."},{"Start":"04:11.015 ","End":"04:12.410","Text":"Now a power to a power,"},{"Start":"04:12.410 ","End":"04:13.969","Text":"you multiply the powers."},{"Start":"04:13.969 ","End":"04:21.045","Text":"The cotangent 1 over n times something has got to equal n."},{"Start":"04:21.045 ","End":"04:30.960","Text":"That something is equal to n divided by cotangent 1 over n,"},{"Start":"04:30.960 ","End":"04:33.975","Text":"but we talked already, 1 over cotangent is tangent,"},{"Start":"04:33.975 ","End":"04:39.375","Text":"so this is just n tangent 1 over n,"},{"Start":"04:39.375 ","End":"04:44.520","Text":"which I can now put here in place of the question mark."},{"Start":"04:47.260 ","End":"04:51.050","Text":"Now, the limit is 2 parts."},{"Start":"04:51.050 ","End":"04:54.980","Text":"There\u0027s this part, which no problem will be e,"},{"Start":"04:54.980 ","End":"04:59.030","Text":"but we won\u0027t know what this limit is when n goes to infinity."},{"Start":"04:59.030 ","End":"05:05.685","Text":"I want to do that as a side exercise also. Let\u0027s see."},{"Start":"05:05.685 ","End":"05:11.570","Text":"We need the limit as n goes to infinity of"},{"Start":"05:11.570 ","End":"05:19.670","Text":"n tangent 1 over n. Now I\u0027m going to use a similar thing that I did here,"},{"Start":"05:19.670 ","End":"05:27.930","Text":"we\u0027ll replace 1 over n by t. If t is 1 over n,"},{"Start":"05:27.930 ","End":"05:30.820","Text":"I\u0027m going to reuse this part."},{"Start":"05:34.730 ","End":"05:41.970","Text":"Now n is also 1 over t. We get the limit t"},{"Start":"05:41.970 ","End":"05:49.950","Text":"goes to 0 from above of 1 over t times tangent t,"},{"Start":"05:49.950 ","End":"05:59.170","Text":"so I can write this as tangent t over t. Now we know the limit of sine t over t,"},{"Start":"05:59.170 ","End":"06:00.310","Text":"it\u0027s a famous limit,"},{"Start":"06:00.310 ","End":"06:02.455","Text":"but we don\u0027t have sine t,"},{"Start":"06:02.455 ","End":"06:08.470","Text":"so I\u0027ll just rewrite this as the limit as t goes to"},{"Start":"06:08.470 ","End":"06:16.390","Text":"0 of sine t over t. But now,"},{"Start":"06:16.390 ","End":"06:20.590","Text":"I need to also multiply it by something to fix it."},{"Start":"06:20.590 ","End":"06:22.765","Text":"Tangent is sine over cosine,"},{"Start":"06:22.765 ","End":"06:26.500","Text":"so I can put here 1 over cosine"},{"Start":"06:26.500 ","End":"06:32.260","Text":"t. Now we have no problem because this thing is a famous limit."},{"Start":"06:32.260 ","End":"06:33.790","Text":"It goes to 1."},{"Start":"06:33.790 ","End":"06:36.880","Text":"When t goes to 0 from either side,"},{"Start":"06:36.880 ","End":"06:38.665","Text":"cosine t goes to 1,"},{"Start":"06:38.665 ","End":"06:40.915","Text":"so this also goes to 1."},{"Start":"06:40.915 ","End":"06:45.350","Text":"This whole limit is equal to 1,"},{"Start":"06:45.350 ","End":"06:47.787","Text":"this bit here now."},{"Start":"06:47.787 ","End":"06:52.075","Text":"Now we can go back here and say, fine."},{"Start":"06:52.075 ","End":"06:57.660","Text":"This here is, I\u0027m going to scroll back up,"},{"Start":"06:57.660 ","End":"07:03.170","Text":"is the Euler limit with cotangent 1 over n in place of smiley,"},{"Start":"07:03.170 ","End":"07:09.460","Text":"so this part goes to e and this,"},{"Start":"07:09.460 ","End":"07:11.075","Text":"we figured out here,"},{"Start":"07:11.075 ","End":"07:14.160","Text":"goes to 1,"},{"Start":"07:14.270 ","End":"07:20.445","Text":"and so the answer to our limit is just"},{"Start":"07:20.445 ","End":"07:27.040","Text":"e to the power of 1 or plain e. That\u0027s the answer."}],"ID":31279}],"Thumbnail":null,"ID":294583},{"Name":"Ratio Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Ratio Test - Introduction","Duration":"14m 38s","ChapterTopicVideoID":29682,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this clip, we\u0027ll be introducing the ratio test,"},{"Start":"00:03.360 ","End":"00:12.630","Text":"I should really say for the convergence of sequences."},{"Start":"00:12.630 ","End":"00:22.990","Text":"Before that, I want to do a review on the mathematical concept of factorial,"},{"Start":"00:23.030 ","End":"00:28.935","Text":"4 factorial, and sometimes it\u0027s pronounced 4 bang."},{"Start":"00:28.935 ","End":"00:33.075","Text":"It\u0027s equal to 4 times 3 times 2 times 1,"},{"Start":"00:33.075 ","End":"00:35.685","Text":"which happens to be 24."},{"Start":"00:35.685 ","End":"00:39.120","Text":"We\u0027ll encounter the factorial a lot in the ratio test,"},{"Start":"00:39.120 ","End":"00:41.985","Text":"so I want to review some properties."},{"Start":"00:41.985 ","End":"00:45.555","Text":"Another example, 10 factorial,"},{"Start":"00:45.555 ","End":"00:54.315","Text":"10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"00:54.315 ","End":"00:57.950","Text":"We include the times 1 even though it doesn\u0027t change the result,"},{"Start":"00:57.950 ","End":"01:00.720","Text":"it just looks better that way."},{"Start":"01:00.910 ","End":"01:07.385","Text":"You could also write a general n factorial as being n,"},{"Start":"01:07.385 ","End":"01:14.120","Text":"then times the next number down and times the next number down if there is one,"},{"Start":"01:14.120 ","End":"01:16.440","Text":"and so on and so on,"},{"Start":"01:16.440 ","End":"01:22.315","Text":"times 3 times 2 times 1."},{"Start":"01:22.315 ","End":"01:27.330","Text":"Another example, let\u0027s say n plus 3 factorial,"},{"Start":"01:27.330 ","End":"01:31.410","Text":"so we\u0027re going to start with n plus 3 and then"},{"Start":"01:31.410 ","End":"01:36.195","Text":"reduce by 1 each time and reduce by another 1,"},{"Start":"01:36.195 ","End":"01:38.430","Text":"and reduce by another 1,"},{"Start":"01:38.430 ","End":"01:42.030","Text":"and reduce by another 1 and so on and so on,"},{"Start":"01:42.030 ","End":"01:44.475","Text":"times 3 times 2 times 1."},{"Start":"01:44.475 ","End":"01:50.225","Text":"Of course, we may not have this many factors and just reduce it appropriately."},{"Start":"01:50.225 ","End":"01:56.450","Text":"Now, I want to show some rules involving factorials because in the exercises,"},{"Start":"01:56.450 ","End":"02:01.890","Text":"we\u0027re going to be canceling and rearranging factorials."},{"Start":"02:02.660 ","End":"02:06.815","Text":"Now, let\u0027s look at this example for a moment."},{"Start":"02:06.815 ","End":"02:10.325","Text":"There\u0027s a rule and I\u0027ll illustrate it."},{"Start":"02:10.325 ","End":"02:13.575","Text":"When I start writing out 10 factorial,"},{"Start":"02:13.575 ","End":"02:17.880","Text":"let\u0027s start off 10 times 9 times 8,"},{"Start":"02:17.880 ","End":"02:20.715","Text":"and at some point I want to stop,"},{"Start":"02:20.715 ","End":"02:27.035","Text":"all I have to do is write factorial here and that will be correct."},{"Start":"02:27.035 ","End":"02:29.195","Text":"Because if you look here,"},{"Start":"02:29.195 ","End":"02:30.890","Text":"what\u0027s after the 8,"},{"Start":"02:30.890 ","End":"02:35.440","Text":"this whole thing is just 7 factorial."},{"Start":"02:35.440 ","End":"02:38.225","Text":"I can stop if I place a factorial."},{"Start":"02:38.225 ","End":"02:40.975","Text":"Another example with 10 factorial,"},{"Start":"02:40.975 ","End":"02:45.300","Text":"it\u0027s 10 times 9 factorial."},{"Start":"02:45.300 ","End":"02:47.085","Text":"I can stop it right here,"},{"Start":"02:47.085 ","End":"02:50.340","Text":"it\u0027s 10, and then from 9 down to 1."},{"Start":"02:50.340 ","End":"02:58.530","Text":"Or I can write a lot 10 times 9 times 8 times 7 times 6 times 5 times 4,"},{"Start":"02:58.530 ","End":"03:01.400","Text":"and here I want to stop factorial."},{"Start":"03:01.400 ","End":"03:04.925","Text":"It\u0027s just the last bit that\u0027s factorial."},{"Start":"03:04.925 ","End":"03:07.489","Text":"In our exercises on the ratio test,"},{"Start":"03:07.489 ","End":"03:15.810","Text":"we\u0027ll typically have expressions like 100 factorial over 97 factorial,"},{"Start":"03:15.810 ","End":"03:20.210","Text":"and we want to somehow cancel stuff like with fractions to get this thing"},{"Start":"03:20.210 ","End":"03:24.650","Text":"to more manageable size and that\u0027s where this comes in."},{"Start":"03:24.650 ","End":"03:34.130","Text":"Because here we would say this is 100 times 99 times 98 times 97,"},{"Start":"03:34.130 ","End":"03:37.555","Text":"and we\u0027d stop here and put a factorial."},{"Start":"03:37.555 ","End":"03:41.645","Text":"Then when it\u0027s over 97 factorial,"},{"Start":"03:41.645 ","End":"03:43.790","Text":"this will cancel with this,"},{"Start":"03:43.790 ","End":"03:48.880","Text":"and we\u0027re left with just 100 times 99 times 98."},{"Start":"03:48.880 ","End":"03:52.550","Text":"Now, do an example with letters."},{"Start":"03:52.550 ","End":"04:00.210","Text":"Suppose I have n plus 2 factorial over n factorial,"},{"Start":"04:00.210 ","End":"04:05.475","Text":"then I\u0027d say this is n plus 2 and go down by 1,"},{"Start":"04:05.475 ","End":"04:07.770","Text":"then go down by another 1."},{"Start":"04:07.770 ","End":"04:09.330","Text":"This is good for me,"},{"Start":"04:09.330 ","End":"04:15.095","Text":"so I put a factorial here and that stops the descent over n factorial."},{"Start":"04:15.095 ","End":"04:19.475","Text":"This cancels and the result is n plus 2 times n plus 1."},{"Start":"04:19.475 ","End":"04:28.970","Text":"Another example, n plus 1 factorial over n plus 4 factorial."},{"Start":"04:28.970 ","End":"04:31.880","Text":"This time, I would like"},{"Start":"04:31.880 ","End":"04:36.650","Text":"to start expanding the denominator because this is the bigger one."},{"Start":"04:36.650 ","End":"04:39.110","Text":"It\u0027s n plus 4,"},{"Start":"04:39.110 ","End":"04:41.365","Text":"n plus 3,"},{"Start":"04:41.365 ","End":"04:43.650","Text":"n plus 2,"},{"Start":"04:43.650 ","End":"04:45.000","Text":"n plus 1,"},{"Start":"04:45.000 ","End":"04:46.565","Text":"and yes, this is what we want,"},{"Start":"04:46.565 ","End":"04:50.410","Text":"put a factorial here that stops it,"},{"Start":"04:50.410 ","End":"04:53.495","Text":"n plus 1 factorial we had here."},{"Start":"04:53.495 ","End":"04:55.130","Text":"This cancels with this,"},{"Start":"04:55.130 ","End":"04:57.560","Text":"and then we\u0027re left with a cubic in n,"},{"Start":"04:57.560 ","End":"04:59.900","Text":"n plus 4 times n plus 3 times n plus 2,"},{"Start":"04:59.900 ","End":"05:03.730","Text":"which we could multiply out if we wanted to."},{"Start":"05:03.730 ","End":"05:08.725","Text":"Enough with factorials, let\u0027s move on to the ratio test."},{"Start":"05:08.725 ","End":"05:18.600","Text":"Let\u0027s say we have a sequence an and we have to have an being positive for all n,"},{"Start":"05:18.600 ","End":"05:22.680","Text":"that\u0027s a definite requirement for the ratio test."},{"Start":"05:23.150 ","End":"05:33.800","Text":"A hint as to when you would want to use the ratio test is when an contains factorials."},{"Start":"05:33.800 ","End":"05:37.430","Text":"Now, it doesn\u0027t guarantee that that\u0027s"},{"Start":"05:37.430 ","End":"05:42.275","Text":"the right test and you could use the ratio test even when there\u0027s not a factorial,"},{"Start":"05:42.275 ","End":"05:47.710","Text":"but this is an indication that you would try the ratio test."},{"Start":"05:47.710 ","End":"05:50.505","Text":"That\u0027s when to use it."},{"Start":"05:50.505 ","End":"05:52.430","Text":"Now, the statement,"},{"Start":"05:52.430 ","End":"05:57.590","Text":"we consider the limit as n goes to infinity of another sequence,"},{"Start":"05:57.590 ","End":"06:02.475","Text":"and that sequence is an plus 1 over an."},{"Start":"06:02.475 ","End":"06:10.415","Text":"You can see now why we requiring an to be positive or at least has to be non-zero,"},{"Start":"06:10.415 ","End":"06:17.640","Text":"and if this limit exists and it\u0027s less than 1,"},{"Start":"06:17.640 ","End":"06:23.550","Text":"then our sequence an tends to 0."},{"Start":"06:23.550 ","End":"06:27.670","Text":"If this limit comes out bigger than 1,"},{"Start":"06:27.670 ","End":"06:32.465","Text":"then an tends to infinity."},{"Start":"06:32.465 ","End":"06:35.180","Text":"You might ask what if it equals 1?"},{"Start":"06:35.180 ","End":"06:37.430","Text":"Then we don\u0027t know,"},{"Start":"06:37.430 ","End":"06:39.800","Text":"at least this test doesn\u0027t help us."},{"Start":"06:39.800 ","End":"06:41.615","Text":"Now, let\u0027s take an example."},{"Start":"06:41.615 ","End":"06:47.165","Text":"Suppose I want the limit as n goes to infinity of"},{"Start":"06:47.165 ","End":"06:52.970","Text":"100^n over n factorial."},{"Start":"06:52.970 ","End":"06:55.445","Text":"Note that there\u0027s a factorial here,"},{"Start":"06:55.445 ","End":"06:58.315","Text":"this is our an,"},{"Start":"06:58.315 ","End":"07:02.870","Text":"and so we\u0027re going to try the ratio test."},{"Start":"07:02.870 ","End":"07:10.935","Text":"We figure out the limit of an plus 1 over an,"},{"Start":"07:10.935 ","End":"07:13.380","Text":"and that is equal to,"},{"Start":"07:13.380 ","End":"07:15.630","Text":"on the denominator,"},{"Start":"07:15.630 ","End":"07:22.080","Text":"I\u0027ll just copy 100^n over n factorial."},{"Start":"07:22.080 ","End":"07:23.595","Text":"On the numerator,"},{"Start":"07:23.595 ","End":"07:28.770","Text":"I just substitute wherever we had n we\u0027ll put n plus 1,"},{"Start":"07:28.770 ","End":"07:33.215","Text":"and here n plus 1 factorial."},{"Start":"07:33.215 ","End":"07:35.560","Text":"When we divide by a fraction,"},{"Start":"07:35.560 ","End":"07:38.170","Text":"we multiply by the inverse fraction,"},{"Start":"07:38.170 ","End":"07:44.395","Text":"so we get 100^n plus 1 over n plus 1 factorial,"},{"Start":"07:44.395 ","End":"07:51.415","Text":"times n factorial over 100^n."},{"Start":"07:51.415 ","End":"07:54.160","Text":"Now I\u0027m just going to rearrange because I"},{"Start":"07:54.160 ","End":"07:57.100","Text":"want the factorials together and the exponents together."},{"Start":"07:57.100 ","End":"08:04.090","Text":"It\u0027s n factorial over n plus 1 factorial"},{"Start":"08:04.090 ","End":"08:12.835","Text":"times 100^n plus 1 over 100^n."},{"Start":"08:12.835 ","End":"08:19.390","Text":"I just realized I forgot to put the limit in each of these,"},{"Start":"08:19.390 ","End":"08:21.145","Text":"n goes to infinity,"},{"Start":"08:21.145 ","End":"08:23.605","Text":"n goes to infinity."},{"Start":"08:23.605 ","End":"08:32.530","Text":"Now, we can do some canceling because we can write this as n plus 1 times n factorial."},{"Start":"08:32.530 ","End":"08:37.810","Text":"Limit n goes to infinity, n factorial,"},{"Start":"08:37.810 ","End":"08:43.195","Text":"and here n plus 1 times n factorial."},{"Start":"08:43.195 ","End":"08:53.845","Text":"Here, I can use the rules of exponents to say it\u0027s 100 times 100^n over 100^n."},{"Start":"08:53.845 ","End":"08:55.900","Text":"Now, a lot of stuff cancels,"},{"Start":"08:55.900 ","End":"09:03.220","Text":"the n factorial cancels with n factorial, 100^n cancels with 100^n,"},{"Start":"09:03.220 ","End":"09:08.920","Text":"and what we\u0027re left with is the limit n goes to infinity of,"},{"Start":"09:08.920 ","End":"09:10.825","Text":"here we have 100,"},{"Start":"09:10.825 ","End":"09:14.080","Text":"here we have n plus 1,"},{"Start":"09:14.080 ","End":"09:18.415","Text":"and this limit is 0, clearly."},{"Start":"09:18.415 ","End":"09:22.390","Text":"Now all we care about is less than 1 or bigger than 1,"},{"Start":"09:22.390 ","End":"09:25.285","Text":"0 happens to be less than 1,"},{"Start":"09:25.285 ","End":"09:29.860","Text":"and when the limit comes out to be less than 1,"},{"Start":"09:29.860 ","End":"09:35.455","Text":"that implies that the original sequence,"},{"Start":"09:35.455 ","End":"09:42.905","Text":"100^n over n factorial is equal to 0."},{"Start":"09:42.905 ","End":"09:44.685","Text":"Remember, less than 1,"},{"Start":"09:44.685 ","End":"09:46.080","Text":"it converges to 0,"},{"Start":"09:46.080 ","End":"09:48.825","Text":"greater than 1 tends to infinity."},{"Start":"09:48.825 ","End":"09:50.795","Text":"That\u0027s this example."},{"Start":"09:50.795 ","End":"09:53.530","Text":"Now let\u0027s do another example and I\u0027ll choose"},{"Start":"09:53.530 ","End":"09:56.485","Text":"an example where it comes out greater than 1."},{"Start":"09:56.485 ","End":"09:58.480","Text":"In this next example,"},{"Start":"09:58.480 ","End":"10:03.415","Text":"we want to compute the limit as n goes to infinity"},{"Start":"10:03.415 ","End":"10:09.505","Text":"of 2 times 5 times 8 times,"},{"Start":"10:09.505 ","End":"10:14.340","Text":"and so on up to 3n plus"},{"Start":"10:14.340 ","End":"10:21.000","Text":"2 over 1 times 3 times 5 times,"},{"Start":"10:21.000 ","End":"10:26.740","Text":"and so on, up to 2n plus 1."},{"Start":"10:26.740 ","End":"10:29.560","Text":"Here, it jumps by 3 every time,"},{"Start":"10:29.560 ","End":"10:32.540","Text":"and here we have a jump of 2."},{"Start":"10:34.200 ","End":"10:41.860","Text":"This is an, a_1 is actually 2 times 5 over 1 times 3,"},{"Start":"10:41.860 ","End":"10:43.315","Text":"because when n is 1,"},{"Start":"10:43.315 ","End":"10:45.925","Text":"this is 5 and this is 3,"},{"Start":"10:45.925 ","End":"10:51.550","Text":"so we start off here and then we keep adding another pair each time."},{"Start":"10:51.550 ","End":"10:54.280","Text":"Ratio test."},{"Start":"10:54.280 ","End":"11:04.060","Text":"We want the limit as n goes to infinity of an plus 1 over an,"},{"Start":"11:04.060 ","End":"11:07.460","Text":"which equals the limit."},{"Start":"11:07.500 ","End":"11:15.055","Text":"Now, an plus 1 is what we get if we substitute n plus 1"},{"Start":"11:15.055 ","End":"11:22.555","Text":"instead of n. In fact I\u0027d like to compute an plus 1 at the side,"},{"Start":"11:22.555 ","End":"11:24.850","Text":"because I want to do some simplification."},{"Start":"11:24.850 ","End":"11:29.605","Text":"It\u0027s 2 times 5 times 8 times,"},{"Start":"11:29.605 ","End":"11:31.840","Text":"and so on up to,"},{"Start":"11:31.840 ","End":"11:41.725","Text":"now here the last term will be 3 times n plus 1 plus 2,"},{"Start":"11:41.725 ","End":"11:45.355","Text":"and I\u0027ll put that in brackets, over,"},{"Start":"11:45.355 ","End":"11:50.740","Text":"here we have 1 times 3 times 5, and so on."},{"Start":"11:50.740 ","End":"11:58.217","Text":"Here we have twice n plus 1 plus 1,"},{"Start":"11:58.217 ","End":"12:07.255","Text":"and this will equal 2 times 5 times 8 times and so on,"},{"Start":"12:07.255 ","End":"12:13.130","Text":"up to, this comes out to be 3n plus 5."},{"Start":"12:13.650 ","End":"12:18.655","Text":"On the denominator 1 times 3 times 5 times,"},{"Start":"12:18.655 ","End":"12:21.234","Text":"and so on, up to,"},{"Start":"12:21.234 ","End":"12:24.860","Text":"comes out 2n plus 3."},{"Start":"12:25.230 ","End":"12:32.980","Text":"It\u0027s more convenient if we actually write the next to last term also."},{"Start":"12:32.980 ","End":"12:34.765","Text":"I\u0027ll do that when I copy it here."},{"Start":"12:34.765 ","End":"12:40.840","Text":"Here we have 2 times 5 times 8 times, et cetera."},{"Start":"12:40.840 ","End":"12:46.165","Text":"Now, the 1 before 3n plus 5 is 3n plus 2,"},{"Start":"12:46.165 ","End":"12:49.345","Text":"and then the 3n plus 5."},{"Start":"12:49.345 ","End":"12:52.450","Text":"On the denominator of the numerator,"},{"Start":"12:52.450 ","End":"12:57.235","Text":"we have 1 times 3 times 5 times, and so on."},{"Start":"12:57.235 ","End":"13:02.410","Text":"The 1 before 2n plus 3 is 2n plus 1,"},{"Start":"13:02.410 ","End":"13:05.590","Text":"and then 2n plus 3."},{"Start":"13:05.590 ","End":"13:10.630","Text":"Over here, we just have to copy this expression."},{"Start":"13:10.630 ","End":"13:14.320","Text":"Now, we\u0027re going to do the fraction division,"},{"Start":"13:14.320 ","End":"13:18.130","Text":"which is multiplying by the inverse fraction."},{"Start":"13:18.130 ","End":"13:28.075","Text":"Here we have over times the upside down of this and here this bit."},{"Start":"13:28.075 ","End":"13:36.775","Text":"Now, notice that this cancels with this,"},{"Start":"13:36.775 ","End":"13:42.385","Text":"and this cancels with this."},{"Start":"13:42.385 ","End":"13:49.555","Text":"All we\u0027re left with is the limit as n goes to infinity of"},{"Start":"13:49.555 ","End":"13:57.130","Text":"3n plus 5 over 2n plus 3."},{"Start":"13:57.130 ","End":"14:02.185","Text":"Now, this is a polynomial over a polynomial same degree top and bottom,"},{"Start":"14:02.185 ","End":"14:09.430","Text":"so we just look at the leading coefficients and the answer is 3/2,"},{"Start":"14:09.430 ","End":"14:12.670","Text":"for those who like decimals, it\u0027s 1.5."},{"Start":"14:12.670 ","End":"14:17.230","Text":"The important thing is that it\u0027s bigger than 1."},{"Start":"14:17.230 ","End":"14:21.850","Text":"Then the limit,"},{"Start":"14:21.850 ","End":"14:23.920","Text":"I don\u0027t want to copy it again,"},{"Start":"14:23.920 ","End":"14:26.305","Text":"I\u0027ll just call it an in our case,"},{"Start":"14:26.305 ","End":"14:28.330","Text":"is equal to infinity."},{"Start":"14:28.330 ","End":"14:30.670","Text":"Remember, less than 1, limit is 0,"},{"Start":"14:30.670 ","End":"14:33.940","Text":"bigger than 1 means infinity."},{"Start":"14:33.940 ","End":"14:36.190","Text":"That\u0027s the answer."},{"Start":"14:36.190 ","End":"14:39.350","Text":"I think we\u0027ll end the clip here."}],"ID":31310},{"Watched":false,"Name":"The Ratio Test - Examples","Duration":"10m 10s","ChapterTopicVideoID":29683,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.480 ","End":"00:05.760","Text":"In this clip, we\u0027ll do threeexamples of using the ratio test"},{"Start":"00:05.760 ","End":"00:07.950","Text":"to find the limit of a sequence."},{"Start":"00:08.550 ","End":"00:13.020","Text":"Uh, first example, the limitas then goes to infinity of n"},{"Start":"00:13.020 ","End":"00:15.960","Text":"factorial over four n factorial."},{"Start":"00:16.560 ","End":"00:19.230","Text":"Note that the sequence is positive."},{"Start":"00:19.710 ","End":"00:24.900","Text":"That\u0027s a must for using the ratiotest, and an indication that we would"},{"Start":"00:24.900 ","End":"00:26.730","Text":"use it is because of the factorial."},{"Start":"00:27.900 ","End":"00:31.049","Text":"So let\u0027s see what n and n plus one are."},{"Start":"00:32.430 ","End":"00:35.849","Text":"N just this copied it."},{"Start":"00:36.180 ","End":"00:40.200","Text":"N plus one, replace N with N plus one."},{"Start":"00:41.099 ","End":"00:44.010","Text":"Now we want the limitof N plus one over N."},{"Start":"00:45.269 ","End":"00:49.440","Text":"And when we find this, what\u0027simportant is whether it\u0027s greater"},{"Start":"00:49.440 ","End":"00:50.910","Text":"than one or less than one."},{"Start":"00:51.330 ","End":"00:53.160","Text":"Anyway, it\u0027s this over this."},{"Start":"00:54.269 ","End":"00:54.900","Text":"So."},{"Start":"00:56.280 ","End":"00:57.330","Text":"Copy it out."},{"Start":"00:57.690 ","End":"00:59.820","Text":"And now we want to, um, rearrange."},{"Start":"00:59.820 ","End":"01:04.739","Text":"First of all, dividing by a fraction islike multiplying by the inverse fraction."},{"Start":"01:04.739 ","End":"01:07.289","Text":"So this is upside down here."},{"Start":"01:07.740 ","End":"01:14.340","Text":"Then we multiply, remember we reviewed theproperties of the factorial, and plus one"},{"Start":"01:14.340 ","End":"01:18.120","Text":"factorial is n plus one times n factorial."},{"Start":"01:18.360 ","End":"01:21.179","Text":"And this, well, I\u0027ll giveyou a numerical example."},{"Start":"01:21.179 ","End":"01:23.940","Text":"Suppose N was two then we\u0027d."},{"Start":"01:24.810 ","End":"01:29.880","Text":"Here, four n plus four is12, and four N is eight."},{"Start":"01:30.150 ","End":"01:35.040","Text":"So what I\u0027d wanna do is write the12 factorial is 12 times, 11 times"},{"Start":"01:35.040 ","End":"01:36.720","Text":"10 times nine, then times eight."},{"Start":"01:36.720 ","End":"01:39.600","Text":"And then when I wanna stop,put the factorial sign."},{"Start":"01:40.020 ","End":"01:41.880","Text":"So this becomes this."},{"Start":"01:42.240 ","End":"01:44.970","Text":"We write out four termsfor factors rather."},{"Start":"01:45.810 ","End":"01:48.660","Text":"Four n plus four, four n plus three,four nm plus two, four N plus one."},{"Start":"01:48.660 ","End":"01:52.620","Text":"And then we get to four N andwe put the factorial and then"},{"Start":"01:52.620 ","End":"01:54.570","Text":"this will cancel with this."},{"Start":"01:55.170 ","End":"01:58.920","Text":"And here the n factorialcancels with the N factorial."},{"Start":"01:59.400 ","End":"02:07.350","Text":"So what we\u0027re left with isn plus one here, and here we"},{"Start":"02:07.350 ","End":"02:09.780","Text":"have the product of these four."},{"Start":"02:11.855 ","End":"02:13.935","Text":"I don\u0027t actually have to multiply it out."},{"Start":"02:14.055 ","End":"02:17.985","Text":"We can see that this is a degreeone polynomial, and this has"},{"Start":"02:17.985 ","End":"02:20.415","Text":"N here, here, here, and here."},{"Start":"02:20.715 ","End":"02:23.895","Text":"If we multiply it out, we\u0027re gonnaget into the fourth something,"},{"Start":"02:24.405 ","End":"02:25.995","Text":"and that\u0027s gonna be degree four."},{"Start":"02:26.295 ","End":"02:30.735","Text":"And when a degree upstairs is lessthan a degree downstairs, then the"},{"Start":"02:30.735 ","End":"02:35.775","Text":"limit at infinity is zero and zero."},{"Start":"02:36.255 ","End":"02:38.475","Text":"All I care about hereis that it\u0027s less than."},{"Start":"02:39.704 ","End":"02:45.704","Text":"And when you have a limit of n plus1:00 AM that\u0027s less than one, then the"},{"Start":"02:46.204 ","End":"02:53.385","Text":"original sequence tends to zero biggerthan one is infinity, less than one zero."},{"Start":"02:53.625 ","End":"02:55.545","Text":"So that\u0027s this example."},{"Start":"02:57.225 ","End":"02:58.994","Text":"And here\u0027s our next example."},{"Start":"02:59.924 ","End":"03:05.505","Text":"Two n factorial over n factorialtimes two N to the power of."},{"Start":"03:07.424 ","End":"03:12.674","Text":"And once again, I note thatit\u0027s a positive sequence in"},{"Start":"03:12.674 ","End":"03:14.415","Text":"order to use the ratio test."},{"Start":"03:14.984 ","End":"03:19.095","Text":"And we would try the ratio testbecause we see the factorial."},{"Start":"03:19.845 ","End":"03:23.415","Text":"Okay, so let\u0027s see whatn and n plus one are."},{"Start":"03:23.595 ","End":"03:26.415","Text":"N I just copied this N plus one."},{"Start":"03:26.415 ","End":"03:28.695","Text":"Replace N with N plus one."},{"Start":"03:28.995 ","End":"03:34.875","Text":"Now we want the ratio of thisover this, so that gives."},{"Start":"03:35.880 ","End":"03:38.340","Text":"This from here, this here."},{"Start":"03:39.090 ","End":"03:45.300","Text":"So I invert this and put it alongside,and now we want to use the properties"},{"Start":"03:45.300 ","End":"03:48.510","Text":"of factorial so we can cancel."},{"Start":"03:48.900 ","End":"03:56.880","Text":"So two n plus two factorial is two n plustwo two n plus one times two N factorial."},{"Start":"03:57.930 ","End":"03:59.850","Text":"And the rest of it stays the."},{"Start":"04:00.450 ","End":"04:06.930","Text":"And here this is using the propertiesof exponents, two n plus two"},{"Start":"04:07.140 ","End":"04:09.600","Text":"times two n plus two to the N."},{"Start":"04:09.810 ","End":"04:14.640","Text":"And now some stuff cancels twon plus two with two n plus two."},{"Start":"04:14.640 ","End":"04:19.740","Text":"The two N factorial with two nfactorial, n factorial with n factorial."},{"Start":"04:20.130 ","End":"04:21.810","Text":"And let\u0027s see what we have now."},{"Start":"04:21.810 ","End":"04:22.880","Text":"And we tie the up."},{"Start":"04:23.685 ","End":"04:26.055","Text":"We have two n plus one from here."},{"Start":"04:26.534 ","End":"04:32.414","Text":"Here is n plus one, which goes here,but here you could think of it as two N"},{"Start":"04:32.835 ","End":"04:37.215","Text":"over two n plus two to the power of N."},{"Start":"04:37.544 ","End":"04:42.405","Text":"And then if I divide top and bottomby two, it\u0027s the same as N over"},{"Start":"04:42.405 ","End":"04:45.135","Text":"n plus one to the power of N."},{"Start":"04:45.914 ","End":"04:47.505","Text":"In any case, this is what I did."},{"Start":"04:49.515 ","End":"04:55.005","Text":"And what we get is a productthis from here and n over n"},{"Start":"04:55.005 ","End":"04:56.354","Text":"plus one to the power of N."},{"Start":"04:56.354 ","End":"05:00.344","Text":"Here i I broke it up the productinto the product of limits."},{"Start":"05:00.795 ","End":"05:03.195","Text":"Now I\u0027ll tell you theanswer to each of these."},{"Start":"05:03.195 ","End":"05:04.305","Text":"I\u0027ll call this one asterisk."},{"Start":"05:04.305 ","End":"05:11.385","Text":"This double asterisk, single asteriskcomes out to be two and double asterisk."},{"Start":"05:11.474 ","End":"05:14.115","Text":"This limit comes out to be one."},{"Start":"05:16.095 ","End":"05:19.935","Text":"Now if you multiply twotimes one E, it\u0027s two E."},{"Start":"05:20.595 ","End":"05:24.225","Text":"And since E is bigger thantwo, it\u0027s 2.718 something."},{"Start":"05:24.255 ","End":"05:26.505","Text":"Two E is less than one."},{"Start":"05:27.255 ","End":"05:33.645","Text":"And when the ratio n plus one Ncomes out less than one, then the"},{"Start":"05:33.650 ","End":"05:36.795","Text":"original sequence tends to zero."},{"Start":"05:37.305 ","End":"05:41.355","Text":"Okay, now I owe you to show youhow I got, uh, these limits."},{"Start":"05:42.750 ","End":"05:46.350","Text":"So let\u0027s start with the single asterisk."},{"Start":"05:46.860 ","End":"05:50.730","Text":"We could have used a shortcut andsaid polynomials of the same degree."},{"Start":"05:50.730 ","End":"05:54.430","Text":"Two over one is two just for the practice."},{"Start":"05:54.435 ","End":"05:56.430","Text":"I, um, did it the long way."},{"Start":"05:56.640 ","End":"06:02.460","Text":"We take n outta the numerator and thatleaves us with two plus one over n."},{"Start":"06:02.460 ","End":"06:06.450","Text":"Take n outta the denominator,N times one plus one over end,"},{"Start":"06:07.020 ","End":"06:10.800","Text":"and then add infinity one over."},{"Start":"06:11.625 ","End":"06:12.195","Text":"Is one over."},{"Start":"06:12.195 ","End":"06:13.664","Text":"Infinity is zero."},{"Start":"06:13.905 ","End":"06:14.325","Text":"One over."},{"Start":"06:14.325 ","End":"06:16.305","Text":"Infinity again is zero."},{"Start":"06:16.635 ","End":"06:18.405","Text":"And cancels with N."},{"Start":"06:19.844 ","End":"06:21.344","Text":"Yeah, and cancels with N."},{"Start":"06:21.344 ","End":"06:23.325","Text":"Like I said, this goes to zero."},{"Start":"06:23.594 ","End":"06:25.185","Text":"This goes to zero."},{"Start":"06:25.575 ","End":"06:29.414","Text":"So we just have two over one,which is two, as we said here."},{"Start":"06:30.075 ","End":"06:33.405","Text":"Now the other one, uh,the double asterisk."},{"Start":"06:34.425 ","End":"06:38.565","Text":"The plan is to use Oilers limit,but we have to do some algebra"},{"Start":"06:38.565 ","End":"06:40.034","Text":"brake manipulation first."},{"Start":"06:40.575 ","End":"06:45.525","Text":"So first thing I\u0027ll do is takethe reciprocal of this, which"},{"Start":"06:45.525 ","End":"06:50.295","Text":"is N plus one over N and put oneover that to the power of Ann."},{"Start":"06:50.805 ","End":"06:54.164","Text":"And now we can simplify this."},{"Start":"06:54.525 ","End":"06:58.155","Text":"N plus one over N is one plus one over n."},{"Start":"06:58.770 ","End":"07:02.580","Text":"And the exponent, I can justput in a denominator because"},{"Start":"07:02.580 ","End":"07:04.440","Text":"the numerator is one to the end."},{"Start":"07:04.980 ","End":"07:06.960","Text":"So this is what we get."},{"Start":"07:08.160 ","End":"07:12.690","Text":"And also I can put thelimit into the denominator."},{"Start":"07:12.780 ","End":"07:18.870","Text":"So we now get this and, oh,I guess I forgot to write."},{"Start":"07:18.930 ","End":"07:20.880","Text":"The end goes to infinity."},{"Start":"07:20.880 ","End":"07:22.020","Text":"Well, that\u0027s obvious."},{"Start":"07:23.010 ","End":"07:27.510","Text":"And this, and the denominator isthe famous Oilers limit, which is E."},{"Start":"07:28.425 ","End":"07:33.855","Text":"Our answer here is one E, andthat was this double asterisk."},{"Start":"07:33.855 ","End":"07:36.165","Text":"So we\u0027ve confirmed both of these."},{"Start":"07:36.525 ","End":"07:43.275","Text":"And so yeah, the result was thatour sequence, uh, converges to zero."},{"Start":"07:44.025 ","End":"07:44.415","Text":"Okay."},{"Start":"07:44.415 ","End":"07:46.005","Text":"I\u0027ll do one more example."},{"Start":"07:47.835 ","End":"07:53.205","Text":"The limit of two n factorialover n factorial squared."},{"Start":"07:54.555 ","End":"07:56.655","Text":"Obviously it\u0027s a positive sequence."},{"Start":"07:57.195 ","End":"08:03.855","Text":"And it contains factorials, which is whywe would naturally choose ratio test."},{"Start":"08:04.335 ","End":"08:11.805","Text":"So this general term is N and n plus onewe get by replacing N with N plus one."},{"Start":"08:12.255 ","End":"08:14.925","Text":"And what we want isthe limit of the ratio."},{"Start":"08:14.955 ","End":"08:21.795","Text":"This over this N plus one overN, which is this over this."},{"Start":"08:22.620 ","End":"08:28.560","Text":"And as usual, we multiply by theupside down denominator, turning"},{"Start":"08:28.560 ","End":"08:34.740","Text":"the division into a multiplicationbreak down the squared here and"},{"Start":"08:34.740 ","End":"08:41.429","Text":"here, uh, to writing this, uh,twice in factorial and factorial."},{"Start":"08:42.180 ","End":"08:44.640","Text":"And also just switch the order around."},{"Start":"08:44.640 ","End":"08:49.920","Text":"Put this one here and here we\u0027ve got Nplus one factorial, N plus one factorial."},{"Start":"08:50.820 ","End":"08:55.740","Text":"And now what I wanna dois expand the factorials."},{"Start":"08:56.280 ","End":"09:02.040","Text":"Now n plus one factorial isN plus one times N factorial."},{"Start":"09:02.040 ","End":"09:03.150","Text":"Do that twice."},{"Start":"09:03.690 ","End":"09:09.030","Text":"And two n plus two factorial is two nplus two two n plus one two N factorial."},{"Start":"09:11.040 ","End":"09:12.060","Text":"things that cancel."},{"Start":"09:12.060 ","End":"09:19.590","Text":"I\u0027ve indicated this with this N factorialcancels twice and what we\u0027re left"},{"Start":"09:19.595 ","End":"09:28.650","Text":"with is two n plus two, two n plus onehere, and n plus one N plus one here."},{"Start":"09:29.550 ","End":"09:37.410","Text":"And this is equal to, uh, Bit ofalgebra, multiply these brackets out."},{"Start":"09:37.980 ","End":"09:39.300","Text":"We didn\u0027t have to do all that."},{"Start":"09:39.300 ","End":"09:41.970","Text":"We really just need theleading coefficients."},{"Start":"09:41.970 ","End":"09:42.780","Text":"We could have seen that."},{"Start":"09:42.780 ","End":"09:46.560","Text":"This is two N times two N is fourn squared and here n squared."},{"Start":"09:46.949 ","End":"09:51.300","Text":"So the limit is fourover one, which is four."},{"Start":"09:51.480 ","End":"09:54.150","Text":"And four obviously is bigger than one."},{"Start":"09:54.630 ","End":"10:01.050","Text":"And when this ratio comes out biggerthan one, than the limit of the."},{"Start":"10:01.485 ","End":"10:11.025","Text":"Um, original sequence is infinity, andthat\u0027s the answer, and we are done."}],"ID":31311},{"Watched":false,"Name":"Exercise 1","Duration":"4m 11s","ChapterTopicVideoID":30482,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.565","Text":"In this exercise, we have to evaluate the limit as n goes to infinity of 2^n factorial."},{"Start":"00:08.565 ","End":"00:14.250","Text":"This will be our general term and this we\u0027ll call this an."},{"Start":"00:14.250 ","End":"00:18.090","Text":"This looks like a case for the ratio test."},{"Start":"00:18.090 ","End":"00:20.930","Text":"I\u0027ll remind you what the ratio test is."},{"Start":"00:20.930 ","End":"00:25.270","Text":"We take the limit as n goes to infinity, not of an,"},{"Start":"00:25.270 ","End":"00:32.945","Text":"but of the ratio of 2 consecutive elements an plus 1/an."},{"Start":"00:32.945 ","End":"00:39.295","Text":"If this exists and comes out bigger than 1,"},{"Start":"00:39.295 ","End":"00:46.365","Text":"then the original sequence an goes to infinity."},{"Start":"00:46.365 ","End":"00:50.780","Text":"If the limit exists and is less than 1,"},{"Start":"00:50.780 ","End":"00:54.650","Text":"then an tends to 0."},{"Start":"00:54.650 ","End":"00:56.314","Text":"As n goes to infinity,"},{"Start":"00:56.314 ","End":"00:59.330","Text":"the limit doesn\u0027t exist or it exists and equals 1,"},{"Start":"00:59.330 ","End":"01:04.280","Text":"we don\u0027t know. Let\u0027s see now."},{"Start":"01:04.280 ","End":"01:10.730","Text":"an plus 1/an."},{"Start":"01:10.730 ","End":"01:14.805","Text":"I don\u0027t want to drag the limits with me everywhere,"},{"Start":"01:14.805 ","End":"01:16.880","Text":"so we\u0027ll just figure out an plus 1/an,"},{"Start":"01:16.880 ","End":"01:18.740","Text":"then we\u0027ll take the limit."},{"Start":"01:18.740 ","End":"01:20.810","Text":"This is equal to,"},{"Start":"01:20.810 ","End":"01:28.850","Text":"the denominator is easier because I just have to copy 2^n factorial,"},{"Start":"01:28.850 ","End":"01:31.720","Text":"and on the numerator,"},{"Start":"01:31.720 ","End":"01:39.990","Text":"2^n plus 1 plus 1 factorial."},{"Start":"01:40.910 ","End":"01:44.995","Text":"Now, I want to simplify this."},{"Start":"01:44.995 ","End":"01:51.830","Text":"I\u0027m going to leave the denominator as is, 2^n factorial."},{"Start":"01:51.830 ","End":"01:58.715","Text":"But the numerator, I want to split the numerator of the numerator."},{"Start":"01:58.715 ","End":"02:04.297","Text":"I can write it as 2^n times 2,"},{"Start":"02:04.297 ","End":"02:09.530","Text":"because that\u0027s what 2^n plus 1 is with the rules of exponents."},{"Start":"02:09.530 ","End":"02:12.365","Text":"n plus 1 factorial,"},{"Start":"02:12.365 ","End":"02:13.520","Text":"it\u0027s well known,"},{"Start":"02:13.520 ","End":"02:18.110","Text":"is equal to n plus 1 times n factorial."},{"Start":"02:18.110 ","End":"02:20.360","Text":"If you\u0027re not sure about this,"},{"Start":"02:20.360 ","End":"02:23.090","Text":"I\u0027ll explain it again with a numerical example."},{"Start":"02:23.090 ","End":"02:26.535","Text":"Suppose I have 6 factorial,"},{"Start":"02:26.535 ","End":"02:31.190","Text":"which is 6 times 5 times 4 times 3 times 2 times"},{"Start":"02:31.190 ","End":"02:39.870","Text":"1 and 6 is my n. If I multiply this by 7,"},{"Start":"02:39.870 ","End":"02:41.400","Text":"which is 6 plus 1,"},{"Start":"02:41.400 ","End":"02:48.810","Text":"7 times 6 factorial is 7 times and then 6 times 4,"},{"Start":"02:48.810 ","End":"02:52.005","Text":"there\u0027s a 5 is missing,"},{"Start":"02:52.005 ","End":"02:54.795","Text":"times 3 times 2 times 1."},{"Start":"02:54.795 ","End":"02:58.140","Text":"This is just 7 factorial."},{"Start":"02:58.140 ","End":"03:02.830","Text":"You can generalize from 6 to n, same idea."},{"Start":"03:03.350 ","End":"03:07.750","Text":"Now, in the denominator,"},{"Start":"03:07.750 ","End":"03:10.755","Text":"I have 2^n factorial."},{"Start":"03:10.755 ","End":"03:17.280","Text":"Here I also have 2^n factorial."},{"Start":"03:17.280 ","End":"03:25.350","Text":"This is equal to 2 plus 1."},{"Start":"03:25.350 ","End":"03:29.140","Text":"The limit as n goes to infinity of"},{"Start":"03:29.140 ","End":"03:38.525","Text":"this an plus 1/an is the limit as n goes to infinity of 2 plus 1."},{"Start":"03:38.525 ","End":"03:41.790","Text":"Obviously this is equal to 0."},{"Start":"03:41.790 ","End":"03:45.355","Text":"0 is less than 1,"},{"Start":"03:45.355 ","End":"03:49.115","Text":"which brings us to this case of less than 1."},{"Start":"03:49.115 ","End":"03:55.640","Text":"Then we conclude that the limit as n goes to infinity of an is"},{"Start":"03:55.640 ","End":"04:02.130","Text":"0 and an is just 2^n factorial."},{"Start":"04:02.130 ","End":"04:04.760","Text":"This is 0 because like I said,"},{"Start":"04:04.760 ","End":"04:07.170","Text":"this was our an."},{"Start":"04:07.270 ","End":"04:11.070","Text":"The answer is 0 and we\u0027re done."}],"ID":32618},{"Watched":false,"Name":"Exercise 2","Duration":"7m 19s","ChapterTopicVideoID":30483,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.320","Text":"In this exercise, we have to evaluate the limit as n goes to"},{"Start":"00:04.320 ","End":"00:09.420","Text":"infinity of n factorial over n to the n. Now,"},{"Start":"00:09.420 ","End":"00:13.215","Text":"we have actually solved this limit before,"},{"Start":"00:13.215 ","End":"00:15.525","Text":"maybe even in the previous clip."},{"Start":"00:15.525 ","End":"00:19.980","Text":"But this is going to be an alternative solution."},{"Start":"00:19.980 ","End":"00:22.770","Text":"There\u0027s often more than 1 way of solving"},{"Start":"00:22.770 ","End":"00:26.640","Text":"something and sometimes the other method is worth knowing."},{"Start":"00:26.640 ","End":"00:33.380","Text":"What I\u0027m going to do in this case is use the ratio test."},{"Start":"00:33.380 ","End":"00:37.995","Text":"In other words, let\u0027s call this general term"},{"Start":"00:37.995 ","End":"00:43.130","Text":"a_n is n factorial over n to the n. In general,"},{"Start":"00:43.130 ","End":"00:46.849","Text":"with a positive sequence,"},{"Start":"00:46.849 ","End":"00:56.255","Text":"we can check what is the limit as n goes to infinity of a_n plus 1 over a_n,"},{"Start":"00:56.255 ","End":"01:00.215","Text":"the ratio of successive terms in the limit."},{"Start":"01:00.215 ","End":"01:07.800","Text":"Now, if this happens to exist and be greater than 1,"},{"Start":"01:07.800 ","End":"01:10.485","Text":"sorry, greater than 1,"},{"Start":"01:10.485 ","End":"01:12.480","Text":"if it exists and greater than 1,"},{"Start":"01:12.480 ","End":"01:18.820","Text":"then a_n goes to infinity."},{"Start":"01:18.940 ","End":"01:23.365","Text":"If this limit exists and is less than 1,"},{"Start":"01:23.365 ","End":"01:26.235","Text":"then a_n goes to 0."},{"Start":"01:26.235 ","End":"01:31.030","Text":"The limit as n goes to infinity of a_n."},{"Start":"01:31.040 ","End":"01:36.040","Text":"If this equals 1, we don\u0027t know."},{"Start":"01:36.100 ","End":"01:39.994","Text":"Now, let\u0027s see what happens in our case."},{"Start":"01:39.994 ","End":"01:42.635","Text":"Let\u0027s see what is,"},{"Start":"01:42.635 ","End":"01:44.531","Text":"before I take the limit of it,"},{"Start":"01:44.531 ","End":"01:48.590","Text":"what is n plus 1 over a_n?"},{"Start":"01:48.590 ","End":"01:54.525","Text":"We\u0027ll get n plus"},{"Start":"01:54.525 ","End":"02:04.075","Text":"1 factorial over n plus 1 to the power of n plus 1."},{"Start":"02:04.075 ","End":"02:11.690","Text":"All this over n factorial over n to the power of"},{"Start":"02:11.690 ","End":"02:15.110","Text":"n. I\u0027m going to rearrange"},{"Start":"02:15.110 ","End":"02:22.400","Text":"this numerator but inverted."},{"Start":"02:22.400 ","End":"02:26.255","Text":"Also, I\u0027m going to collect them in the right order that I need them."},{"Start":"02:26.255 ","End":"02:35.520","Text":"I want to put n plus 1 factorial next to the n factorial."},{"Start":"02:35.520 ","End":"02:39.769","Text":"I want to put this n to the n"},{"Start":"02:39.769 ","End":"02:47.890","Text":"opposite n plus 1 to the power of n plus 1."},{"Start":"02:49.720 ","End":"02:52.685","Text":"Now I\u0027m going to simplify this."},{"Start":"02:52.685 ","End":"03:03.380","Text":"There\u0027s a rule with factorials that n plus 1 factorial is n plus 1 times n factorial."},{"Start":"03:03.380 ","End":"03:05.914","Text":"I\u0027ll explain that in a moment."},{"Start":"03:05.914 ","End":"03:15.990","Text":"Over n factorial times n to the power of n over,"},{"Start":"03:15.990 ","End":"03:17.835","Text":"now if we go to the n plus 1."},{"Start":"03:17.835 ","End":"03:21.350","Text":"I\u0027ll take n plus 1 just to the power of n,"},{"Start":"03:21.350 ","End":"03:24.950","Text":"and I\u0027ll throw in that extra factor."},{"Start":"03:24.950 ","End":"03:27.785","Text":"Now I said I\u0027d explain about this."},{"Start":"03:27.785 ","End":"03:38.220","Text":"Given numerical example, 5 factorial is 5 times 4 times 3 times 2 times 1."},{"Start":"03:38.270 ","End":"03:42.450","Text":"This is equal to 5 times,"},{"Start":"03:42.450 ","End":"03:44.040","Text":"and I can put it into brackets,"},{"Start":"03:44.040 ","End":"03:47.460","Text":"4 times 3 times 2 times 1,"},{"Start":"03:47.460 ","End":"03:49.260","Text":"which is 5 times,"},{"Start":"03:49.260 ","End":"03:52.935","Text":"and this is 4 factorial."},{"Start":"03:52.935 ","End":"03:57.930","Text":"5 factorial, you think of like n being 4,"},{"Start":"03:57.930 ","End":"03:59.095","Text":"in this case,"},{"Start":"03:59.095 ","End":"04:06.305","Text":"4 factorial times 4 plus 1 gives me 4 plus 1 factorial."},{"Start":"04:06.305 ","End":"04:15.365","Text":"It\u0027s just peeling off the leading term and that\u0027s how we got from here to here."},{"Start":"04:15.365 ","End":"04:17.915","Text":"Now some things cancel."},{"Start":"04:17.915 ","End":"04:22.085","Text":"N factorial cancels with n factorial,"},{"Start":"04:22.085 ","End":"04:27.600","Text":"and this n plus 1 cancels with this n plus 1."},{"Start":"04:28.790 ","End":"04:37.470","Text":"What we\u0027re left with is this and we have to now take the limit as n goes to infinity."},{"Start":"04:37.490 ","End":"04:45.035","Text":"We want the limit as n goes to infinity of a_n plus 1"},{"Start":"04:45.035 ","End":"04:52.790","Text":"over a_n is the limit as n goes to infinity."},{"Start":"04:52.790 ","End":"04:55.850","Text":"I\u0027m going to do something that might appear strange."},{"Start":"04:55.850 ","End":"04:57.575","Text":"Instead of writing this,"},{"Start":"04:57.575 ","End":"05:04.295","Text":"I\u0027ll write 1 over and I\u0027ll write this upside down."},{"Start":"05:04.295 ","End":"05:09.470","Text":"N plus 1 to the power of n"},{"Start":"05:09.470 ","End":"05:15.170","Text":"over n to the power of n. There\u0027s a reason I\u0027m doing this,"},{"Start":"05:15.170 ","End":"05:18.680","Text":"I\u0027ll tell you where I\u0027m heading to."},{"Start":"05:18.680 ","End":"05:27.320","Text":"There is a famous limit that the limit as n goes to infinity of 1 plus 1"},{"Start":"05:27.320 ","End":"05:35.630","Text":"over n to the power of n is equal to e. If you think about it,"},{"Start":"05:35.630 ","End":"05:44.910","Text":"this bit here is n plus 1 over n. Maybe you can see where I\u0027m heading now."},{"Start":"05:45.790 ","End":"05:53.855","Text":"This equals the limit n goes to infinity of 1 over,"},{"Start":"05:53.855 ","End":"05:59.780","Text":"and this is 1 plus 1 over n to the power"},{"Start":"05:59.780 ","End":"06:05.300","Text":"of n. Because I can take the power of n that\u0027s common to both of them outside."},{"Start":"06:05.300 ","End":"06:12.080","Text":"Then this n plus 1 over n is 1 plus 1 over n. Now,"},{"Start":"06:12.080 ","End":"06:14.345","Text":"if I have the limit of 1 over something,"},{"Start":"06:14.345 ","End":"06:17.519","Text":"I can make it as 1 over the limit of that something,"},{"Start":"06:17.519 ","End":"06:20.875","Text":"so that\u0027s 1 over the limit."},{"Start":"06:20.875 ","End":"06:27.360","Text":"As n goes to infinity of 1 plus 1 over n to the power of"},{"Start":"06:27.360 ","End":"06:37.690","Text":"n. This is equal to 1 over e by this and so,"},{"Start":"06:38.060 ","End":"06:44.649","Text":"we\u0027ve shown that this thing"},{"Start":"06:46.070 ","End":"06:50.745","Text":"in the limit is 1 over e,"},{"Start":"06:50.745 ","End":"06:53.610","Text":"which is less than 1."},{"Start":"06:53.610 ","End":"06:55.470","Text":"Because e is bigger than 1,"},{"Start":"06:55.470 ","End":"06:59.020","Text":"it\u0027s 2.718 or whatever."},{"Start":"06:59.150 ","End":"07:07.370","Text":"We\u0027re in the case now where it\u0027s less than 1 and so a_n goes to 0,"},{"Start":"07:07.370 ","End":"07:10.950","Text":"so our original series goes to 0."},{"Start":"07:12.560 ","End":"07:19.420","Text":"This limit is equal to 0. We\u0027re done."}],"ID":32619},{"Watched":false,"Name":"Exercise 3","Duration":"1m 58s","ChapterTopicVideoID":30484,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":32620},{"Watched":false,"Name":"Exercise 4","Duration":"4m 2s","ChapterTopicVideoID":30485,"CourseChapterTopicPlaylistID":294584,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":32621},{"Watched":false,"Name":"Exercise 5","Duration":"3m 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could be one ormore, and a recurrence relation,"},{"Start":"00:17.010 ","End":"00:18.090","Text":"and we\u0027ll see this in a moment."},{"Start":"00:18.090 ","End":"00:22.470","Text":"In the examples, recursivelydefined sequences are infinite,"},{"Start":"00:23.460 ","End":"00:27.000","Text":"and one of the questions weoften ask about such a sequence."},{"Start":"00:27.389 ","End":"00:29.130","Text":"Is, does it converge?"},{"Start":"00:29.160 ","End":"00:31.290","Text":"And if so, find its limit."},{"Start":"00:31.529 ","End":"00:33.690","Text":"Okay, that\u0027s the preface."},{"Start":"00:33.870 ","End":"00:35.519","Text":"Let\u0027s start with the examples."},{"Start":"00:35.580 ","End":"00:39.120","Text":"We\u0027ll define a sequenced,an recursively as follows."},{"Start":"00:39.489 ","End":"00:43.620","Text":"Remember, there are two things we haveto give the initial condition, and"},{"Start":"00:43.625 ","End":"00:49.769","Text":"that will be that a one is one andthe recurrence relation were given."},{"Start":"00:50.565 ","End":"00:53.175","Text":"An plus one equals two minus a."},{"Start":"00:53.835 ","End":"00:58.455","Text":"What we use this for is anytime weknow an we can use it to find an"},{"Start":"00:58.455 ","End":"01:02.685","Text":"plus one, and then the followinga and the following and so on."},{"Start":"01:03.135 ","End":"01:08.235","Text":"So it\u0027s kind of like an inductionmethod of defining all the terms"},{"Start":"01:08.325 ","End":"01:13.005","Text":"by defining each one in terms ofthe previous one or previous ones."},{"Start":"01:13.185 ","End":"01:16.815","Text":"Now let\u0027s write some of themembers of this sequence."},{"Start":"01:17.415 ","End":"01:19.575","Text":"First one we can get fromthe initial condition."},{"Start":"01:19.575 ","End":"01:21.315","Text":"We know what a one is, it\u0027s one."},{"Start":"01:21.615 ","End":"01:23.385","Text":"Then we\u0027ll use this to find A two."},{"Start":"01:23.385 ","End":"01:28.035","Text":"If you let N equals one, we getthat A two is two minus a one, which"},{"Start":"01:28.035 ","End":"01:29.865","Text":"is two minus one, which is one."},{"Start":"01:29.985 ","End":"01:34.305","Text":"And then if we let N equals two, weget that a three is two minus a two."},{"Start":"01:34.995 ","End":"01:36.675","Text":"And that also happens to be one."},{"Start":"01:37.425 ","End":"01:38.235","Text":"And so on."},{"Start":"01:38.355 ","End":"01:43.965","Text":"A four is one and looks like it\u0027sa constant sequence of all ones."},{"Start":"01:44.115 ","End":"01:48.465","Text":"We could also write this in anon recursive way by just saying"},{"Start":"01:48.615 ","End":"01:53.565","Text":"A is equal to one for all N oran identically equal to one."},{"Start":"01:53.685 ","End":"01:57.705","Text":"Might just add that the alternativeto defining a sequence recursively."},{"Start":"01:58.245 ","End":"02:02.595","Text":"Is by giving a formula a Nis some function of N here."},{"Start":"02:02.595 ","End":"02:04.035","Text":"It\u0027s the constant function."},{"Start":"02:04.275 ","End":"02:06.555","Text":"Okay, now let\u0027s go to example two."},{"Start":"02:06.615 ","End":"02:09.225","Text":"I chose this one because it\u0027s so famous."},{"Start":"02:09.845 ","End":"02:12.495","Text":"Everyone\u0027s heard of the fiat sequence."},{"Start":"02:12.645 ","End":"02:15.405","Text":"Usually we just describe it asfollows, not mathematically."},{"Start":"02:15.405 ","End":"02:21.165","Text":"We start with one and one, and then we sayone plus one is two, and one plus two is."},{"Start":"02:22.350 ","End":"02:24.450","Text":"Two plus three is five, and so on."},{"Start":"02:24.450 ","End":"02:28.830","Text":"Each time we take the last two,add them and get a new member."},{"Start":"02:29.070 ","End":"02:33.000","Text":"Let\u0027s call it FN insteadof a n F for fiat."},{"Start":"02:33.480 ","End":"02:38.220","Text":"And the recursive definitionparallels what we just said in words."},{"Start":"02:39.030 ","End":"02:41.460","Text":"We start with one and one."},{"Start":"02:41.640 ","End":"02:45.990","Text":"That\u0027s the first and secondmember, and then each following"},{"Start":"02:45.990 ","End":"02:51.015","Text":"member, Is the sum of the twoprevious, or could say it this way."},{"Start":"02:51.525 ","End":"02:56.445","Text":"The member in the n place plus then plus first place gives us the"},{"Start":"02:56.445 ","End":"03:00.825","Text":"one in the N plus second place, andwe have two consecutive indices."},{"Start":"03:00.829 ","End":"03:02.685","Text":"We can get the following index."},{"Start":"03:02.745 ","End":"03:06.375","Text":"We\u0027re adding these twomembers, like the three plus."},{"Start":"03:06.375 ","End":"03:10.785","Text":"The five is the eight, Ffour plus F five is F six."},{"Start":"03:11.865 ","End":"03:13.605","Text":"We have two initial conditions."},{"Start":"03:13.605 ","End":"03:15.585","Text":"F one is one, and F two is one."},{"Start":"03:16.230 ","End":"03:17.730","Text":"Now we start applying this."},{"Start":"03:17.730 ","End":"03:20.190","Text":"If N is one, we get F one plus."},{"Start":"03:20.195 ","End":"03:24.780","Text":"F two is F three, or F three is F oneplus F two, so that\u0027s one plus one is two."},{"Start":"03:25.230 ","End":"03:27.720","Text":"Then F four is F two plus F three."},{"Start":"03:28.140 ","End":"03:33.750","Text":"So one plus two is three, andthen two plus three is five."},{"Start":"03:33.780 ","End":"03:34.980","Text":"Three plus five is eight."},{"Start":"03:36.375 ","End":"03:42.015","Text":"That gives us the beginning partup to eight of the fiat sequence."},{"Start":"03:42.165 ","End":"03:47.834","Text":"If we ask about convergence ordivergence, this actually diverges"},{"Start":"03:47.834 ","End":"03:53.265","Text":"to infinity because whenever youhave a strictly increasing integer"},{"Start":"03:53.269 ","End":"03:56.415","Text":"sequence, it always goes to infinity."},{"Start":"03:56.655 ","End":"03:59.595","Text":"It\u0027s important that it\u0027s aninteger sequence, cuz in general"},{"Start":"03:59.595 ","End":"04:01.234","Text":"we can have increasing sequence."},{"Start":"04:01.710 ","End":"04:05.880","Text":"Don\u0027t go to infinity, and just bythe way, not important, just for"},{"Start":"04:06.380 ","End":"04:11.970","Text":"interest\u0027s sake, there is actuallya formula, a function that gives you"},{"Start":"04:12.359 ","End":"04:19.320","Text":"straight away the nth member of thefiat sequence, this complicated thing."},{"Start":"04:20.160 ","End":"04:27.900","Text":"So theoretically, if we put N equal sixhere, we should get F six, which is eight."},{"Start":"04:28.050 ","End":"04:29.160","Text":"I\u0027m not gonna do it."},{"Start":"04:29.250 ","End":"04:30.600","Text":"It\u0027s just for interest sake."},{"Start":"04:31.050 ","End":"04:36.810","Text":"Now, a third example, we\u0027re gonnadefine a recursive sequence as follows."},{"Start":"04:36.810 ","End":"04:39.420","Text":"The initial conditionwill be that A one is one."},{"Start":"04:40.109 ","End":"04:46.590","Text":"And the current relation will be thatan plus one is one plus one over a."},{"Start":"04:47.219 ","End":"04:50.369","Text":"So let\u0027s start building our sequence."},{"Start":"04:50.669 ","End":"04:55.739","Text":"We start off with a one equals one,the initial, then put N equals one"},{"Start":"04:55.739 ","End":"05:00.570","Text":"here, and we get a two is one plusone over a, one that comes out two."},{"Start":"05:01.380 ","End":"05:05.659","Text":"Then we get a three is one plusone over a two, which we get from."},{"Start":"05:06.510 ","End":"05:07.950","Text":"Comes out three over two."},{"Start":"05:08.310 ","End":"05:12.780","Text":"An A four is one plus oneover a three from here."},{"Start":"05:13.650 ","End":"05:16.620","Text":"That\u0027s five over three, andthen we get eight over five."},{"Start":"05:17.370 ","End":"05:22.800","Text":"If you look at it, denominatorsare actually the fiat sequence."},{"Start":"05:22.860 ","End":"05:27.320","Text":"The numerator are just thefiat sequence moved by one."},{"Start":"05:28.185 ","End":"05:33.224","Text":"This is the ratio of consecutivemembers of the Fiac sequence."},{"Start":"05:33.344 ","End":"05:37.005","Text":"And that\u0027s just, by the way,let\u0027s talk about convergence."},{"Start":"05:37.245 ","End":"05:42.525","Text":"It does converge, but it\u0027s a lotof work to prove that it converges."},{"Start":"05:42.525 ","End":"05:43.844","Text":"So let\u0027s do the following."},{"Start":"05:44.265 ","End":"05:47.174","Text":"I\u0027ll give you the fact that AAN converges."},{"Start":"05:47.550 ","End":"05:49.200","Text":"We\u0027ll call the limit L."},{"Start":"05:49.500 ","End":"05:52.080","Text":"Our task is just to find what L is."},{"Start":"05:52.080 ","End":"05:54.600","Text":"We\u0027ll skip the part aboutproving that it converges."},{"Start":"05:54.600 ","End":"05:58.440","Text":"We\u0027ll just find out what it convergesto, assuming that it does, and"},{"Start":"05:58.440 ","End":"06:02.940","Text":"there\u0027s a standard trick thatis used for recursive sequences."},{"Start":"06:03.090 ","End":"06:03.870","Text":"The limit."},{"Start":"06:04.455 ","End":"06:09.344","Text":"Aen goes to infinity of A,is the same as the li limit."},{"Start":"06:09.525 ","End":"06:12.284","Text":"Aen goes to infinity of an plus one."},{"Start":"06:12.645 ","End":"06:16.724","Text":"If you move a sequence by one, ifyou start with the second term and"},{"Start":"06:16.724 ","End":"06:19.125","Text":"onwards, it doesn\u0027t make any difference."},{"Start":"06:19.185 ","End":"06:25.725","Text":"So using that fact and using therecurrence relationship for an plus one,"},{"Start":"06:26.055 ","End":"06:28.475","Text":"this is the limit of one plus one over an."},{"Start":"06:29.250 ","End":"06:31.800","Text":"The sum of limits is limit of the sum."},{"Start":"06:31.950 ","End":"06:34.890","Text":"The limit of a reciprocal isthe reciprocal of the limit."},{"Start":"06:35.190 ","End":"06:38.550","Text":"Here this is L, and here this is L."},{"Start":"06:38.760 ","End":"06:41.970","Text":"So we get L equals one plus one over L."},{"Start":"06:42.610 ","End":"06:46.410","Text":"Multiply both sides and rearrange,and we get the quadratic equation."},{"Start":"06:46.860 ","End":"06:49.980","Text":"L squared minus L minus one equals zero."},{"Start":"06:51.075 ","End":"06:56.715","Text":"The solution to the quadratic usingthe formula comes out to be one plus"},{"Start":"06:56.715 ","End":"06:59.115","Text":"or minus square root to five over two."},{"Start":"06:59.235 ","End":"07:03.375","Text":"Now we\u0027re only working withpositive numbers here, and"},{"Start":"07:03.375 ","End":"07:04.905","Text":"one of these is negative."},{"Start":"07:05.145 ","End":"07:06.974","Text":"The limit is not gonna be negative."},{"Start":"07:06.974 ","End":"07:08.115","Text":"We\u0027ll take the plus."},{"Start":"07:08.585 ","End":"07:12.705","Text":"So the limit is one plusroute five over two."},{"Start":"07:12.854 ","End":"07:19.094","Text":"This actually has a name, thisis called Upper Greek Phi."},{"Start":"07:19.935 ","End":"07:27.615","Text":"Which is also the golden ratio or goldensection and is roughly equal to 1.618."},{"Start":"07:28.005 ","End":"07:31.575","Text":"One last example, wellactually it\u0027s two in one."},{"Start":"07:32.175 ","End":"07:35.805","Text":"We define a recursivesequence X 10 as follows."},{"Start":"07:36.435 ","End":"07:41.115","Text":"The initial condition is the next oneis one, and the recurrence relation is"},{"Start":"07:41.115 ","End":"07:44.885","Text":"that each XN plus one is a times xn."},{"Start":"07:44.890 ","End":"07:48.255","Text":"You get the following member bymultiplying the current member by."},{"Start":"07:49.260 ","End":"07:52.950","Text":"But I haven\u0027t given you what Ais and I said it\u0027s two in one."},{"Start":"07:52.950 ","End":"07:59.070","Text":"So we\u0027ll take once A equals two and oncea equals minus one and see what happens."},{"Start":"07:59.190 ","End":"08:04.110","Text":"Well, first we\u0027ll do the common partregardless of what A is X one is one"},{"Start":"08:04.320 ","End":"08:09.450","Text":"x two, we multiply by A, so it\u0027s alet we multiply by a again, we get a."},{"Start":"08:10.470 ","End":"08:15.110","Text":"And then a cubed, and then a to thefourth sequence starts out one A,"},{"Start":"08:15.115 ","End":"08:16.230","Text":"a squared, a cube, A to the fourth."},{"Start":"08:16.230 ","End":"08:20.640","Text":"It\u0027s a geometric sequence withfirst term one and common ratio A."},{"Start":"08:20.790 ","End":"08:24.360","Text":"It\u0027s another one of those cases wherewe can actually write the nth term"},{"Start":"08:24.630 ","End":"08:26.610","Text":"immediately without a recursion."},{"Start":"08:26.940 ","End":"08:30.780","Text":"We know that X N is A, to the N minus one."},{"Start":"08:30.990 ","End":"08:32.100","Text":"You can just see it from here."},{"Start":"08:32.100 ","End":"08:35.069","Text":"This is the fifth element,and it\u0027s A to the four."},{"Start":"08:35.075 ","End":"08:36.539","Text":"Cause it\u0027s five minus one."},{"Start":"08:36.720 ","End":"08:40.350","Text":"That\u0027s just again, by the way,now it\u0027s time to split into two."},{"Start":"08:40.830 ","End":"08:44.039","Text":"So we take A equals two,and then A equals minus one."},{"Start":"08:44.310 ","End":"08:51.540","Text":"If A equals two, then this comes out to be1, 2, 4, 8, 16 each time multiplying by."},{"Start":"08:52.650 ","End":"08:56.790","Text":"And if A is minus one, again,we start with one and multiply"},{"Start":"08:56.795 ","End":"09:01.680","Text":"by minus one each time, so itoscillates one minus one, and so on."},{"Start":"09:02.699 ","End":"09:06.689","Text":"As for convergence, theyboth diverge but differently."},{"Start":"09:07.680 ","End":"09:11.969","Text":"This is like we had above a strictlyincreasing integer sequence."},{"Start":"09:12.360 ","End":"09:15.990","Text":"So it tends to infinityor diverges to infinity."},{"Start":"09:16.199 ","End":"09:21.630","Text":"This one doesn\u0027t have any limit, and itdoesn\u0027t tend to plus or minus infinity."},{"Start":"09:21.635 ","End":"09:23.970","Text":"It just oscillatesbetween plus or minus one."},{"Start":"09:24.209 ","End":"09:26.069","Text":"It diverges and that\u0027s it."},{"Start":"09:26.550 ","End":"09:29.850","Text":"And by the way, it can sometimes converge."},{"Start":"09:30.000 ","End":"09:35.100","Text":"Depends on the value of A, it turnsout that if A is strictly bigger than"},{"Start":"09:35.100 ","End":"09:37.980","Text":"minus one and less than or equal to one."},{"Start":"09:38.610 ","End":"09:42.390","Text":"Then the sequence convergesand otherwise it diverges."},{"Start":"09:43.200 ","End":"09:47.730","Text":"Okay, with that, we\u0027ll endthis introductory tutorial."},{"Start":"09:48.180 ","End":"09:51.490","Text":"There are severalexercises following this."}],"ID":32607},{"Watched":false,"Name":"Exercise 1","Duration":"10m 8s","ChapterTopicVideoID":29618,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.690","Text":"In this exercise,"},{"Start":"00:01.690 ","End":"00:05.700","Text":"we have a sequence that\u0027s defined recursively."},{"Start":"00:05.700 ","End":"00:12.140","Text":"Remember, recursively means that we\u0027re given some starting conditions"},{"Start":"00:12.140 ","End":"00:18.515","Text":"and then we define each successive term in terms of the previous one,"},{"Start":"00:18.515 ","End":"00:22.480","Text":"or sometimes by several previous ones."},{"Start":"00:23.140 ","End":"00:27.400","Text":"We have to show that this converges and to find its limit."},{"Start":"00:27.400 ","End":"00:31.085","Text":"Let\u0027s, first of all, see what\u0027s going on with this recursion."},{"Start":"00:31.085 ","End":"00:34.890","Text":"We\u0027re told that a_1 is equal to 1."},{"Start":"00:34.890 ","End":"00:36.720","Text":"We want to find a few elements."},{"Start":"00:36.720 ","End":"00:38.615","Text":"What would a_2 be?"},{"Start":"00:38.615 ","End":"00:40.580","Text":"Well, to find a_2,"},{"Start":"00:40.580 ","End":"00:42.190","Text":"we let n equal 1 here,"},{"Start":"00:42.190 ","End":"00:47.795","Text":"and we see that this is equal to the square root of 2 plus a_1,"},{"Start":"00:47.795 ","End":"00:50.115","Text":"but we have a_1,"},{"Start":"00:50.115 ","End":"00:59.620","Text":"and this is equal to the square root of 2 plus square root of 2."},{"Start":"01:00.560 ","End":"01:07.075","Text":"Then a_3, we get by letting n equals 2 here."},{"Start":"01:07.075 ","End":"01:13.950","Text":"We get that this is the square root of 2 plus a_2,"},{"Start":"01:13.950 ","End":"01:16.050","Text":"but a_2 we have here,"},{"Start":"01:16.050 ","End":"01:21.315","Text":"so it\u0027s the square root of 2 plus the square root of 2,"},{"Start":"01:21.315 ","End":"01:25.630","Text":"plus the square root of 2, and so on."},{"Start":"01:25.630 ","End":"01:29.550","Text":"We just keep getting these nested square roots of 2."},{"Start":"01:31.490 ","End":"01:36.460","Text":"Back to our problem to show that it converted them to find its limit."},{"Start":"01:36.460 ","End":"01:38.150","Text":"I\u0027m going to do it in reverse order."},{"Start":"01:38.150 ","End":"01:39.320","Text":"I\u0027m going to, first of all,"},{"Start":"01:39.320 ","End":"01:42.580","Text":"assume it converges and then find its limit."},{"Start":"01:42.580 ","End":"01:44.540","Text":"That\u0027ll be the easy part."},{"Start":"01:44.540 ","End":"01:49.690","Text":"The hard part is to show that it converges or not hard but harder."},{"Start":"01:49.690 ","End":"01:55.700","Text":"Let\u0027s assume that a_n converges"},{"Start":"01:55.700 ","End":"02:02.050","Text":"and it converges to some limit L as n goes to infinity."},{"Start":"02:02.570 ","End":"02:07.045","Text":"Let\u0027s take the recursion relation, this one,"},{"Start":"02:07.045 ","End":"02:11.790","Text":"a_n plus 1 equals,"},{"Start":"02:11.790 ","End":"02:13.730","Text":"and I\u0027m leaving a space here,"},{"Start":"02:13.730 ","End":"02:18.400","Text":"the square root of 2 plus a_n."},{"Start":"02:18.400 ","End":"02:22.535","Text":"Now I\u0027m going to put the limit in front of each one of them."},{"Start":"02:22.535 ","End":"02:29.840","Text":"Take the limit as n goes to infinity equals the limit as n goes to infinity."},{"Start":"02:30.360 ","End":"02:35.700","Text":"Now, the limit as n goes to infinity of a_n plus"},{"Start":"02:35.700 ","End":"02:43.085","Text":"1 is just the same as the limit as n goes to infinity of a_n."},{"Start":"02:43.085 ","End":"02:45.700","Text":"This is just starting one term later,"},{"Start":"02:45.700 ","End":"02:49.205","Text":"but it\u0027s still the same sequence."},{"Start":"02:49.205 ","End":"02:53.115","Text":"Here, I can put the limit under the square root,"},{"Start":"02:53.115 ","End":"02:57.035","Text":"so I\u0027ve got the square root of 2 plus"},{"Start":"02:57.035 ","End":"03:04.050","Text":"the limit of a_n as n goes to infinity."},{"Start":"03:04.050 ","End":"03:06.140","Text":"Now, this limit,"},{"Start":"03:06.140 ","End":"03:07.950","Text":"we\u0027ve assumed that it\u0027s L,"},{"Start":"03:07.950 ","End":"03:10.005","Text":"so we have an equation for L,"},{"Start":"03:10.005 ","End":"03:14.300","Text":"that L equals the square root of 2 plus"},{"Start":"03:14.300 ","End":"03:19.360","Text":"L. Now let\u0027s solve this equation now go over here."},{"Start":"03:19.360 ","End":"03:29.380","Text":"Square both sides and we get L^2 equals 2 plus L. That gives us a quadratic equation."},{"Start":"03:29.380 ","End":"03:32.310","Text":"Sorry, this is an equals."},{"Start":"03:33.420 ","End":"03:38.820","Text":"L^2 minus L minus 2 equals 0,"},{"Start":"03:38.820 ","End":"03:49.350","Text":"so L equals using the formula minus b plus or minus the square root of b^2 minus 4ac,"},{"Start":"03:49.350 ","End":"03:51.995","Text":"4 times 1 times 2,"},{"Start":"03:51.995 ","End":"03:54.565","Text":"all of these over 2a."},{"Start":"03:54.565 ","End":"03:56.600","Text":"What we get,"},{"Start":"03:56.600 ","End":"04:02.680","Text":"4 times 2 is 8 plus 1 is 9."},{"Start":"04:02.680 ","End":"04:07.570","Text":"Square root of 9 is 3,"},{"Start":"04:07.640 ","End":"04:12.655","Text":"1 plus or minus 3 over 2."},{"Start":"04:12.655 ","End":"04:14.885","Text":"If we take the plus,"},{"Start":"04:14.885 ","End":"04:17.060","Text":"we get 4 over 2 is 2."},{"Start":"04:17.060 ","End":"04:18.715","Text":"If we take the minus,"},{"Start":"04:18.715 ","End":"04:21.145","Text":"we get minus 1."},{"Start":"04:21.145 ","End":"04:27.455","Text":"Now, this is a positive sequence."},{"Start":"04:27.455 ","End":"04:30.070","Text":"It\u0027s easy to see that everything\u0027s positive."},{"Start":"04:30.070 ","End":"04:31.805","Text":"The first one is positive."},{"Start":"04:31.805 ","End":"04:35.060","Text":"If I take the square root of something positive,"},{"Start":"04:35.060 ","End":"04:36.160","Text":"it\u0027ll stay positive,"},{"Start":"04:36.160 ","End":"04:38.080","Text":"so every term is positive."},{"Start":"04:38.080 ","End":"04:41.025","Text":"I can rule this one out."},{"Start":"04:41.025 ","End":"04:46.525","Text":"We can say that the limit is equal to 2."},{"Start":"04:46.525 ","End":"04:55.380","Text":"Or in other words, the limit of a_n as n goes to infinity is 2."},{"Start":"04:55.380 ","End":"04:58.415","Text":"That\u0027s the easy part."},{"Start":"04:58.415 ","End":"05:05.240","Text":"Now we have to show that the sequence a_n converges."},{"Start":"05:05.240 ","End":"05:11.060","Text":"I\u0027m going to show that a_n converges."},{"Start":"05:12.880 ","End":"05:17.480","Text":"I\u0027ll show it by showing 2 things."},{"Start":"05:17.480 ","End":"05:22.250","Text":"That a_n is increasing,"},{"Start":"05:22.250 ","End":"05:29.110","Text":"and a_n is bounded."},{"Start":"05:30.330 ","End":"05:37.160","Text":"I only have to show that it\u0027s bounded from above if it\u0027s increasing."},{"Start":"05:37.170 ","End":"05:42.420","Text":"It is bounded from below also by 0 because it\u0027s positive series,"},{"Start":"05:42.420 ","End":"05:46.800","Text":"but increasing and bounded from above implies that it converges."},{"Start":"05:46.800 ","End":"05:52.090","Text":"We\u0027re going to show this by induction."},{"Start":"05:53.240 ","End":"06:00.470","Text":"We\u0027ll show each of the parts that it\u0027s increasing and that it\u0027s bounded by induction."},{"Start":"06:00.920 ","End":"06:04.100","Text":"First, we\u0027ll prove this,"},{"Start":"06:04.100 ","End":"06:06.100","Text":"that a_n is bounded from above."},{"Start":"06:06.100 ","End":"06:16.260","Text":"In fact, I\u0027m going to prove that a_n is less than or equal to 2 for every n. Then,"},{"Start":"06:16.260 ","End":"06:19.239","Text":"we\u0027ll prove that a_n is increasing."},{"Start":"06:19.239 ","End":"06:27.300","Text":"That is, we\u0027ll show that a_n is less than or equal to a_n plus 1 for all n. As I said,"},{"Start":"06:27.300 ","End":"06:29.195","Text":"we\u0027ll prove both of them by induction,"},{"Start":"06:29.195 ","End":"06:34.230","Text":"but we\u0027ll start with the bounded from above."},{"Start":"06:34.310 ","End":"06:37.200","Text":"We prove this one first."},{"Start":"06:37.200 ","End":"06:39.030","Text":"Let\u0027s start with n equals 1."},{"Start":"06:39.030 ","End":"06:41.105","Text":"I said we\u0027re doing it by induction."},{"Start":"06:41.105 ","End":"06:51.375","Text":"a_1 is equal to square root of 2,"},{"Start":"06:51.375 ","End":"06:56.350","Text":"and certainly, the square root of 2 is less than or equal to 2."},{"Start":"06:56.350 ","End":"06:58.390","Text":"This is 1.4 or something,"},{"Start":"06:58.390 ","End":"07:00.985","Text":"it\u0027s quite a bit less than 2."},{"Start":"07:00.985 ","End":"07:10.120","Text":"Now, the induction step is that we assume that a_n is less than or equal to 2."},{"Start":"07:10.120 ","End":"07:12.970","Text":"We have to show that if this is true,"},{"Start":"07:12.970 ","End":"07:16.700","Text":"then a_n plus 1 is also less than or equal to 2."},{"Start":"07:16.700 ","End":"07:19.030","Text":"That\u0027s the induction step."},{"Start":"07:19.040 ","End":"07:22.080","Text":"Let\u0027s assume this."},{"Start":"07:22.080 ","End":"07:24.555","Text":"This is like now given."},{"Start":"07:24.555 ","End":"07:29.435","Text":"So a_n plus 1 is equal to,"},{"Start":"07:29.435 ","End":"07:31.365","Text":"by the recursion rule,"},{"Start":"07:31.365 ","End":"07:36.175","Text":"the square root of 2 plus a_n,"},{"Start":"07:36.175 ","End":"07:39.475","Text":"and this is less than or equal to,"},{"Start":"07:39.475 ","End":"07:43.320","Text":"if I replace a_n by something bigger or equal to,"},{"Start":"07:43.320 ","End":"07:46.900","Text":"it\u0027s the square root of 2 plus 2,"},{"Start":"07:46.900 ","End":"07:49.095","Text":"because this is less than or equal to this,"},{"Start":"07:49.095 ","End":"07:53.160","Text":"and this equals the square root of 4, which is 2."},{"Start":"07:53.160 ","End":"08:02.428","Text":"Altogether, we have that a_n plus 1 is less than or equal to 2 because of the induction."},{"Start":"08:02.428 ","End":"08:08.430","Text":"Because this is less than or equal to this by the induction hypothesis."},{"Start":"08:08.430 ","End":"08:12.595","Text":"Now, we\u0027ve got the bounded part."},{"Start":"08:12.595 ","End":"08:15.765","Text":"Now let\u0027s do the increasing part."},{"Start":"08:15.765 ","End":"08:18.285","Text":"If we take n equals 1,"},{"Start":"08:18.285 ","End":"08:22.835","Text":"this says that a_1 is less than or equal to a_2."},{"Start":"08:22.835 ","End":"08:26.400","Text":"But we\u0027ve already computed a_1 and a_2,"},{"Start":"08:26.400 ","End":"08:29.265","Text":"a_1 we know is square root of 2,"},{"Start":"08:29.265 ","End":"08:35.095","Text":"and a_2 is the square root of 2 plus square root of 2."},{"Start":"08:35.095 ","End":"08:41.395","Text":"Obviously, a_2 is bigger than a_1 because it\u0027s the square root of something more than 2."},{"Start":"08:41.395 ","End":"08:44.395","Text":"This is okay for n equals 1."},{"Start":"08:44.395 ","End":"08:49.155","Text":"Then, we have to show the induction step that"},{"Start":"08:49.155 ","End":"08:54.755","Text":"if a_n is less than or equal to a_n plus 1,"},{"Start":"08:54.755 ","End":"09:02.355","Text":"then a_n plus 1 is less than or equal to a_n plus 1 plus 1,"},{"Start":"09:02.355 ","End":"09:05.060","Text":"which is n plus 2."},{"Start":"09:05.060 ","End":"09:08.820","Text":"Let\u0027s see how we go about that."},{"Start":"09:10.260 ","End":"09:19.925","Text":"Okay, a_n plus 1 by the recursive definition is equal to the square root of 2 plus a_n."},{"Start":"09:19.925 ","End":"09:23.345","Text":"Now, this is less than or equal to,"},{"Start":"09:23.345 ","End":"09:26.905","Text":"I can use the induction hypothesis."},{"Start":"09:26.905 ","End":"09:28.795","Text":"This is now assumed to be given."},{"Start":"09:28.795 ","End":"09:32.810","Text":"This is less than or equal to the square root of 2 plus,"},{"Start":"09:32.810 ","End":"09:36.370","Text":"instead of a_n, I put a_n plus 1."},{"Start":"09:36.370 ","End":"09:39.400","Text":"Now what is the square root of 2 plus a_n plus 1?"},{"Start":"09:39.400 ","End":"09:41.720","Text":"By the recursive definition,"},{"Start":"09:41.720 ","End":"09:44.395","Text":"with n plus 1 instead of n,"},{"Start":"09:44.395 ","End":"09:48.885","Text":"we get that this is equal to a_n plus 1 plus 1,"},{"Start":"09:48.885 ","End":"09:50.960","Text":"which is a_n plus 2,"},{"Start":"09:50.960 ","End":"09:53.420","Text":"which is what we have here."},{"Start":"09:54.070 ","End":"09:56.950","Text":"That basically concludes the proof."},{"Start":"09:56.950 ","End":"09:59.375","Text":"We\u0027ve shown that a_n is increasing here,"},{"Start":"09:59.375 ","End":"10:09.380","Text":"we\u0027ve shown here that it\u0027s bounded above by 2 and therefore it converges. We\u0027re done."}],"ID":31202},{"Watched":false,"Name":"Exercise 2","Duration":"3m 42s","ChapterTopicVideoID":29619,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"In this exercise, we have a recursively defined sequence."},{"Start":"00:04.515 ","End":"00:09.270","Text":"The first term a_1 is 2 and the recursion relation is that a_n"},{"Start":"00:09.270 ","End":"00:14.130","Text":"plus 1 is equal to the square root of 2a_n minus 1."},{"Start":"00:14.130 ","End":"00:19.905","Text":"We have to show that this sequence a_n converges and also to find the limit."},{"Start":"00:19.905 ","End":"00:22.950","Text":"We\u0027ll start by listing first few terms."},{"Start":"00:22.950 ","End":"00:24.060","Text":"A_1 is 2,"},{"Start":"00:24.060 ","End":"00:27.435","Text":"that\u0027s given and then using this, when n=1,"},{"Start":"00:27.435 ","End":"00:34.320","Text":"we get that a_2 is the square root of 2a_1 minus 1 and that comes out to be about 1.73."},{"Start":"00:34.320 ","End":"00:43.200","Text":"When n=2, we get a_3 in terms of a_2 and the computation gives it at 1.57."},{"Start":"00:43.200 ","End":"00:46.890","Text":"So far it looks like it\u0027s decreasing and in fact,"},{"Start":"00:46.890 ","End":"00:50.705","Text":"we\u0027ll show that it\u0027s decreasing and bounded below."},{"Start":"00:50.705 ","End":"00:52.670","Text":"We\u0027ll look at things backwards for a change,"},{"Start":"00:52.670 ","End":"00:59.000","Text":"let\u0027s first compute the limit assuming that it exists and then we\u0027ll show that it exists."},{"Start":"00:59.000 ","End":"01:02.975","Text":"Let\u0027s say that the limit of a_n is L. Now of course,"},{"Start":"01:02.975 ","End":"01:05.255","Text":"the limit of a_n plus 1 is also L,"},{"Start":"01:05.255 ","End":"01:09.740","Text":"starting the sequence later on doesn\u0027t change the limit."},{"Start":"01:09.740 ","End":"01:11.840","Text":"From the recursion relation,"},{"Start":"01:11.840 ","End":"01:15.845","Text":"we can apply the limit to both sides and get this,"},{"Start":"01:15.845 ","End":"01:18.205","Text":"but the square root is continuous."},{"Start":"01:18.205 ","End":"01:20.570","Text":"So here\u0027s the function 2x minus 1,"},{"Start":"01:20.570 ","End":"01:24.588","Text":"so we can throw the limit inside so we have the following."},{"Start":"01:24.588 ","End":"01:26.660","Text":"Now this is L and this is L,"},{"Start":"01:26.660 ","End":"01:30.075","Text":"so we get L equals square root of 2L minus 1,"},{"Start":"01:30.075 ","End":"01:33.405","Text":"squared both sides L^2 is 12 minus 1."},{"Start":"01:33.405 ","End":"01:36.390","Text":"Bring everything on the left and we have this."},{"Start":"01:36.390 ","End":"01:39.585","Text":"You recognize this as L minus 1^2."},{"Start":"01:39.585 ","End":"01:43.850","Text":"So there\u0027s only one solution and that is that L=1."},{"Start":"01:43.850 ","End":"01:46.445","Text":"If the limit exists and it\u0027s equal to 1."},{"Start":"01:46.445 ","End":"01:48.475","Text":"Now we\u0027ll do what we said earlier,"},{"Start":"01:48.475 ","End":"01:51.410","Text":"we show that the sequence is decreasing and bounded below,"},{"Start":"01:51.410 ","End":"01:56.255","Text":"and this guarantees that it has a limit and therefore the limit is 1, just write that."},{"Start":"01:56.255 ","End":"01:57.890","Text":"It\u0027s convergence of the limit exists,"},{"Start":"01:57.890 ","End":"02:01.525","Text":"and as we showed, if it exists then it must equal 1."},{"Start":"02:01.525 ","End":"02:07.475","Text":"Now we have to show that it\u0027s bounded below and decreasing."},{"Start":"02:07.475 ","End":"02:11.020","Text":"Bounded below will show that a_n is bigger or equal to 1."},{"Start":"02:11.020 ","End":"02:13.600","Text":"By induction, when n is 1,"},{"Start":"02:13.600 ","End":"02:19.010","Text":"this says that a_1 is bigger or equal to 1 which is true because a_1 is 2,"},{"Start":"02:19.010 ","End":"02:23.690","Text":"and the induction step from n to n plus 1 means that we take it as"},{"Start":"02:23.690 ","End":"02:30.825","Text":"given that a_n is bigger than 1 and we have to show that a_n plus 1 is bigger than 1."},{"Start":"02:30.825 ","End":"02:35.140","Text":"So a_n plus 1 is the square root of 2a_n minus 1."},{"Start":"02:35.140 ","End":"02:37.190","Text":"By the induction hypothesis,"},{"Start":"02:37.190 ","End":"02:41.070","Text":"a_n is bigger or equal to 1 and this is equal to 1,"},{"Start":"02:41.070 ","End":"02:44.335","Text":"so a_n plus 1 is bigger or equal to 1."},{"Start":"02:44.335 ","End":"02:46.295","Text":"Now the decreasing part,"},{"Start":"02:46.295 ","End":"02:50.440","Text":"we do this by showing that a_n plus 1 is less than or equal to a_n."},{"Start":"02:50.440 ","End":"02:53.040","Text":"Again by induction, when n=1,"},{"Start":"02:53.040 ","End":"02:55.825","Text":"this says a_2 less than or equal to a_1,"},{"Start":"02:55.825 ","End":"02:59.405","Text":"which is true because the square root of 3 is less than or equal to 2."},{"Start":"02:59.405 ","End":"03:01.430","Text":"Square root of 3, what did we say it was?"},{"Start":"03:01.430 ","End":"03:04.540","Text":"About 1.73, anyway, less than 2."},{"Start":"03:04.540 ","End":"03:07.400","Text":"Now the induction step."},{"Start":"03:07.400 ","End":"03:11.360","Text":"The induction hypothesis is that a_n plus 1 is less than or equal to a_n,"},{"Start":"03:11.360 ","End":"03:15.890","Text":"and we have to show that a_n plus 2 is less than or equal to a_n plus 1."},{"Start":"03:15.890 ","End":"03:21.690","Text":"A_n plus 2 is the square root of twice a_n plus 1 minus 1 and plus"},{"Start":"03:21.690 ","End":"03:27.680","Text":"1 in place of n. Given by the induction hypothesis,"},{"Start":"03:27.680 ","End":"03:30.830","Text":"this is less than a_n and the square root of 2a_n plus"},{"Start":"03:30.830 ","End":"03:34.040","Text":"1 minus 1 is less than or equal to square root of twice a_n minus 1,"},{"Start":"03:34.040 ","End":"03:35.800","Text":"but this is a n plus 1,"},{"Start":"03:35.800 ","End":"03:38.520","Text":"so this is less than or equal to this and that\u0027s what we had to"},{"Start":"03:38.520 ","End":"03:42.790","Text":"show and that actually concludes the proof."}],"ID":31203},{"Watched":false,"Name":"Exercise 3","Duration":"14m 25s","ChapterTopicVideoID":29620,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.335","Text":"In this exercise, we\u0027re given a sequence that\u0027s defined recursively, a_1 is 2."},{"Start":"00:07.335 ","End":"00:12.675","Text":"If we have a_n, we compute a_n plus 1 by this formula,"},{"Start":"00:12.675 ","End":"00:16.050","Text":"1/2 of a_n plus 1 over a_n."},{"Start":"00:16.050 ","End":"00:20.065","Text":"We have to show that a_n converges and to find its limit."},{"Start":"00:20.065 ","End":"00:23.270","Text":"What we\u0027ll do is do the second part."},{"Start":"00:23.270 ","End":"00:26.060","Text":"First, we\u0027ll find the limit assuming that it"},{"Start":"00:26.060 ","End":"00:29.990","Text":"converges just because it\u0027s easier and get it out of the way,"},{"Start":"00:29.990 ","End":"00:32.790","Text":"and then we\u0027ll show that it converges."},{"Start":"00:33.280 ","End":"00:43.490","Text":"Let\u0027s say that a_n converges to some limit L as n goes to infinity."},{"Start":"00:43.490 ","End":"00:47.330","Text":"Then, we\u0027ll find what L is."},{"Start":"00:47.330 ","End":"00:55.100","Text":"Now, the recursion says that a_n plus 1 is"},{"Start":"00:55.100 ","End":"01:02.875","Text":"equal to 1/2 of a_n plus 1 over a_n."},{"Start":"01:02.875 ","End":"01:11.370","Text":"If that\u0027s true, I can write the limit in front of each as n goes to infinity."},{"Start":"01:11.390 ","End":"01:16.700","Text":"Now, the limit of a_n plus 1 is the same as the limit of a_n."},{"Start":"01:16.700 ","End":"01:21.409","Text":"It\u0027s the same sequence just missing the first term."},{"Start":"01:21.409 ","End":"01:24.405","Text":"Just off by 1."},{"Start":"01:24.405 ","End":"01:30.595","Text":"This is equal to the limit of a_n."},{"Start":"01:30.595 ","End":"01:36.500","Text":"Here, I can put the limit inside and say that this is 1/2."},{"Start":"01:36.500 ","End":"01:39.125","Text":"The limit of this plus the limit of this,"},{"Start":"01:39.125 ","End":"01:41.430","Text":"which is the limit of a_n,"},{"Start":"01:41.430 ","End":"01:46.245","Text":"and the limit of 1 over is 1 over the limit of a_n."},{"Start":"01:46.245 ","End":"01:48.120","Text":"N goes to infinity,"},{"Start":"01:48.120 ","End":"01:49.970","Text":"a_n goes to infinity."},{"Start":"01:49.970 ","End":"01:54.995","Text":"Now we can substitute this limit as L. We get the equation that"},{"Start":"01:54.995 ","End":"02:00.385","Text":"L equals 1/2 of L plus 1 over L."},{"Start":"02:00.385 ","End":"02:06.390","Text":"We can multiply by 2L and then we\u0027ll get 2L"},{"Start":"02:06.390 ","End":"02:15.285","Text":"squared is equal to L squared plus 1,"},{"Start":"02:15.285 ","End":"02:22.065","Text":"which means that L squared equals 1."},{"Start":"02:22.065 ","End":"02:26.115","Text":"L is plus or minus 1."},{"Start":"02:26.115 ","End":"02:33.505","Text":"Now, L can\u0027t be minus 1 because the elements of the sequence a_n are all positive."},{"Start":"02:33.505 ","End":"02:35.455","Text":"A_1 is positive,"},{"Start":"02:35.455 ","End":"02:38.510","Text":"and if a_n is positive,"},{"Start":"02:38.510 ","End":"02:40.969","Text":"then this expression is positive."},{"Start":"02:40.969 ","End":"02:42.755","Text":"Actually, this is an induction."},{"Start":"02:42.755 ","End":"02:47.000","Text":"A_1 is positive and if a _n is positive then a_n plus 1 is positive,"},{"Start":"02:47.000 ","End":"02:50.130","Text":"so all the elements are positive and"},{"Start":"02:50.130 ","End":"02:54.060","Text":"so the limit can\u0027t be minus 1 so a_n is bigger than 0."},{"Start":"02:54.060 ","End":"02:59.095","Text":"We just did an informal induction proof of that."},{"Start":"02:59.095 ","End":"03:02.520","Text":"We\u0027ve done part of it that if a_n converges,"},{"Start":"03:02.520 ","End":"03:06.990","Text":"we found its limit and that limit is equal to 1."},{"Start":"03:06.990 ","End":"03:10.230","Text":"Now we have to show that a_n converges."},{"Start":"03:10.230 ","End":"03:15.495","Text":"The way we show that a_n converges is if we show two things,"},{"Start":"03:15.495 ","End":"03:23.120","Text":"that 1, that a_n is decreasing."},{"Start":"03:23.120 ","End":"03:24.619","Text":"Or in other words,"},{"Start":"03:24.619 ","End":"03:31.490","Text":"if I show that each term is bigger or equal to the successive term,"},{"Start":"03:31.490 ","End":"03:33.845","Text":"then that means it\u0027s decreasing."},{"Start":"03:33.845 ","End":"03:41.710","Text":"Secondly, we\u0027ll show that a_n is bounded, bounded below."},{"Start":"03:41.710 ","End":"03:46.080","Text":"In fact, I\u0027ll show that a_n is bigger or equal to"},{"Start":"03:46.080 ","End":"03:50.915","Text":"1 for every n. Decreasing and bounded from below."},{"Start":"03:50.915 ","End":"03:54.820","Text":"There\u0027s a theorem that says that then the sequence converges."},{"Start":"03:54.820 ","End":"03:58.770","Text":"Now we\u0027ll prove 1 and 2."},{"Start":"03:58.770 ","End":"04:01.440","Text":"I\u0027ll start with 1."},{"Start":"04:01.440 ","End":"04:07.590","Text":"We have to show that a_n is decreasing i.e.,"},{"Start":"04:07.590 ","End":"04:12.900","Text":"that a_n is bigger or equal to a_n plus 1 for all n. I\u0027ll"},{"Start":"04:12.900 ","End":"04:18.950","Text":"start with the a_n plus 1 and show that it\u0027s less than or equal to a_n."},{"Start":"04:18.950 ","End":"04:28.190","Text":"Now, this is equal to 1/2 of a_n plus 1 over a_n."},{"Start":"04:28.190 ","End":"04:33.650","Text":"I just realized I need to use the fact that a_n is bigger or equal to 1."},{"Start":"04:33.650 ","End":"04:38.555","Text":"Let me go for 2 first and then I\u0027ll come back to 1."},{"Start":"04:38.555 ","End":"04:42.380","Text":"We need to show that a_n is bigger or equal to 1."},{"Start":"04:42.380 ","End":"04:44.045","Text":"Well, if n is 1,"},{"Start":"04:44.045 ","End":"04:47.730","Text":"a_1 is 2,"},{"Start":"04:47.730 ","End":"04:51.005","Text":"and 2 is certainly bigger or equal to 1."},{"Start":"04:51.005 ","End":"04:55.930","Text":"Everything else can be written as a_n plus 1."},{"Start":"04:55.930 ","End":"04:58.300","Text":"So a_n plus 1."},{"Start":"04:58.300 ","End":"05:00.620","Text":"I\u0027ll show that this is bigger or equal to 1."},{"Start":"05:00.620 ","End":"05:10.530","Text":"It\u0027s equal to 1/2 of a_n plus 1 over a_n."},{"Start":"05:10.990 ","End":"05:16.129","Text":"At this point, I need to introduce a trick."},{"Start":"05:16.129 ","End":"05:19.715","Text":"It\u0027s not so easy otherwise."},{"Start":"05:19.715 ","End":"05:24.440","Text":"Notice that 1/2 of this plus this is the mean."},{"Start":"05:24.440 ","End":"05:26.510","Text":"It\u0027s called the arithmetical mean."},{"Start":"05:26.510 ","End":"05:34.635","Text":"In general, if I have A and B then the arithmetic mean is A plus B over 2."},{"Start":"05:34.635 ","End":"05:36.830","Text":"There is another kind of mean."},{"Start":"05:36.830 ","End":"05:40.970","Text":"I\u0027m assuming A and B are positive and all our a_n\u0027s are positive here."},{"Start":"05:40.970 ","End":"05:45.890","Text":"At the geometric mean is the square root of A times B."},{"Start":"05:45.890 ","End":"05:47.540","Text":"It\u0027s also defined both positive,"},{"Start":"05:47.540 ","End":"05:49.985","Text":"so square root is okay."},{"Start":"05:49.985 ","End":"05:57.920","Text":"The theorem or proposition that\u0027s well known is that the geometric mean"},{"Start":"05:57.920 ","End":"06:01.820","Text":"is less than or equal to the arithmetic mean for"},{"Start":"06:01.820 ","End":"06:06.560","Text":"positive numbers A and B. I\u0027ll just give an example."},{"Start":"06:06.560 ","End":"06:14.595","Text":"Suppose we take that A is 4 and B is 6,"},{"Start":"06:14.595 ","End":"06:21.100","Text":"then A plus B over 2."},{"Start":"06:21.230 ","End":"06:24.495","Text":"Sorry, let\u0027s make this 9."},{"Start":"06:24.495 ","End":"06:31.110","Text":"A plus B over 2 is 4 plus 9 over 2 is 6.5,"},{"Start":"06:31.110 ","End":"06:37.185","Text":"whereas the square root of A times B 4 times 9 is 36,"},{"Start":"06:37.185 ","End":"06:40.080","Text":"square root of 36 is 6,"},{"Start":"06:40.080 ","End":"06:43.760","Text":"and 6 is certainly less than or equal to 6.5."},{"Start":"06:43.760 ","End":"06:46.850","Text":"This is the trick we\u0027re going to use."},{"Start":"06:46.850 ","End":"06:52.910","Text":"Here, the arithmetic mean is bigger or equal to the geometric mean,"},{"Start":"06:52.910 ","End":"07:00.300","Text":"which is the square root of this times this of a_n times 1 over a_n."},{"Start":"07:00.410 ","End":"07:04.110","Text":"This is just equal to 1."},{"Start":"07:04.110 ","End":"07:07.860","Text":"A_n plus 1 is bigger or equal to 1."},{"Start":"07:07.860 ","End":"07:10.144","Text":"Now back here."},{"Start":"07:10.144 ","End":"07:17.510","Text":"Now, I know that a_n is bigger or equal to 1 because we\u0027ve just proven it here."},{"Start":"07:17.700 ","End":"07:24.910","Text":"If I take a number that\u0027s bigger or equal to 1 and I take the reciprocal 1 over a_n,"},{"Start":"07:24.910 ","End":"07:28.225","Text":"then it\u0027s going to be less than or equal to 1."},{"Start":"07:28.225 ","End":"07:30.280","Text":"I am dividing by at least 1,"},{"Start":"07:30.280 ","End":"07:31.945","Text":"so it can only get smaller."},{"Start":"07:31.945 ","End":"07:37.315","Text":"In particular, this and this together imply"},{"Start":"07:37.315 ","End":"07:42.805","Text":"that 1 over a_n is less than or equal to a_n,"},{"Start":"07:42.805 ","End":"07:44.305","Text":"because there\u0027s 1 in the middle."},{"Start":"07:44.305 ","End":"07:46.975","Text":"This is less than or equal to 1, bigger or equal to 1."},{"Start":"07:46.975 ","End":"07:48.985","Text":"This is than or equal to this."},{"Start":"07:48.985 ","End":"07:51.280","Text":"Now if I plug that in here,"},{"Start":"07:51.280 ","End":"07:59.905","Text":"I get that a_n plus 1 is less than or equal to 1/2 of a_n."},{"Start":"07:59.905 ","End":"08:09.565","Text":"I\u0027m replacing this now by a_n because of this and half of a_n plus a_n is equal to a_n."},{"Start":"08:09.565 ","End":"08:14.365","Text":"I\u0027ve shown that a_n plus 1 is less than or equal to a_n."},{"Start":"08:14.365 ","End":"08:17.655","Text":"So a_n is decreasing."},{"Start":"08:17.655 ","End":"08:20.610","Text":"We have that a_n is decreasing and we have"},{"Start":"08:20.610 ","End":"08:24.165","Text":"that a_n is bigger or equal to 1, so it converges."},{"Start":"08:24.165 ","End":"08:26.580","Text":"The first part shows that if it converges,"},{"Start":"08:26.580 ","End":"08:32.785","Text":"then the limit is equal to 1. We\u0027re done."},{"Start":"08:32.785 ","End":"08:34.300","Text":"But don\u0027t go yet,"},{"Start":"08:34.300 ","End":"08:37.944","Text":"I\u0027d like to show you an alternative ending if you\u0027re interested,"},{"Start":"08:37.944 ","End":"08:40.120","Text":"you\u0027re welcome to stay."},{"Start":"08:40.120 ","End":"08:46.630","Text":"Let\u0027s get rid of this and I\u0027d like to remind you that a_n plus"},{"Start":"08:46.630 ","End":"08:53.455","Text":"1 is 1/2 of a_n plus 1 over a_n."},{"Start":"08:53.455 ","End":"09:04.420","Text":"This will equal f(a_n) if I define a function f on the real numbers and positive numbers,"},{"Start":"09:04.420 ","End":"09:11.515","Text":"that f(x) is 1/2 of x plus 1 over x."},{"Start":"09:11.515 ","End":"09:17.815","Text":"Each member of the sequence is f of the previous member,"},{"Start":"09:17.815 ","End":"09:21.340","Text":"a_n plus 1 is f(a_n)."},{"Start":"09:21.340 ","End":"09:25.660","Text":"Let\u0027s see what f\u0027(x) is."},{"Start":"09:25.660 ","End":"09:33.595","Text":"f\u0027(x) is a 1/2 minus 1 over x^2."},{"Start":"09:33.595 ","End":"09:37.510","Text":"I could alternatively write this as 1/2"},{"Start":"09:37.510 ","End":"09:44.605","Text":"of x^2 minus 1 over x^2."},{"Start":"09:44.605 ","End":"09:49.510","Text":"Now, if x is bigger or equal to 1,"},{"Start":"09:49.510 ","End":"09:57.080","Text":"then x^2 minus 1 is going to be bigger or equal to 0."},{"Start":"09:57.570 ","End":"10:02.890","Text":"So f is increasing."},{"Start":"10:02.890 ","End":"10:04.870","Text":"This will be very useful."},{"Start":"10:04.870 ","End":"10:13.150","Text":"Now let\u0027s start on proving that a_n is decreasing and bounded below by 1."},{"Start":"10:13.150 ","End":"10:20.470","Text":"Start with 1 to show that a_n is bigger or equal to a_n plus 1."},{"Start":"10:20.470 ","End":"10:24.535","Text":"We\u0027ll prove this by induction."},{"Start":"10:24.535 ","End":"10:33.490","Text":"If n=1, we have to show that a_1 is bigger or equal to a_2."},{"Start":"10:33.490 ","End":"10:40.390","Text":"Well, a_1 was given as 2 and a_2 is 1/2"},{"Start":"10:40.390 ","End":"10:49.690","Text":"of 2 plus 1 over 2 using the recursion formula and this is 1 and 1/4."},{"Start":"10:49.690 ","End":"10:54.355","Text":"Indeed 2 is bigger than 1 and 1/4."},{"Start":"10:54.355 ","End":"10:56.665","Text":"This is true."},{"Start":"10:56.665 ","End":"11:00.730","Text":"Now, let\u0027s go to the induction phase."},{"Start":"11:00.730 ","End":"11:03.460","Text":"We show that if it\u0027s true for particular n,"},{"Start":"11:03.460 ","End":"11:05.140","Text":"it\u0027s true for n plus 1."},{"Start":"11:05.140 ","End":"11:07.705","Text":"In other words, we have to show that if,"},{"Start":"11:07.705 ","End":"11:12.220","Text":"and this time we\u0027re taking this as given for a particular n,"},{"Start":"11:12.220 ","End":"11:17.485","Text":"this is true, but that implies that is true for n plus 1."},{"Start":"11:17.485 ","End":"11:21.835","Text":"Here we get n plus 1 and here if we put n plus 1,"},{"Start":"11:21.835 ","End":"11:24.190","Text":"we get n plus 2."},{"Start":"11:24.190 ","End":"11:26.830","Text":"This is given, we have to show this."},{"Start":"11:26.830 ","End":"11:29.230","Text":"Well, if this is true,"},{"Start":"11:29.230 ","End":"11:34.465","Text":"I can apply f to both sides because f is increasing."},{"Start":"11:34.465 ","End":"11:36.760","Text":"If f is increasing,"},{"Start":"11:36.760 ","End":"11:40.675","Text":"it means that if x is bigger or equal to y,"},{"Start":"11:40.675 ","End":"11:44.890","Text":"that f(x) is bigger or equal to f(y)"},{"Start":"11:44.890 ","End":"11:50.210","Text":"for a pair of x and y as long as it\u0027s bigger or equal to 1."},{"Start":"11:50.340 ","End":"11:54.700","Text":"Now I realize I should have done Part 2 first."},{"Start":"11:54.700 ","End":"12:00.670","Text":"Just take it on trust that will show Part 2 in a minute."},{"Start":"12:00.670 ","End":"12:03.220","Text":"We\u0027ll base on that,"},{"Start":"12:03.220 ","End":"12:05.170","Text":"that a_n is bigger or equal to 1,"},{"Start":"12:05.170 ","End":"12:07.345","Text":"in which case f is increasing."},{"Start":"12:07.345 ","End":"12:11.230","Text":"We can apply f to both sides and get"},{"Start":"12:11.230 ","End":"12:18.855","Text":"that f(a_n) is bigger or equal to f(a_n) plus 1."},{"Start":"12:18.855 ","End":"12:27.540","Text":"We have this formula that a_n plus 1 is f(a_n)."},{"Start":"12:27.540 ","End":"12:30.570","Text":"So f(a_n) is a_n plus 1."},{"Start":"12:30.570 ","End":"12:31.860","Text":"I think I\u0027ll highlight this."},{"Start":"12:31.860 ","End":"12:33.410","Text":"It\u0027s so important."},{"Start":"12:33.410 ","End":"12:34.990","Text":"Now, if I apply this,"},{"Start":"12:34.990 ","End":"12:38.930","Text":"this is true for any n. If I let n equal n plus 1,"},{"Start":"12:38.930 ","End":"12:48.550","Text":"then it will give me f(a_n) plus 1 is a_n plus 2 and that\u0027s precisely this."},{"Start":"12:48.550 ","End":"12:53.605","Text":"We\u0027ve done the n equals 1 case and we\u0027ve done the induction step."},{"Start":"12:53.605 ","End":"12:56.320","Text":"That proves Part 1."},{"Start":"12:56.320 ","End":"12:59.905","Text":"I\u0027m conditioned that I show you Part 2,"},{"Start":"12:59.905 ","End":"13:08.770","Text":"so 2 to show that a_n is bigger or equal to 1 for all n. Also,"},{"Start":"13:08.770 ","End":"13:12.070","Text":"we\u0027re going to do this by induction."},{"Start":"13:12.070 ","End":"13:14.710","Text":"If n equals 1,"},{"Start":"13:14.710 ","End":"13:17.500","Text":"then we get a_1,"},{"Start":"13:17.500 ","End":"13:21.055","Text":"which is 2 is certainly bigger or equal to 1."},{"Start":"13:21.055 ","End":"13:23.170","Text":"We\u0027re okay for n=1."},{"Start":"13:23.170 ","End":"13:29.425","Text":"Now, we have to show the inductive step that if a_n is bigger or equal to 1,"},{"Start":"13:29.425 ","End":"13:35.020","Text":"then we can derive that a_n plus 1 is also bigger or equal to 1."},{"Start":"13:35.020 ","End":"13:37.480","Text":"If a_n is bigger or equal to 1,"},{"Start":"13:37.480 ","End":"13:40.180","Text":"and also 1 is bigger or equal to 1."},{"Start":"13:40.180 ","End":"13:44.230","Text":"Since f is increasing on x bigger or equal to 1,"},{"Start":"13:44.230 ","End":"13:53.380","Text":"we can apply f to both sides and say that f(a_n) is bigger or equal to f(1)."},{"Start":"13:53.380 ","End":"13:58.330","Text":"Now, f(a_n) is a_n plus 1,"},{"Start":"13:58.330 ","End":"14:06.745","Text":"and f(1) is 1/2 of 1 plus 1 over 1,"},{"Start":"14:06.745 ","End":"14:09.295","Text":"but that is equal to 1."},{"Start":"14:09.295 ","End":"14:12.550","Text":"We\u0027ve shown that a_n plus 1 is bigger or equal to 1,"},{"Start":"14:12.550 ","End":"14:13.930","Text":"which is just this."},{"Start":"14:13.930 ","End":"14:15.310","Text":"We\u0027ve d1 the inductive step,"},{"Start":"14:15.310 ","End":"14:16.825","Text":"we\u0027ve done n equals 1."},{"Start":"14:16.825 ","End":"14:19.510","Text":"That proves 1 and 2."},{"Start":"14:19.510 ","End":"14:21.970","Text":"We should have done 2 before 1,"},{"Start":"14:21.970 ","End":"14:26.570","Text":"but that\u0027s okay. We\u0027re done."}],"ID":31204},{"Watched":false,"Name":"Exercise 4","Duration":"6m 19s","ChapterTopicVideoID":29621,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:05.205","Text":"In this exercise, we a have a recursively defined sequence"},{"Start":"00:05.205 ","End":"00:10.230","Text":"where x_n plus 1 is defined in terms of x_n as in this formula,"},{"Start":"00:10.230 ","End":"00:14.280","Text":"a is some positive constant, and x_1,"},{"Start":"00:14.280 ","End":"00:18.735","Text":"the first term of the sequence is also positive."},{"Start":"00:18.735 ","End":"00:24.300","Text":"We have to prove that the sequence x_n converges to the square root of a."},{"Start":"00:24.300 ","End":"00:28.410","Text":"This is actually an ancient technique for finding the square root,"},{"Start":"00:28.410 ","End":"00:30.750","Text":"x_1 is an initial guess."},{"Start":"00:30.750 ","End":"00:33.300","Text":"Then you keep finding x_2, x_3,"},{"Start":"00:33.300 ","End":"00:37.210","Text":"x_4 closer and closer approximations each time,"},{"Start":"00:37.210 ","End":"00:40.250","Text":"substituting the previous approximation in"},{"Start":"00:40.250 ","End":"00:45.365","Text":"this formula and getting a closer approximation, anyway."},{"Start":"00:45.365 ","End":"00:48.455","Text":"Going to divide the solution up into steps."},{"Start":"00:48.455 ","End":"00:51.515","Text":"In the first step we\u0027ll prove that x_n is"},{"Start":"00:51.515 ","End":"00:55.445","Text":"positive for every n. We\u0027ll do this by induction."},{"Start":"00:55.445 ","End":"00:58.210","Text":"For n=1, it\u0027s just given."},{"Start":"00:58.210 ","End":"01:02.255","Text":"It just says that X_1 is bigger than 0 and we\u0027re given that."},{"Start":"01:02.255 ","End":"01:08.165","Text":"We need the inductive step that if it\u0027s true for n,"},{"Start":"01:08.165 ","End":"01:10.580","Text":"it\u0027s also true for n plus 1."},{"Start":"01:10.580 ","End":"01:14.689","Text":"What this means is that if x_n is bigger than 0,"},{"Start":"01:14.689 ","End":"01:18.295","Text":"then x_n plus 1 is bigger than 0."},{"Start":"01:18.295 ","End":"01:21.605","Text":"Recall, that a is bigger than 0,"},{"Start":"01:21.605 ","End":"01:23.165","Text":"a over x_n,"},{"Start":"01:23.165 ","End":"01:27.110","Text":"this part here is positive,"},{"Start":"01:27.110 ","End":"01:30.635","Text":"it\u0027s a positive over a positive, so positive."},{"Start":"01:30.635 ","End":"01:36.815","Text":"So x_n plus 1 is a 1/2 of something positive plus something positive,"},{"Start":"01:36.815 ","End":"01:39.500","Text":"so it\u0027s also positive."},{"Start":"01:39.500 ","End":"01:42.565","Text":"Rephrasing that x_n plus 1 is bigger than 0."},{"Start":"01:42.565 ","End":"01:45.795","Text":"That\u0027s the induction step, and it\u0027s okay."},{"Start":"01:45.795 ","End":"01:48.545","Text":"We\u0027ve proven this by induction."},{"Start":"01:48.545 ","End":"01:51.490","Text":"Now go on to step 2."},{"Start":"01:51.490 ","End":"01:57.705","Text":"We\u0027ll prove that all the x_n are bigger or equal to square root of a, well,"},{"Start":"01:57.705 ","End":"02:00.030","Text":"not all the x_n but from 2 onwards,"},{"Start":"02:00.030 ","End":"02:02.750","Text":"which won\u0027t make any difference because for limits,"},{"Start":"02:02.750 ","End":"02:04.520","Text":"the head of a sequence is not important,"},{"Start":"02:04.520 ","End":"02:06.440","Text":"only the tail is important."},{"Start":"02:06.440 ","End":"02:16.535","Text":"We start with the recursion formula and apply the AM-GM inequality to this."},{"Start":"02:16.535 ","End":"02:20.975","Text":"We get the square root of x_n times a over x_n,"},{"Start":"02:20.975 ","End":"02:22.100","Text":"the x_n cancels,"},{"Start":"02:22.100 ","End":"02:24.035","Text":"this as the square root of a."},{"Start":"02:24.035 ","End":"02:26.975","Text":"To remind you, the AM-GM inequality,"},{"Start":"02:26.975 ","End":"02:29.045","Text":"Arithmetic Mean, Geometric Mean,"},{"Start":"02:29.045 ","End":"02:34.430","Text":"says that the arithmetic mean is bigger or equal to the geometric mean."},{"Start":"02:34.430 ","End":"02:39.780","Text":"In fact, we can also say that it\u0027s strictly bigger than if a is not equal to b."},{"Start":"02:39.780 ","End":"02:45.710","Text":"We have x_n plus 1 is bigger or equal to square root of a."},{"Start":"02:45.710 ","End":"02:48.370","Text":"That\u0027s for n=1, 2, 3, etc."},{"Start":"02:48.370 ","End":"02:50.300","Text":"By changing the index,"},{"Start":"02:50.300 ","End":"02:55.940","Text":"we can say that x_n is bigger or equal to a for n=2, 3, 4."},{"Start":"02:55.940 ","End":"02:59.770","Text":"That\u0027s why we have this bigger or equal to 2 to be precise."},{"Start":"02:59.770 ","End":"03:01.820","Text":"Now in step 3,"},{"Start":"03:01.820 ","End":"03:05.420","Text":"we\u0027re going to show that the sequence x_n is decreasing."},{"Start":"03:05.420 ","End":"03:06.874","Text":"Again, to be precise,"},{"Start":"03:06.874 ","End":"03:10.010","Text":"I wrote that x_n is only from 2 to infinity."},{"Start":"03:10.010 ","End":"03:16.940","Text":"Decreasing means that each subsequent term is less than or equal to the current term."},{"Start":"03:16.940 ","End":"03:25.130","Text":"We usually show this by saying that this minus this is less than or equal to 0."},{"Start":"03:25.130 ","End":"03:28.890","Text":"Let\u0027s start with x_n plus 1 minus x_n."},{"Start":"03:30.080 ","End":"03:36.635","Text":"The first term by the recursion formula is this minus this,"},{"Start":"03:36.635 ","End":"03:38.945","Text":"then a bit of simplification."},{"Start":"03:38.945 ","End":"03:42.445","Text":"It a 1/2 of a over x_n, that\u0027s here,"},{"Start":"03:42.445 ","End":"03:47.380","Text":"and 1/2 x_n minus x_n is minus 1/2 x_n."},{"Start":"03:47.380 ","End":"03:50.945","Text":"Now we can put this over a common denominator,"},{"Start":"03:50.945 ","End":"03:52.340","Text":"take the half out,"},{"Start":"03:52.340 ","End":"03:54.155","Text":"put it over x_n,"},{"Start":"03:54.155 ","End":"03:58.630","Text":"and we get a minus x_n squared."},{"Start":"03:58.630 ","End":"04:06.210","Text":"Now here, the numerator is negative or less than or equal to 0 because,"},{"Start":"04:06.470 ","End":"04:10.770","Text":"x_n we showed is bigger or equal to square root of a,"},{"Start":"04:10.770 ","End":"04:13.040","Text":"then x_n squared is bigger or equal to a,"},{"Start":"04:13.040 ","End":"04:18.410","Text":"so a minus x_n squared is less than or equal to 0,"},{"Start":"04:18.410 ","End":"04:22.405","Text":"and x_n is bigger than 0."},{"Start":"04:22.405 ","End":"04:28.590","Text":"This over this is less than or equal to 0 and the 0.5 doesn\u0027t affect that."},{"Start":"04:32.810 ","End":"04:35.465","Text":"From the steps above,"},{"Start":"04:35.465 ","End":"04:41.225","Text":"we know that the sequence is decreasing and bounded from below."},{"Start":"04:41.225 ","End":"04:48.120","Text":"That means that it converges to some limit L. Note,"},{"Start":"04:48.120 ","End":"04:54.485","Text":"that because all the x_n are bigger or equal to root a,"},{"Start":"04:54.485 ","End":"04:58.880","Text":"then the limit L is also bigger or equal to route a,"},{"Start":"04:58.880 ","End":"05:01.725","Text":"which is bigger than 0."},{"Start":"05:01.725 ","End":"05:05.920","Text":"The limit L is positive."},{"Start":"05:06.710 ","End":"05:15.430","Text":"What we have left to show is that that limit is strictly equal to root a."},{"Start":"05:18.980 ","End":"05:22.085","Text":"Here\u0027s the recursion formula."},{"Start":"05:22.085 ","End":"05:28.020","Text":"Now take the limit as n goes to infinity, and of course,"},{"Start":"05:28.020 ","End":"05:30.210","Text":"the limit of x_n plus 1,"},{"Start":"05:30.210 ","End":"05:33.225","Text":"is the same as the limit of x_n,"},{"Start":"05:33.225 ","End":"05:35.460","Text":"it\u0027s just offset by 1,"},{"Start":"05:35.460 ","End":"05:38.070","Text":"but the limit\u0027s the same."},{"Start":"05:38.070 ","End":"05:41.295","Text":"This is L,"},{"Start":"05:41.295 ","End":"05:42.430","Text":"this is L,"},{"Start":"05:42.430 ","End":"05:50.775","Text":"and this is L. We get the following equation in L. Multiply both sides by 2L."},{"Start":"05:50.775 ","End":"05:52.895","Text":"This is what we get."},{"Start":"05:52.895 ","End":"05:56.930","Text":"Then, we have L^2 equals a."},{"Start":"05:56.930 ","End":"05:58.970","Text":"Take the square root,"},{"Start":"05:58.970 ","End":"06:01.160","Text":"could be plus or minus,"},{"Start":"06:01.160 ","End":"06:09.815","Text":"but we only take the plus because we just saw that L is bigger or equal to 0."},{"Start":"06:09.815 ","End":"06:13.025","Text":"We actually saw that it is strictly bigger than 0."},{"Start":"06:13.025 ","End":"06:15.395","Text":"Anyway, the correct answer is the plus."},{"Start":"06:15.395 ","End":"06:19.740","Text":"This is what we had to show and we are done."}],"ID":31205},{"Watched":false,"Name":"Exercise 5","Duration":"7m 13s","ChapterTopicVideoID":29622,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have a recursive sequence x_n,"},{"Start":"00:04.965 ","End":"00:08.070","Text":"but given that x_1 is equal to a,"},{"Start":"00:08.070 ","End":"00:12.239","Text":"which is some non-negative number and we\u0027re given the recurrence relation"},{"Start":"00:12.239 ","End":"00:16.965","Text":"that x_n plus 1 equals this expression which is in terms of x_n."},{"Start":"00:16.965 ","End":"00:19.170","Text":"Then we\u0027re asked to prove 3 things."},{"Start":"00:19.170 ","End":"00:22.935","Text":"First of all, that all the x_n are non-negative."},{"Start":"00:22.935 ","End":"00:27.645","Text":"Secondly, the question to find out for which values of"},{"Start":"00:27.645 ","End":"00:33.435","Text":"a the sequence is increasing and for which values of a is the sequence decreasing."},{"Start":"00:33.435 ","End":"00:39.455","Text":"Thirdly, given that a is a number strictly between 3 and 3.5,"},{"Start":"00:39.455 ","End":"00:42.575","Text":"prove that the sequence does not converge."},{"Start":"00:42.575 ","End":"00:45.065","Text":"Start with a, which is easy."},{"Start":"00:45.065 ","End":"00:49.040","Text":"First of all, note that x_1 is bigger or equal to 0 because it\u0027s equal"},{"Start":"00:49.040 ","End":"00:52.970","Text":"to a and it\u0027s given and for n equals 2 onwards,"},{"Start":"00:52.970 ","End":"00:55.115","Text":"we can use the recursion formula."},{"Start":"00:55.115 ","End":"00:56.540","Text":"When n is 1 here,"},{"Start":"00:56.540 ","End":"00:58.550","Text":"we get x_2 here,"},{"Start":"00:58.550 ","End":"01:02.460","Text":"so we\u0027re going to get from x_2 onwards by putting values of"},{"Start":"01:02.460 ","End":"01:08.620","Text":"n. We see basically everything on the right-hand side is positive,"},{"Start":"01:08.620 ","End":"01:14.150","Text":"5th is a positive number times 6 plus something non-negative."},{"Start":"01:14.150 ","End":"01:15.955","Text":"It\u0027s always positive,"},{"Start":"01:15.955 ","End":"01:20.535","Text":"so x_1 is positive and all the x_n plus 1 are positive."},{"Start":"01:20.535 ","End":"01:24.235","Text":"This means that x_n is positive or"},{"Start":"01:24.235 ","End":"01:27.700","Text":"non-negative for n bigger equal to 2 and here we have the n equals 1."},{"Start":"01:27.700 ","End":"01:29.930","Text":"That covers all the cases."},{"Start":"01:29.930 ","End":"01:33.685","Text":"Now we\u0027ll see when it\u0027s decreasing and when it\u0027s increasing."},{"Start":"01:33.685 ","End":"01:37.480","Text":"We\u0027ll define decreasing not in the strict sense,"},{"Start":"01:37.480 ","End":"01:39.585","Text":"less than or equal to."},{"Start":"01:39.585 ","End":"01:43.390","Text":"We can also rewrite that by bringing x_n over to"},{"Start":"01:43.390 ","End":"01:47.725","Text":"the other side and saying x_n plus 1 minus x_n less than or equal to 0."},{"Start":"01:47.725 ","End":"01:53.395","Text":"Similarly, it\u0027s increasing when x_n plus 1 minus x_n is bigger or equal to 0,"},{"Start":"01:53.395 ","End":"01:56.620","Text":"so then x_n plus 1 is bigger or equal to x_n."},{"Start":"01:56.620 ","End":"01:57.910","Text":"Just as a by the way,"},{"Start":"01:57.910 ","End":"02:03.400","Text":"if x_n plus 1 minus x_n is exactly equal to 0,"},{"Start":"02:03.400 ","End":"02:06.900","Text":"so it\u0027s both less than or equal to n bigger than or equal to,"},{"Start":"02:06.900 ","End":"02:12.395","Text":"then it\u0027s a constant sequence because the differences are 0."},{"Start":"02:12.395 ","End":"02:16.295","Text":"You could say it\u0027s both increasing and decreasing in the non-strict sense."},{"Start":"02:16.295 ","End":"02:17.830","Text":"That\u0027s just a, by the way."},{"Start":"02:17.830 ","End":"02:24.000","Text":"We can compute x_n plus 1 minus x_n from the recursion formula here."},{"Start":"02:24.000 ","End":"02:28.530","Text":"First of all, we\u0027ll just copy x_n plus 1 from here."},{"Start":"02:28.530 ","End":"02:32.260","Text":"Then for x_n will just reduce n by 1."},{"Start":"02:32.260 ","End":"02:36.260","Text":"Of course, this won\u0027t work if n equals 1 and has to be at least 2,"},{"Start":"02:36.260 ","End":"02:39.580","Text":"so this only works for n bigger or equal to 2."},{"Start":"02:39.580 ","End":"02:42.735","Text":"Now some algebra, the 6 cancels."},{"Start":"02:42.735 ","End":"02:48.210","Text":"Then we have 1/5 x_n squared minus x_n minus 1 squared."},{"Start":"02:48.210 ","End":"02:52.585","Text":"Then we can expand this as a difference of squares,"},{"Start":"02:52.585 ","End":"02:56.100","Text":"so it\u0027s x_n plus x_n minus 1,"},{"Start":"02:56.100 ","End":"02:59.320","Text":"times x_n minus x_n minus 1."},{"Start":"02:59.320 ","End":"03:03.770","Text":"Now we know that this is bigger or equal to 0 because we showed that all"},{"Start":"03:03.770 ","End":"03:10.130","Text":"the x_n\u0027s are bigger or equal to 0 in part A."},{"Start":"03:10.130 ","End":"03:12.185","Text":"As for this, well,"},{"Start":"03:12.185 ","End":"03:16.795","Text":"we don\u0027t really know and we\u0027re going to have to distinguish cases."},{"Start":"03:16.795 ","End":"03:24.139","Text":"What this does show is that if this is bigger or equal to 0,"},{"Start":"03:24.139 ","End":"03:26.350","Text":"then so is this."},{"Start":"03:26.350 ","End":"03:32.540","Text":"This implies this and if x_n minus x_n minus 1 is less than or equal to 0,"},{"Start":"03:32.540 ","End":"03:34.655","Text":"then so is the next difference,"},{"Start":"03:34.655 ","End":"03:36.850","Text":"x_n plus 1 minus x_n."},{"Start":"03:36.850 ","End":"03:38.760","Text":"Actually, if you think about this,"},{"Start":"03:38.760 ","End":"03:44.390","Text":"if you can keep adding 1 or you can work your way back to x_2 minus x_1,"},{"Start":"03:44.390 ","End":"03:46.220","Text":"which will determine everything."},{"Start":"03:46.220 ","End":"03:49.095","Text":"Let\u0027s just divide that into 2 cases."},{"Start":"03:49.095 ","End":"03:54.240","Text":"First case, x_2 minus x_1 is bigger or equal to 0."},{"Start":"03:54.240 ","End":"03:58.090","Text":"Then we\u0027ll do case 2 where it\u0027s less than or equal to 0."},{"Start":"03:58.090 ","End":"04:04.055","Text":"By induction, x_n plus 1 minus x_n is bigger or equal to 0."},{"Start":"04:04.055 ","End":"04:07.490","Text":"Now, it\u0027s true for n equals"},{"Start":"04:07.490 ","End":"04:12.950","Text":"1 because we\u0027re assuming x_2 minus x_1 is bigger or equal to 0."},{"Start":"04:12.950 ","End":"04:20.030","Text":"Also, we have the induction step from n minus 1 to n above."},{"Start":"04:20.030 ","End":"04:23.125","Text":"If n is bigger than 1, we have this."},{"Start":"04:23.125 ","End":"04:27.575","Text":"Similarly, in case 2, same thing happens."},{"Start":"04:27.575 ","End":"04:33.440","Text":"We by induction, prove in this case that it\u0027s less than or equal to 0."},{"Start":"04:33.440 ","End":"04:36.290","Text":"It\u0027s the same as above just by using the"},{"Start":"04:36.290 ","End":"04:40.310","Text":"less than or equal to instead of the bigger than or equal to case."},{"Start":"04:40.310 ","End":"04:45.829","Text":"Now we\u0027re going to determine when x_2 minus x_1 is bigger or equal to 0,"},{"Start":"04:45.829 ","End":"04:51.155","Text":"and when x_2 minus x_1 is less than or equal to 0 in terms of the value of a."},{"Start":"04:51.155 ","End":"04:56.310","Text":"So x_2 minus x_1 is, x_1 is a,"},{"Start":"04:56.310 ","End":"05:02.385","Text":"and x_2 from the recursion is 1/5 x_1 squared plus 6."},{"Start":"05:02.385 ","End":"05:07.685","Text":"We get this minus this and then putting the a inside it becomes"},{"Start":"05:07.685 ","End":"05:13.775","Text":"minus 5a and this factorizes as a minus 3, a minus 2."},{"Start":"05:13.775 ","End":"05:18.690","Text":"Here\u0027s the graph of the parabola for this function."},{"Start":"05:18.690 ","End":"05:23.040","Text":"You see it\u0027s 0 when a is 2 or 3 and it\u0027s bigger or equal to"},{"Start":"05:23.040 ","End":"05:28.065","Text":"0 when a is bigger or equal to 3 or less than or equal to 2."},{"Start":"05:28.065 ","End":"05:35.775","Text":"Similarly, it\u0027s less than or equal to 0 when a is between 2 and 3 including the ends."},{"Start":"05:35.775 ","End":"05:38.900","Text":"We have these 2 cases for a."},{"Start":"05:38.900 ","End":"05:43.955","Text":"Now summarizing, when a is less than 2 or bigger than 3,"},{"Start":"05:43.955 ","End":"05:46.460","Text":"then x_n is increasing."},{"Start":"05:46.460 ","End":"05:51.240","Text":"When a is between 2 and 3, x_n is decreasing."},{"Start":"05:51.240 ","End":"05:56.045","Text":"Part c, we were given that a is between 3 and 3.5,"},{"Start":"05:56.045 ","End":"05:58.340","Text":"we have to prove a sequence doesn\u0027t converge."},{"Start":"05:58.340 ","End":"06:04.225","Text":"From part b we see that when a is bigger than 3,"},{"Start":"06:04.225 ","End":"06:05.690","Text":"it can\u0027t come from here,"},{"Start":"06:05.690 ","End":"06:07.430","Text":"has to come from here,"},{"Start":"06:07.430 ","End":"06:10.385","Text":"so it\u0027s strictly increasing."},{"Start":"06:10.385 ","End":"06:11.780","Text":"We don\u0027t need the strictly,"},{"Start":"06:11.780 ","End":"06:14.270","Text":"but we need to see that it\u0027s increasing."},{"Start":"06:14.270 ","End":"06:17.480","Text":"Now we\u0027ll prove that it doesn\u0027t have a limit by contradiction,"},{"Start":"06:17.480 ","End":"06:24.545","Text":"so suppose it does converge and form the limit L. From the recursion relation,"},{"Start":"06:24.545 ","End":"06:27.560","Text":"we take the limit as n goes to infinity."},{"Start":"06:27.560 ","End":"06:31.340","Text":"This tends to L, This tends to L squared."},{"Start":"06:31.340 ","End":"06:32.900","Text":"We get the equation,"},{"Start":"06:32.900 ","End":"06:36.190","Text":"L equals 1/5 x squared plus 6."},{"Start":"06:36.190 ","End":"06:37.710","Text":"If you expand this,"},{"Start":"06:37.710 ","End":"06:41.015","Text":"it\u0027s the same quadratic that we had before."},{"Start":"06:41.015 ","End":"06:43.610","Text":"L squared minus 5L plus 6."},{"Start":"06:43.610 ","End":"06:46.835","Text":"The solutions are L equals 2 or 3,"},{"Start":"06:46.835 ","End":"06:53.330","Text":"but neither of these is any good because x_1 is a and x_n is increasing,"},{"Start":"06:53.330 ","End":"06:56.480","Text":"so all the x_n\u0027s are bigger or equal to a."},{"Start":"06:56.480 ","End":"06:58.280","Text":"If they\u0027re all bigger or equal to a,"},{"Start":"06:58.280 ","End":"07:00.955","Text":"then the limit is bigger or equal to a."},{"Start":"07:00.955 ","End":"07:03.600","Text":"But a is bigger than 3,"},{"Start":"07:03.600 ","End":"07:05.100","Text":"so L is bigger than 3,"},{"Start":"07:05.100 ","End":"07:08.280","Text":"so it can\u0027t be equal to 2 or 3 and"},{"Start":"07:08.280 ","End":"07:13.990","Text":"that contradiction proves our claim and that concludes this clip."}],"ID":31206},{"Watched":false,"Name":"Exercise 6","Duration":"6m 9s","ChapterTopicVideoID":29623,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.705","Text":"In this exercise, we have 2 recursive sequences: a_n, and b_n."},{"Start":"00:06.705 ","End":"00:11.130","Text":"We\u0027re given a_1 and b_1 arbitrary positive numbers,"},{"Start":"00:11.130 ","End":"00:13.575","Text":"but a_1 is bigger than b_1."},{"Start":"00:13.575 ","End":"00:18.525","Text":"This is the recursion formula for the n plus 1 term."},{"Start":"00:18.525 ","End":"00:20.565","Text":"They each depend on the other,"},{"Start":"00:20.565 ","End":"00:25.095","Text":"a_n plus 1 is the arithmetic mean of a_n and b_n,"},{"Start":"00:25.095 ","End":"00:29.655","Text":"and b_n plus 1 is the geometric mean of a_n and b_n."},{"Start":"00:29.655 ","End":"00:33.390","Text":"Our task is to prove that each of the sequences a_n and b_n"},{"Start":"00:33.390 ","End":"00:38.100","Text":"converges and that they converge to the same limit."},{"Start":"00:38.100 ","End":"00:41.870","Text":"We\u0027re given a hint to use the inequality of the means,"},{"Start":"00:41.870 ","End":"00:46.580","Text":"that the arithmetic mean is bigger or equal to the geometric mean."},{"Start":"00:46.580 ","End":"00:47.900","Text":"Just to get a feel for it,"},{"Start":"00:47.900 ","End":"00:48.950","Text":"we\u0027ll take an example."},{"Start":"00:48.950 ","End":"00:51.860","Text":"Let\u0027s say a_1 is 8 and b_1 is 2,"},{"Start":"00:51.860 ","End":"00:55.205","Text":"suddenly they\u0027re both positive and a_1 is bigger than b_1."},{"Start":"00:55.205 ","End":"00:57.335","Text":"Let\u0027s compute the next level."},{"Start":"00:57.335 ","End":"01:01.950","Text":"We have that a_2 is the arithmetic mean of 8 and 2,"},{"Start":"01:01.950 ","End":"01:06.330","Text":"and that\u0027s 5, and b_2 is the geometric mean of 8 and 2,"},{"Start":"01:06.330 ","End":"01:08.190","Text":"and that comes out to be 4."},{"Start":"01:08.190 ","End":"01:10.170","Text":"Let\u0027s take another level,"},{"Start":"01:10.170 ","End":"01:15.015","Text":"a_3 is the arithmetic mean of 5 and 4,"},{"Start":"01:15.015 ","End":"01:20.275","Text":"that\u0027s 4.5, and b_3 is the geometric mean of 5 and 4,"},{"Start":"01:20.275 ","End":"01:25.265","Text":"which is the square root of 20, about 4.47."},{"Start":"01:25.265 ","End":"01:32.640","Text":"Now it looks like at least in this case that the sequence a_n is decreasing,"},{"Start":"01:32.640 ","End":"01:35.595","Text":"and the sequence b_n is increasing."},{"Start":"01:35.595 ","End":"01:38.960","Text":"It also looks like this stays bigger than this."},{"Start":"01:38.960 ","End":"01:43.900","Text":"This is bigger than this but they get closer."},{"Start":"01:43.900 ","End":"01:47.405","Text":"In fact, this is true in general,"},{"Start":"01:47.405 ","End":"01:49.070","Text":"and we\u0027ll prove it."},{"Start":"01:49.070 ","End":"01:50.960","Text":"We\u0027ll prove 3 things,"},{"Start":"01:50.960 ","End":"01:56.665","Text":"this inequality that the a_n are decreasing and the b_n are increasing."},{"Start":"01:56.665 ","End":"02:00.800","Text":"Let\u0027s start with the first step that this is true."},{"Start":"02:00.800 ","End":"02:02.330","Text":"We\u0027ll prove it by induction,"},{"Start":"02:02.330 ","End":"02:04.625","Text":"for n is 1, it\u0027s just given."},{"Start":"02:04.625 ","End":"02:06.470","Text":"Now let\u0027s take the induction step."},{"Start":"02:06.470 ","End":"02:09.290","Text":"If it\u0027s true for n, it\u0027s true for n plus 1."},{"Start":"02:09.290 ","End":"02:16.835","Text":"We take the case n as a given and we have to prove the case n plus 1."},{"Start":"02:16.835 ","End":"02:21.590","Text":"B_n plus 1 is the square root of a_n b_n,"},{"Start":"02:21.590 ","End":"02:26.540","Text":"and is certainly positive because these are both positive."},{"Start":"02:26.540 ","End":"02:27.995","Text":"That\u0027s the positivity part."},{"Start":"02:27.995 ","End":"02:32.300","Text":"We still have to show that a_n plus 1 is bigger than b_n plus 1."},{"Start":"02:32.300 ","End":"02:34.235","Text":"B_n plus 1,"},{"Start":"02:34.235 ","End":"02:36.050","Text":"which is the square root of a_n b_n,"},{"Start":"02:36.050 ","End":"02:38.885","Text":"by the inequality of the means,"},{"Start":"02:38.885 ","End":"02:43.530","Text":"is less than a_n plus b_n over 2 which is a_n plus 1."},{"Start":"02:43.530 ","End":"02:46.095","Text":"Now, you might ask why less than"},{"Start":"02:46.095 ","End":"02:50.435","Text":"because the inequality of the mean says less than or equal to."},{"Start":"02:50.435 ","End":"02:54.005","Text":"Well, it turns out that there\u0027s more than this."},{"Start":"02:54.005 ","End":"03:00.400","Text":"The inequality is actually equality if and only if x equals y."},{"Start":"03:00.400 ","End":"03:03.570","Text":"Here a_n does not equal b_n,"},{"Start":"03:03.570 ","End":"03:06.030","Text":"so we get a strict inequality."},{"Start":"03:06.030 ","End":"03:08.340","Text":"That proves the first of the 3."},{"Start":"03:08.340 ","End":"03:14.910","Text":"Now step 2, we\u0027ll show that a_n is decreasing. Here goes."},{"Start":"03:14.910 ","End":"03:19.950","Text":"A_n plus 1 is a_n plus b_n over 2,"},{"Start":"03:19.950 ","End":"03:22.870","Text":"b_n is less than a_n,"},{"Start":"03:22.870 ","End":"03:25.655","Text":"because that\u0027s what we showed in step 1,"},{"Start":"03:25.655 ","End":"03:27.020","Text":"b_n less than a_n,"},{"Start":"03:27.020 ","End":"03:28.730","Text":"so this is less than this."},{"Start":"03:28.730 ","End":"03:30.950","Text":"A_n plus a_n over 2 is a_n,"},{"Start":"03:30.950 ","End":"03:33.830","Text":"so a_n plus 1 is less than a_n."},{"Start":"03:33.830 ","End":"03:35.935","Text":"That\u0027s step 2."},{"Start":"03:35.935 ","End":"03:41.685","Text":"What remains is the last 1 of the 3 that b_n is increasing."},{"Start":"03:41.685 ","End":"03:43.680","Text":"B_n plus 1,"},{"Start":"03:43.680 ","End":"03:47.180","Text":"which is the square root of a_n b_n is bigger"},{"Start":"03:47.180 ","End":"03:51.260","Text":"than the square root of b_n b_n because a_n is bigger than b_n,"},{"Start":"03:51.260 ","End":"03:53.645","Text":"and the square root of b_n b_n is just b_n,"},{"Start":"03:53.645 ","End":"03:59.865","Text":"so b_n plus 1 is bigger than b_n and b_n is decreasing."},{"Start":"03:59.865 ","End":"04:04.170","Text":"Step 4, let me just go back to the example."},{"Start":"04:04.310 ","End":"04:09.560","Text":"In the example, not only is this decreasing and this increasing,"},{"Start":"04:09.560 ","End":"04:13.805","Text":"but it looks like all of these are bigger than all of these."},{"Start":"04:13.805 ","End":"04:15.800","Text":"This seems to be bounded below,"},{"Start":"04:15.800 ","End":"04:19.150","Text":"it doesn\u0027t decrease indefinitely."},{"Start":"04:19.150 ","End":"04:22.200","Text":"This seems to be bounded above,"},{"Start":"04:22.200 ","End":"04:24.335","Text":"it doesn\u0027t increase indefinitely."},{"Start":"04:24.335 ","End":"04:28.550","Text":"It\u0027ll never get beyond 4.5, for example."},{"Start":"04:28.550 ","End":"04:32.105","Text":"Let\u0027s prove this. This is what it looks like from the example."},{"Start":"04:32.105 ","End":"04:33.350","Text":"That\u0027s our step 4,"},{"Start":"04:33.350 ","End":"04:36.800","Text":"that a_n is bounded below and b_n is bounded above."},{"Start":"04:36.800 ","End":"04:40.415","Text":"Just look at the following, which we showed."},{"Start":"04:40.415 ","End":"04:42.770","Text":"We showed that the b_n\u0027s are increasing,"},{"Start":"04:42.770 ","End":"04:44.495","Text":"so b_1 is less than or equal to b."},{"Start":"04:44.495 ","End":"04:46.430","Text":"We showed that b_n is less than a_n,"},{"Start":"04:46.430 ","End":"04:50.225","Text":"and we showed that the a_n\u0027s are decreasing so that a_n is less than or equal to a_1."},{"Start":"04:50.225 ","End":"04:55.235","Text":"From this, you can see that all the b_n\u0027s are less than a_1,"},{"Start":"04:55.235 ","End":"04:58.340","Text":"and all the a_n\u0027s are bigger than b_1."},{"Start":"04:58.340 ","End":"05:00.170","Text":"That\u0027s step 4."},{"Start":"05:00.170 ","End":"05:04.690","Text":"Now I want to stress the important stuff for the following steps:"},{"Start":"05:04.690 ","End":"05:10.680","Text":"a_n is decreasing and a_n is bounded below."},{"Start":"05:10.680 ","End":"05:16.875","Text":"Whereas b_n is increasing and b_n is bounded from above."},{"Start":"05:16.875 ","End":"05:20.270","Text":"It follows that a_n and b_n are convergent,"},{"Start":"05:20.270 ","End":"05:22.235","Text":"decreasing and bounded from below,"},{"Start":"05:22.235 ","End":"05:26.585","Text":"or increasing and bounded from above implies convergent."},{"Start":"05:26.585 ","End":"05:30.230","Text":"Let\u0027s say that the limit of a_n is a and the limit of b_n is b."},{"Start":"05:30.230 ","End":"05:34.160","Text":"All that remains to show now is that a is equal to b,"},{"Start":"05:34.160 ","End":"05:37.045","Text":"I put a question mark meanwhile, also we\u0027re going to show."},{"Start":"05:37.045 ","End":"05:39.270","Text":"Take the recursion equation,"},{"Start":"05:39.270 ","End":"05:42.240","Text":"a_n plus 1 is a_n plus b_n over 2."},{"Start":"05:42.240 ","End":"05:45.530","Text":"I suppose it would work if we took the other recursion equation."},{"Start":"05:45.530 ","End":"05:46.730","Text":"Anyways, take this one."},{"Start":"05:46.730 ","End":"05:49.100","Text":"Now let\u0027s take the limit as n goes to infinity."},{"Start":"05:49.100 ","End":"05:51.830","Text":"We can let a_n plus 1 go to infinity,"},{"Start":"05:51.830 ","End":"05:54.650","Text":"and a_n go to infinity and b_n go to infinity."},{"Start":"05:54.650 ","End":"05:55.915","Text":"This, and this go to a,"},{"Start":"05:55.915 ","End":"05:57.345","Text":"and this goes to b."},{"Start":"05:57.345 ","End":"06:00.530","Text":"We get a equals a plus b over 2."},{"Start":"06:00.530 ","End":"06:02.700","Text":"2 a equals a plus b,"},{"Start":"06:02.700 ","End":"06:04.740","Text":"so a equals b."},{"Start":"06:04.740 ","End":"06:06.363","Text":"That\u0027s what we had to show,"},{"Start":"06:06.363 ","End":"06:09.610","Text":"and that completes this exercise."}],"ID":31207},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"7m 42s","ChapterTopicVideoID":29624,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.745","Text":"In this exercise, we have a sequence a_n which is defined recursively."},{"Start":"00:05.745 ","End":"00:12.555","Text":"Each member depends on the two previous members,"},{"Start":"00:12.555 ","End":"00:16.665","Text":"which is not what we usually encounter."},{"Start":"00:16.665 ","End":"00:25.005","Text":"We\u0027re given a_1 and we\u0027re given a_1 and then from n=2 onwards,"},{"Start":"00:25.005 ","End":"00:28.290","Text":"we have this recursion relation."},{"Start":"00:28.290 ","End":"00:35.440","Text":"The first one would be like a_3 is twice a_2 plus 3 times a_1 and so on."},{"Start":"00:35.440 ","End":"00:43.485","Text":"Now, we define a new sequence b_n as b_n=a_n over a_n plus 1,"},{"Start":"00:43.485 ","End":"00:47.470","Text":"where n is bigger or equal to 1."},{"Start":"00:47.560 ","End":"00:54.320","Text":"Our task is assuming that the limit exists to compute the value"},{"Start":"00:54.320 ","End":"01:01.950","Text":"of the limit of b_n and the proof of the existence of this will be in the next exercise."},{"Start":"01:03.220 ","End":"01:11.405","Text":"Then we have to use the result from Part 1 to prove that a_n tends to infinity."},{"Start":"01:11.405 ","End":"01:14.000","Text":"Here\u0027s what we do."},{"Start":"01:14.000 ","End":"01:16.355","Text":"In Part 1,"},{"Start":"01:16.355 ","End":"01:19.595","Text":"copy the recursion formula."},{"Start":"01:19.595 ","End":"01:26.035","Text":"N plus 1 is twice a_n plus 3 times a_n minus 1."},{"Start":"01:26.035 ","End":"01:31.600","Text":"Now what we do is we divide this expression by the middle one,"},{"Start":"01:31.600 ","End":"01:34.170","Text":"then plus 1 is a_n and a_1 minus 1,"},{"Start":"01:34.170 ","End":"01:41.955","Text":"we divide by the a_n and we get a_n plus 1 over"},{"Start":"01:41.955 ","End":"01:47.025","Text":"a_n=2 plus"},{"Start":"01:47.025 ","End":"01:55.090","Text":"3a_n minus 1 over a_n."},{"Start":"01:55.090 ","End":"02:01.325","Text":"Now, notice that this is equal"},{"Start":"02:01.325 ","End":"02:07.850","Text":"to 1 over b_n because if b_n is a_n over a_n plus 1,"},{"Start":"02:07.850 ","End":"02:09.965","Text":"we flip it, we get this."},{"Start":"02:09.965 ","End":"02:17.690","Text":"Note also that if I put n minus 1 here,"},{"Start":"02:17.690 ","End":"02:24.590","Text":"then we get that b_n minus 1 is equal"},{"Start":"02:24.590 ","End":"02:31.850","Text":"to a_n minus 1 and this is a_n."},{"Start":"02:31.850 ","End":"02:37.475","Text":"Since we are taking this for n bigger or equal to 2,"},{"Start":"02:37.475 ","End":"02:43.680","Text":"this will always make sense because we won\u0027t get b_0 or anything."},{"Start":"02:43.680 ","End":"02:50.249","Text":"If n is at least 2, then n minus 1 is at least 1 so we\u0027re okay as far as the indexing."},{"Start":"02:50.420 ","End":"02:54.240","Text":"This is now equal to this."},{"Start":"02:54.240 ","End":"03:01.170","Text":"We\u0027ve got 2 plus 3 times"},{"Start":"03:01.170 ","End":"03:05.310","Text":"b_n minus 1."},{"Start":"03:05.310 ","End":"03:11.955","Text":"This is a recursive relation for b_n but we\u0027d better take the reciprocal first."},{"Start":"03:11.955 ","End":"03:14.070","Text":"We get actually b_n."},{"Start":"03:14.070 ","End":"03:23.850","Text":"So b_n is 1 over 2 plus 3 time b_n minus 1."},{"Start":"03:23.850 ","End":"03:29.730","Text":"This is the recursive formula for b_n."},{"Start":"03:29.730 ","End":"03:32.410","Text":"I\u0027ll put it in a box."},{"Start":"03:32.720 ","End":"03:36.375","Text":"Now let\u0027s compute the limit."},{"Start":"03:36.375 ","End":"03:39.385","Text":"We\u0027re assuming that the limit exists."},{"Start":"03:39.385 ","End":"03:43.780","Text":"We\u0027ve got the limit as n goes to infinity,"},{"Start":"03:43.780 ","End":"03:49.490","Text":"b_n is equal to the limit of this."},{"Start":"03:49.490 ","End":"03:51.305","Text":"Now I can put the limit inside,"},{"Start":"03:51.305 ","End":"03:57.770","Text":"so it\u0027s 1 over 2 plus 3 times"},{"Start":"03:57.770 ","End":"04:05.875","Text":"the limit of b_n minus 1 as N goes to infinity."},{"Start":"04:05.875 ","End":"04:16.560","Text":"Now, let\u0027s suppose that this limit is L. Then we get that L=1"},{"Start":"04:16.560 ","End":"04:23.970","Text":"over 2 plus 3 L. The limit of b_n minus 1"},{"Start":"04:23.970 ","End":"04:33.635","Text":"is also L because it\u0027s the same sequence just off by 1."},{"Start":"04:33.635 ","End":"04:36.860","Text":"That would be the same limit."},{"Start":"04:36.860 ","End":"04:40.730","Text":"We just have to solve this equation,"},{"Start":"04:40.730 ","End":"04:44.600","Text":"which will give us a quadratic. Let\u0027s see."},{"Start":"04:44.600 ","End":"04:48.660","Text":"We get L times this equals 1."},{"Start":"04:48.740 ","End":"04:57.760","Text":"L times 2 plus 3L=1."},{"Start":"04:58.070 ","End":"05:10.140","Text":"Let\u0027s see, that gives us 3L^2 plus 2L minus 1=0."},{"Start":"05:10.140 ","End":"05:13.235","Text":"If we use the formula,"},{"Start":"05:13.235 ","End":"05:23.260","Text":"we get that L equals minus 2 plus or minus the square root of b^2,"},{"Start":"05:23.260 ","End":"05:31.210","Text":"which is 4 minus 4 ac is plus 4 times 3 times that 1 is 12,"},{"Start":"05:31.210 ","End":"05:35.050","Text":"all this over 6."},{"Start":"05:35.050 ","End":"05:42.480","Text":"That gives us the square root of 4 plus 12 is 4."},{"Start":"05:42.480 ","End":"05:44.955","Text":"I have 2 possibilities."},{"Start":"05:44.955 ","End":"05:50.025","Text":"I have minus 2 plus 4 over 6,"},{"Start":"05:50.025 ","End":"05:56.365","Text":"and minus 2 minus 4 over 6."},{"Start":"05:56.365 ","End":"06:02.420","Text":"Now the limit is not going to be negative because it starts"},{"Start":"06:02.420 ","End":"06:08.900","Text":"out a_1 and a_2 are positive and this will always give us a positive quantity."},{"Start":"06:08.900 ","End":"06:16.950","Text":"This is ruled out, we get 1/3."},{"Start":"06:16.950 ","End":"06:22.060","Text":"The limit as n goes to infinity,"},{"Start":"06:22.490 ","End":"06:30.100","Text":"now b_n is just, where is it?"},{"Start":"06:33.560 ","End":"06:38.595","Text":"There it is, a_n over a_n plus 1."},{"Start":"06:38.595 ","End":"06:43.848","Text":"We have that the limit as n goes to infinity of a_n"},{"Start":"06:43.848 ","End":"06:50.770","Text":"over a_n plus 1 is equal to 1/3."},{"Start":"06:51.440 ","End":"06:55.799","Text":"Now, if I flip this,"},{"Start":"06:55.799 ","End":"07:06.270","Text":"I will get that limit as n goes to infinity of a_n plus 1 over a_n is equal to 3."},{"Start":"07:06.270 ","End":"07:12.365","Text":"The reason I did this is because this is what we would do in the ratio test."},{"Start":"07:12.365 ","End":"07:15.455","Text":"This expression, a_n plus 1 over a_n,"},{"Start":"07:15.455 ","End":"07:19.270","Text":"we know from the ratio test that if it\u0027s bigger than 1,"},{"Start":"07:19.270 ","End":"07:23.150","Text":"then the original sequence goes to infinity and if it\u0027s less than 1,"},{"Start":"07:23.150 ","End":"07:27.215","Text":"the sequence goes to 0, but this is definitely bigger than 1."},{"Start":"07:27.215 ","End":"07:29.515","Text":"By the ratio test,"},{"Start":"07:29.515 ","End":"07:36.340","Text":"the limit as n goes to infinity of a_n is infinity."},{"Start":"07:36.800 ","End":"07:42.700","Text":"That solves this. We\u0027re done."}],"ID":31208},{"Watched":false,"Name":"Exercise 7 - Part b","Duration":"14m 27s","ChapterTopicVideoID":29625,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.795","Text":"This exercise is a companion exercise for the previous one."},{"Start":"00:06.795 ","End":"00:10.860","Text":"It\u0027s related, I\u0027m not sure if it\u0027s a sequel or a prequel."},{"Start":"00:10.860 ","End":"00:14.640","Text":"Anyway, we are given the same sequence as there,"},{"Start":"00:14.640 ","End":"00:17.505","Text":"which is defined recursively."},{"Start":"00:17.505 ","End":"00:26.235","Text":"This time we have to find a closed meaning not with recursion expression for a_n."},{"Start":"00:26.235 ","End":"00:32.938","Text":"Then using this to prove that this limit of a_n over n plus 1 exists,"},{"Start":"00:32.938 ","End":"00:34.780","Text":"and to compute the limit."},{"Start":"00:34.780 ","End":"00:36.679","Text":"In the previous exercise,"},{"Start":"00:36.679 ","End":"00:39.560","Text":"we showed that this limit is a third,"},{"Start":"00:39.560 ","End":"00:43.460","Text":"but here we\u0027ll do it differently using the closed expression."},{"Start":"00:43.460 ","End":"00:45.695","Text":"This will be a confirmation."},{"Start":"00:45.695 ","End":"00:49.220","Text":"Then the last step will be to prove by induction that what we"},{"Start":"00:49.220 ","End":"00:54.490","Text":"found in Part 1 is actually correct."},{"Start":"00:54.490 ","End":"00:58.720","Text":"Let\u0027s see. In Part 1,"},{"Start":"00:58.720 ","End":"01:01.970","Text":"there is a standard approach when we"},{"Start":"01:01.970 ","End":"01:08.525","Text":"have each member recursively defined in terms of previous members."},{"Start":"01:08.525 ","End":"01:12.440","Text":"Specifically some constant times the previous one,"},{"Start":"01:12.440 ","End":"01:16.490","Text":"and then another constant times the one before the previous."},{"Start":"01:16.490 ","End":"01:17.840","Text":"There could be more of these,"},{"Start":"01:17.840 ","End":"01:21.245","Text":"but here we\u0027re just taking the 2 previous ones."},{"Start":"01:21.245 ","End":"01:31.220","Text":"Then we look for a solution of the form a_n is some number q to the power of n. a_n"},{"Start":"01:31.220 ","End":"01:35.550","Text":"plus 1 will be q^n plus"},{"Start":"01:35.550 ","End":"01:42.975","Text":"1 and a_n minus 1 is q^n minus 1."},{"Start":"01:42.975 ","End":"01:46.930","Text":"Then we get the equation."},{"Start":"01:47.780 ","End":"01:53.370","Text":"q^n plus 1 is equal to 2q ^n,"},{"Start":"01:53.370 ","End":"02:02.050","Text":"plus 3q^n minus 1. n is bigger or equal to 2."},{"Start":"02:02.050 ","End":"02:05.720","Text":"Now we can divide everything by q^n minus 1."},{"Start":"02:05.720 ","End":"02:07.790","Text":"It\u0027s the smallest one."},{"Start":"02:07.790 ","End":"02:15.110","Text":"We get q^2 equals 2q plus 3,"},{"Start":"02:15.110 ","End":"02:16.835","Text":"or we could bring this to the other side,"},{"Start":"02:16.835 ","End":"02:23.720","Text":"minus 2q minus 3=0 divided by q^n minus 1,"},{"Start":"02:23.720 ","End":"02:25.385","Text":"and then I brought stuff over."},{"Start":"02:25.385 ","End":"02:27.515","Text":"Now if we solve this,"},{"Start":"02:27.515 ","End":"02:30.510","Text":"this gives us 2 solutions."},{"Start":"02:30.510 ","End":"02:37.630","Text":"We\u0027ve got q=3 or minus 1."},{"Start":"02:37.630 ","End":"02:40.370","Text":"I know you know how to solve quadratic equations."},{"Start":"02:40.370 ","End":"02:44.750","Text":"The idea is to look for a general solution."},{"Start":"02:44.750 ","End":"02:47.990","Text":"Now, as a combination of these,"},{"Start":"02:47.990 ","End":"02:51.500","Text":"when I mean is we let look for a_n equals"},{"Start":"02:51.500 ","End":"02:57.500","Text":"some constant times 3^n"},{"Start":"02:57.500 ","End":"03:03.415","Text":"plus another constant times minus 1^n."},{"Start":"03:03.415 ","End":"03:09.880","Text":"This 3 is this 3 and this minus 1 is this minus 1."},{"Start":"03:09.880 ","End":"03:15.780","Text":"I want to point out that if we had a_n plus 1 or n plus 2,"},{"Start":"03:15.780 ","End":"03:18.635","Text":"depending on the 3 previous ones,"},{"Start":"03:18.635 ","End":"03:21.950","Text":"then we would have tried the same thing."},{"Start":"03:21.950 ","End":"03:25.265","Text":"We would have gotten here a cubic equation."},{"Start":"03:25.265 ","End":"03:28.250","Text":"We would have got maybe 3 solutions,"},{"Start":"03:28.250 ","End":"03:29.570","Text":"3 minus 1,"},{"Start":"03:29.570 ","End":"03:30.995","Text":"let\u0027s say, and 4."},{"Start":"03:30.995 ","End":"03:37.320","Text":"Then we\u0027d put here plus C times 4^n, whatever."},{"Start":"03:37.320 ","End":"03:39.045","Text":"It works in general."},{"Start":"03:39.045 ","End":"03:41.150","Text":"But beyond 2,"},{"Start":"03:41.150 ","End":"03:43.200","Text":"it gets a bit more complicated."},{"Start":"03:43.200 ","End":"03:48.385","Text":"Now the way we find A and B is using the information we haven\u0027t used yet,"},{"Start":"03:48.385 ","End":"03:52.715","Text":"that we have the initial conditions for a_1 and a_2."},{"Start":"03:52.715 ","End":"03:58.575","Text":"That means that we can substitute n equals 1,"},{"Start":"03:58.575 ","End":"04:00.465","Text":"and n equals 2,"},{"Start":"04:00.465 ","End":"04:04.380","Text":"and then we get 2 equations and 2 unknowns."},{"Start":"04:04.380 ","End":"04:11.415","Text":"a_1 equals 1 gives us that"},{"Start":"04:11.415 ","End":"04:18.660","Text":"1 equals a times 3^1,"},{"Start":"04:18.660 ","End":"04:24.015","Text":"plus B times minus 1 to the 1."},{"Start":"04:24.015 ","End":"04:33.395","Text":"In other words, 3A minus B=1."},{"Start":"04:33.395 ","End":"04:39.495","Text":"The other one we had was that a_2 equals 1,"},{"Start":"04:39.495 ","End":"04:43.290","Text":"which gives us the 1 equals A"},{"Start":"04:43.290 ","End":"04:50.865","Text":"times 3^2 plus B times minus 1^2 squared,"},{"Start":"04:50.865 ","End":"04:58.760","Text":"which means that 9A plus B is equal to 1."},{"Start":"04:58.760 ","End":"05:03.745","Text":"Now we have 2 equations and 2 unknowns, A and B."},{"Start":"05:03.745 ","End":"05:06.765","Text":"If we add these 2 equations,"},{"Start":"05:06.765 ","End":"05:12.430","Text":"we get 12A= 2."},{"Start":"05:13.160 ","End":"05:18.165","Text":"A = 2 over 12, which is 1/6."},{"Start":"05:18.165 ","End":"05:22.780","Text":"Now if we put A equals 1/6 here,"},{"Start":"05:22.780 ","End":"05:30.810","Text":"B is 3A minus 1 minus 0.5."},{"Start":"05:31.520 ","End":"05:35.805","Text":"Now we can get a_n explicitly,"},{"Start":"05:35.805 ","End":"05:42.060","Text":"a_n equals a is 1/6 times 3^n,"},{"Start":"05:42.060 ","End":"05:51.375","Text":"and B is minus 0.5 minus 1^n."},{"Start":"05:51.375 ","End":"05:55.530","Text":"This answers the first part."},{"Start":"05:55.530 ","End":"06:00.310","Text":"Now, let\u0027s go on to Part 2."},{"Start":"06:00.310 ","End":"06:10.565","Text":"We have to compute the limit as n goes to infinity of a_n over a_n plus 1."},{"Start":"06:10.565 ","End":"06:14.810","Text":"Let\u0027s see. We have the closed expression."},{"Start":"06:14.810 ","End":"06:16.580","Text":"We can substitute n,"},{"Start":"06:16.580 ","End":"06:18.410","Text":"and then n plus 1."},{"Start":"06:18.410 ","End":"06:23.765","Text":"On the numerator we have 1/6 times 3^n,"},{"Start":"06:23.765 ","End":"06:27.515","Text":"minus 0.5 times minus 1^n,"},{"Start":"06:27.515 ","End":"06:32.960","Text":"all these over 1/6th"},{"Start":"06:32.960 ","End":"06:42.855","Text":"times 3^n plus 1 minus 0.5 minus 1 to the n plus 1."},{"Start":"06:42.855 ","End":"06:44.865","Text":"Want to simplify this."},{"Start":"06:44.865 ","End":"06:49.775","Text":"Let\u0027s take 3^n outside the brackets,"},{"Start":"06:49.775 ","End":"06:51.820","Text":"on the top and on the bottom."},{"Start":"06:51.820 ","End":"07:00.245","Text":"Here we have 3^n 1/6 minus"},{"Start":"07:00.245 ","End":"07:05.090","Text":"1.5 of minus"},{"Start":"07:05.090 ","End":"07:11.305","Text":"1^n over 3^n."},{"Start":"07:11.305 ","End":"07:17.300","Text":"On the denominator, we can take the 3 out of here."},{"Start":"07:17.300 ","End":"07:22.205","Text":"The 3 goes with the 1/6 to give us 0.5."},{"Start":"07:22.205 ","End":"07:25.800","Text":"Then the 3^n comes out."},{"Start":"07:27.810 ","End":"07:31.960","Text":"Here, we have pretty much the same as here."},{"Start":"07:31.960 ","End":"07:37.105","Text":"We have 1/2 only we have minus 1 to the n plus"},{"Start":"07:37.105 ","End":"07:44.275","Text":"1/3 to the n. Let\u0027s look at this and this."},{"Start":"07:44.275 ","End":"07:51.850","Text":"Minus 1 to the n is plus or minus 1 and also minus 1 to the n plus 1 is plus or minus 1."},{"Start":"07:51.850 ","End":"07:54.805","Text":"If this is 1, this is minus 1 and the other way round."},{"Start":"07:54.805 ","End":"08:01.585","Text":"In event, we have plus or minus 1/3 to the n,"},{"Start":"08:01.585 ","End":"08:05.050","Text":"and when n goes to infinity,"},{"Start":"08:05.050 ","End":"08:08.755","Text":"this goes to 0."},{"Start":"08:08.755 ","End":"08:12.115","Text":"With the 1/2 it\u0027ll still goes to 0."},{"Start":"08:12.115 ","End":"08:13.900","Text":"This bit goes to 0,"},{"Start":"08:13.900 ","End":"08:15.430","Text":"this bit goes to 0."},{"Start":"08:15.430 ","End":"08:20.980","Text":"This cancels with this so the limit is"},{"Start":"08:20.980 ","End":"08:28.705","Text":"just 1/6 over 1/2 and that is equal to 1/3,"},{"Start":"08:28.705 ","End":"08:33.895","Text":"1/6 over 1/2 is 2/6 which is a 1/3."},{"Start":"08:33.895 ","End":"08:38.380","Text":"This is the answer to the second part."},{"Start":"08:38.380 ","End":"08:44.049","Text":"This is what we expected to get because in the previous exercise,"},{"Start":"08:44.049 ","End":"08:50.200","Text":"we got 1/3 that\u0027s 2/3."},{"Start":"08:50.200 ","End":"08:56.695","Text":"In Part 3 we had to prove that the formula we obtained and I just copied it,"},{"Start":"08:56.695 ","End":"09:00.895","Text":"that this is correct and to do it by induction."},{"Start":"09:00.895 ","End":"09:06.550","Text":"Also I want to remind you that a_1 is equal to 1,"},{"Start":"09:06.550 ","End":"09:10.405","Text":"a_2 equals 1 and"},{"Start":"09:10.405 ","End":"09:17.935","Text":"a_(n+1) is equal to twice a_n plus 3 times a_(n-1)."},{"Start":"09:17.935 ","End":"09:20.875","Text":"This is for n bigger or equal to 2."},{"Start":"09:20.875 ","End":"09:24.325","Text":"If we let n equals 1,"},{"Start":"09:24.325 ","End":"09:31.375","Text":"then we will get a_1 equals 1/6 of"},{"Start":"09:31.375 ","End":"09:39.010","Text":"3 to the 1 minus 1/2 of minus 1 to the 1,"},{"Start":"09:39.010 ","End":"09:47.400","Text":"and this is equal to 3/6 is a 1/2 minus times minus is plus 1/2,"},{"Start":"09:47.400 ","End":"09:48.945","Text":"which is equal to 1."},{"Start":"09:48.945 ","End":"09:50.265","Text":"This is true."},{"Start":"09:50.265 ","End":"09:52.535","Text":"When n equals 2,"},{"Start":"09:52.535 ","End":"09:58.540","Text":"we get a_2 equals 1/6 times"},{"Start":"09:58.540 ","End":"10:03.460","Text":"3^2 minus 1/2 times"},{"Start":"10:03.460 ","End":"10:09.115","Text":"minus 1^2 and this is equal to 3^2 is 9,"},{"Start":"10:09.115 ","End":"10:12.070","Text":"9/6 is 1 and 1/2."},{"Start":"10:12.070 ","End":"10:14.440","Text":"This comes out minus 1/2,"},{"Start":"10:14.440 ","End":"10:16.870","Text":"which is also equal to 1."},{"Start":"10:16.870 ","End":"10:19.660","Text":"Now we want to apply our special induction."},{"Start":"10:19.660 ","End":"10:21.670","Text":"It\u0027s called strong induction."},{"Start":"10:21.670 ","End":"10:26.188","Text":"That if this is true for 1, 2,"},{"Start":"10:26.188 ","End":"10:29.800","Text":"3 all the way up to n,"},{"Start":"10:29.800 ","End":"10:34.090","Text":"then it\u0027s also true for n plus 1."},{"Start":"10:34.090 ","End":"10:36.880","Text":"Now I don\u0027t need all of these,"},{"Start":"10:36.880 ","End":"10:42.234","Text":"I just need n minus 1 and n. If I show"},{"Start":"10:42.234 ","End":"10:49.315","Text":"that truth for n minus 1 and n implies truth for m plus 1, then I\u0027m done."},{"Start":"10:49.315 ","End":"10:51.730","Text":"What does this say?"},{"Start":"10:51.730 ","End":"10:58.330","Text":"A_n minus 1 is equal to 1/6 times 3 to the n"},{"Start":"10:58.330 ","End":"11:05.335","Text":"minus 1 minus 1/2 of minus 1 to the n minus 1."},{"Start":"11:05.335 ","End":"11:10.150","Text":"When I put n in it\u0027s just that a_n is equal"},{"Start":"11:10.150 ","End":"11:17.725","Text":"to 1/6 times 3 to the n minus 1/2 of minus 1^n."},{"Start":"11:17.725 ","End":"11:25.480","Text":"I have to show that these 2 together imply that a_n plus"},{"Start":"11:25.480 ","End":"11:34.915","Text":"1 is 1/6 times 3 to the n plus 1 minus 1/2,"},{"Start":"11:34.915 ","End":"11:39.040","Text":"minus 1 to the n plus 1."},{"Start":"11:39.040 ","End":"11:41.935","Text":"These 2 we\u0027re taking as true,"},{"Start":"11:41.935 ","End":"11:46.360","Text":"and we have to show that this equals this."},{"Start":"11:46.360 ","End":"11:48.850","Text":"A_n plus 1,"},{"Start":"11:48.850 ","End":"11:52.525","Text":"according to the recursion formula,"},{"Start":"11:52.525 ","End":"11:58.585","Text":"is twice a_n plus 3 times a_(n-1),"},{"Start":"11:58.585 ","End":"12:05.395","Text":"which is equal to now I can substitute a_n is 1/6,"},{"Start":"12:05.395 ","End":"12:14.890","Text":"3^n minus 1/2 minus 1^n plus 3 times this,"},{"Start":"12:14.890 ","End":"12:21.745","Text":"which is 1/6 3 to the n minus 1 minus 1/2,"},{"Start":"12:21.745 ","End":"12:26.035","Text":"minus 1 to the n minus 1."},{"Start":"12:26.035 ","End":"12:28.540","Text":"Now we have a bit of algebra to do."},{"Start":"12:28.540 ","End":"12:33.670","Text":"Let\u0027s collect the terms with the exponents of 3."},{"Start":"12:33.670 ","End":"12:41.800","Text":"We have, from this and from this well there\u0027s 1/6 in common and then we"},{"Start":"12:41.800 ","End":"12:51.190","Text":"have 2 times 3^n plus 3 times 3 to the n minus 1."},{"Start":"12:51.190 ","End":"12:55.090","Text":"Then for the other bit, we have,"},{"Start":"12:55.090 ","End":"13:02.500","Text":"allow me to write this bit as minus 1 to the n minus 1."},{"Start":"13:02.500 ","End":"13:05.995","Text":"I wanted to equalize the exponents here."},{"Start":"13:05.995 ","End":"13:14.440","Text":"What we have is 2 times 1/2 is 1 and the minus minus cancels."},{"Start":"13:14.440 ","End":"13:21.445","Text":"I get minus 1 to the n minus 1."},{"Start":"13:21.445 ","End":"13:28.200","Text":"Here, 3 times minus 1/2 is minus 3?"},{"Start":"13:28.200 ","End":"13:33.190","Text":"2 minus 1 to the n minus 1."},{"Start":"13:33.190 ","End":"13:41.500","Text":"This simplifies 3 times 3 to the n minus 1 is just 3^n plus twice"},{"Start":"13:41.500 ","End":"13:50.110","Text":"3^n is 3 times 3^n and 3 times 3 to the n is 3 to the n plus 1."},{"Start":"13:50.110 ","End":"13:52.420","Text":"Now here we have,"},{"Start":"13:52.420 ","End":"13:55.900","Text":"if we take minus 1 to the n minus 1 out,"},{"Start":"13:55.900 ","End":"13:59.005","Text":"we have 1 minus 1 and 1/2,"},{"Start":"13:59.005 ","End":"14:01.375","Text":"so it\u0027s minus 1/2,"},{"Start":"14:01.375 ","End":"14:05.905","Text":"minus 1 to the n minus 1."},{"Start":"14:05.905 ","End":"14:08.235","Text":"It\u0027s almost what we want."},{"Start":"14:08.235 ","End":"14:14.320","Text":"Here there\u0027s a minus and here there\u0027s a plus but if we multiply by minus 1^2,"},{"Start":"14:14.320 ","End":"14:19.330","Text":"which is 1, it\u0027s clear that we can just make this a plus."},{"Start":"14:19.330 ","End":"14:21.985","Text":"This is what we had to show."},{"Start":"14:21.985 ","End":"14:27.770","Text":"This is the inductive step and we are done."}],"ID":31209},{"Watched":false,"Name":"Exercise 8","Duration":"3m 30s","ChapterTopicVideoID":29626,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.695","Text":"In this exercise, we have to prove that this sequence defined recursively as a_1 is 2."},{"Start":"00:07.695 ","End":"00:11.295","Text":"Then from n=1 onwards,"},{"Start":"00:11.295 ","End":"00:16.380","Text":"an plus 1 equals the square root of 11 minus an^2."},{"Start":"00:16.380 ","End":"00:18.780","Text":"To prove that this sequence doesn\u0027t have a limit."},{"Start":"00:18.780 ","End":"00:26.565","Text":"Our usual way of doing that is finding 2 sub-sequences that converge to different limits."},{"Start":"00:26.565 ","End":"00:31.610","Text":"But let\u0027s first write out a few members of the sequence."},{"Start":"00:31.610 ","End":"00:33.240","Text":"Well, a_1 is 2,"},{"Start":"00:33.240 ","End":"00:35.700","Text":"we\u0027re given now let\u0027s see we put n=1."},{"Start":"00:35.700 ","End":"00:40.530","Text":"We get that a2 is the square root of 11 minus a_1^2,"},{"Start":"00:40.530 ","End":"00:43.035","Text":"which comes out to be root 7."},{"Start":"00:43.035 ","End":"00:48.900","Text":"Then a3 is the square root of 11 minus root 7^2."},{"Start":"00:48.900 ","End":"00:52.145","Text":"That comes out to be square root of 4 is 2."},{"Start":"00:52.145 ","End":"00:55.475","Text":"Then get root 7 again and so on."},{"Start":"00:55.475 ","End":"01:04.095","Text":"It looks like that for n odd we get 2, 2."},{"Start":"01:04.095 ","End":"01:06.770","Text":"For uneven we\u0027re going to get root 7."},{"Start":"01:06.770 ","End":"01:08.945","Text":"It\u0027s going to repeat itself."},{"Start":"01:08.945 ","End":"01:12.120","Text":"If that\u0027s the case,"},{"Start":"01:14.510 ","End":"01:19.435","Text":"then well, I\u0027m going to write it and then we\u0027ll do a formal proof of it."},{"Start":"01:19.435 ","End":"01:24.340","Text":"Looks like the sub-sequence a_2n minus 1 is 2."},{"Start":"01:24.340 ","End":"01:27.430","Text":"The sub-sequence a_2n is root 7."},{"Start":"01:27.430 ","End":"01:32.795","Text":"This would be the odds and this would be the evens index."},{"Start":"01:32.795 ","End":"01:37.335","Text":"We have 2 sub-sequence that converge to different limits."},{"Start":"01:37.335 ","End":"01:39.670","Text":"The original sequence doesn\u0027t converge."},{"Start":"01:39.670 ","End":"01:44.900","Text":"I\u0027m just going to return to this to prove it properly by induction."},{"Start":"01:44.990 ","End":"01:49.315","Text":"Here we are for the more pedantic of us."},{"Start":"01:49.315 ","End":"01:55.760","Text":"Let\u0027s see if we can prove them together as a pair by induction if n=1,"},{"Start":"01:55.760 ","End":"02:02.370","Text":"what this says is that a_2n minus 1 is going to be 2 and a_2n has got to be root 7."},{"Start":"02:02.370 ","End":"02:04.335","Text":"Well 2n minus 1 is 1,"},{"Start":"02:04.335 ","End":"02:06.700","Text":"and 2n is 2."},{"Start":"02:06.700 ","End":"02:10.115","Text":"If I take these indices 1 and 2 here and here,"},{"Start":"02:10.115 ","End":"02:14.090","Text":"we already showed this because we wrote out the first 4 members of the sequence."},{"Start":"02:14.090 ","End":"02:19.805","Text":"Now let\u0027s do the inductive step and assume it\u0027s true for n that"},{"Start":"02:19.805 ","End":"02:25.705","Text":"a_2n minus 1 is 2 and a_2n is root 7."},{"Start":"02:25.705 ","End":"02:29.735","Text":"We have to prove that it\u0027s true for n plus 1."},{"Start":"02:29.735 ","End":"02:37.110","Text":"In other words, we have to show that this is true,"},{"Start":"02:37.110 ","End":"02:41.880","Text":"a_2n plus 1 and a_2n plus 2 are 2n and root 7,"},{"Start":"02:41.880 ","End":"02:46.770","Text":"given that 2n minus 1 and 2n are this way."},{"Start":"02:46.770 ","End":"02:51.880","Text":"a_2n plus 1 by"},{"Start":"02:51.880 ","End":"02:57.025","Text":"the recursion is the square root of 11 minus the previous elements squared."},{"Start":"02:57.025 ","End":"03:01.975","Text":"This equals square root of 11 minus root 7^2."},{"Start":"03:01.975 ","End":"03:04.670","Text":"That comes out to be a."},{"Start":"03:04.670 ","End":"03:07.760","Text":"For a_2n plus 2,"},{"Start":"03:07.760 ","End":"03:12.445","Text":"we get also the square root of 11 minus the previous 1^2."},{"Start":"03:12.445 ","End":"03:15.460","Text":"By the induction hypothesis,"},{"Start":"03:15.460 ","End":"03:20.935","Text":"this is 2,"},{"Start":"03:20.935 ","End":"03:22.700","Text":"and then 11 minus 2^2,"},{"Start":"03:22.700 ","End":"03:26.275","Text":"11 minus 47 so root 7."},{"Start":"03:26.275 ","End":"03:30.870","Text":"That completes the proof by induction and we\u0027re done."}],"ID":31210},{"Watched":false,"Name":"Exercise 9","Duration":"4m 29s","ChapterTopicVideoID":29627,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.880","Text":"In this exercise, we\u0027re given a recursively defined sequence a_n is defined as a_1=2."},{"Start":"00:08.880 ","End":"00:14.250","Text":"Then we get from a_n to a_n plus 1 using this formula."},{"Start":"00:14.250 ","End":"00:17.190","Text":"A_n plus 1 is 1 over the square root of a_n."},{"Start":"00:17.190 ","End":"00:20.715","Text":"We have to prove that this sequence converges."},{"Start":"00:20.715 ","End":"00:25.335","Text":"Let\u0027s use exponential notation so that"},{"Start":"00:25.335 ","End":"00:31.935","Text":"this a_n plus 1 is a_n^minus 1/2."},{"Start":"00:31.935 ","End":"00:35.340","Text":"Now we\u0027ll write out a few members of the sequence."},{"Start":"00:35.340 ","End":"00:38.355","Text":"Let\u0027s see, we start off with 2."},{"Start":"00:38.355 ","End":"00:42.090","Text":"Then we raise it to the power of minus 1/2."},{"Start":"00:42.090 ","End":"00:45.630","Text":"Then we raise it to the power of minus 1/2 again."},{"Start":"00:45.630 ","End":"00:47.240","Text":"Using the rules for exponents,"},{"Start":"00:47.240 ","End":"00:53.720","Text":"we just keep multiplying the exponent by minus 1/2 times minus 1/2 is 1 over 2^4."},{"Start":"00:53.720 ","End":"00:56.525","Text":"We notice that there\u0027s an alternating sign."},{"Start":"00:56.525 ","End":"01:00.965","Text":"It looks like if n is odd and those are the green ones,"},{"Start":"01:00.965 ","End":"01:05.540","Text":"then we have 2^1 over 2^n minus 1."},{"Start":"01:05.540 ","End":"01:07.895","Text":"Similarly, if n is even,"},{"Start":"01:07.895 ","End":"01:10.235","Text":"let\u0027s just take for example,"},{"Start":"01:10.235 ","End":"01:13.335","Text":"if n is 6,"},{"Start":"01:13.335 ","End":"01:16.215","Text":"then 6 minus 1 is 5."},{"Start":"01:16.215 ","End":"01:20.790","Text":"2^minus 1 over 2^5 which is this then it really is the 1,"},{"Start":"01:20.790 ","End":"01:23.355","Text":"2, 3, 4, 5, 6th member."},{"Start":"01:23.355 ","End":"01:25.250","Text":"It looks like this,"},{"Start":"01:25.250 ","End":"01:28.650","Text":"and we\u0027ll prove this by induction soon."},{"Start":"01:28.930 ","End":"01:37.655","Text":"But I\u0027ll just write it for now that we can rephrase this with the odd uneven and saying,"},{"Start":"01:37.655 ","End":"01:44.220","Text":"odd is characterized by 2n minus 1 and even as characterized by 2n."},{"Start":"01:44.330 ","End":"01:49.295","Text":"Each of these sub-sequences converges to 1."},{"Start":"01:49.295 ","End":"01:51.680","Text":"2n minus 2 goes to infinity,"},{"Start":"01:51.680 ","End":"01:53.390","Text":"2 to infinity is infinity,"},{"Start":"01:53.390 ","End":"01:55.880","Text":"1 over infinity is 0."},{"Start":"01:55.880 ","End":"01:57.240","Text":"2^0 is 1."},{"Start":"01:57.240 ","End":"01:59.435","Text":"Similarly here this goes to infinity,"},{"Start":"01:59.435 ","End":"02:01.415","Text":"1 over it goes to 0."},{"Start":"02:01.415 ","End":"02:05.440","Text":"Both of these sequences converge to 1."},{"Start":"02:05.440 ","End":"02:08.090","Text":"Now we\u0027re going to use a theorem."},{"Start":"02:08.090 ","End":"02:13.475","Text":"It\u0027s one of the few that reverse theorems about sub-sequences."},{"Start":"02:13.475 ","End":"02:17.915","Text":"These two sub-sequences are not just any sub-sequences."},{"Start":"02:17.915 ","End":"02:21.960","Text":"They cover the original sequence."}],"ID":31211},{"Watched":false,"Name":"Exercise 10","Duration":"11m 38s","ChapterTopicVideoID":29628,"CourseChapterTopicPlaylistID":294567,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"In this exercise,"},{"Start":"00:01.755 ","End":"00:03.720","Text":"we\u0027re given a sequence a_n,"},{"Start":"00:03.720 ","End":"00:05.955","Text":"which is defined recursively."},{"Start":"00:05.955 ","End":"00:13.110","Text":"We\u0027re told that a_1 is 0 and each n can be gotten from the previous n with this formula,"},{"Start":"00:13.110 ","End":"00:15.960","Text":"a_n plus 1 is 1/1 plus a_n."},{"Start":"00:15.960 ","End":"00:20.265","Text":"We have to prove that the sequence converges. You know what?"},{"Start":"00:20.265 ","End":"00:25.065","Text":"Also, let\u0027s find the limit."},{"Start":"00:25.065 ","End":"00:26.805","Text":"I want to prepare you."},{"Start":"00:26.805 ","End":"00:31.815","Text":"This is a rather lengthy exercise, so beware."},{"Start":"00:31.815 ","End":"00:37.905","Text":"Let\u0027s start by writing out a few members of the sequence."},{"Start":"00:37.905 ","End":"00:43.640","Text":"We start off with a_1 is 0 and then 1/1 plus 0 is 1,"},{"Start":"00:43.640 ","End":"00:48.695","Text":"1/1 plus 1/2, 1/1/2 is 2/3."},{"Start":"00:48.695 ","End":"00:50.510","Text":"Then if you check,"},{"Start":"00:50.510 ","End":"00:53.580","Text":"we\u0027ll get 3/5, 5/8, 8/13."},{"Start":"00:53.580 ","End":"00:55.700","Text":"Let\u0027s just separate the terms with"},{"Start":"00:55.700 ","End":"01:00.700","Text":"the odd and even places and I\u0027ve already colored them differently to make it easier."},{"Start":"01:00.700 ","End":"01:03.480","Text":"For the odd terms,"},{"Start":"01:03.480 ","End":"01:09.030","Text":"we get 0 or 1/2, 3/8, 8/13."},{"Start":"01:09.030 ","End":"01:11.540","Text":"If you check with a calculator,"},{"Start":"01:11.540 ","End":"01:13.415","Text":"it appears to be increasing."},{"Start":"01:13.415 ","End":"01:20.340","Text":"Oh, I forgot to mention that the original sequence seems to be bounded between 0 and 1,"},{"Start":"01:20.340 ","End":"01:23.055","Text":"and the even terms,"},{"Start":"01:23.055 ","End":"01:27.210","Text":"we just look 1, 2/3, 5/8."},{"Start":"01:27.210 ","End":"01:29.610","Text":"If you check in the calculator,"},{"Start":"01:29.610 ","End":"01:32.850","Text":"this is 1.67,"},{"Start":"01:32.850 ","End":"01:38.555","Text":"0.625, these are actually decreasing."},{"Start":"01:38.555 ","End":"01:42.505","Text":"It appears to be at this stage, allegedly."},{"Start":"01:42.505 ","End":"01:47.125","Text":"We\u0027re going to prove the following and there is 5 things here."},{"Start":"01:47.125 ","End":"01:51.945","Text":"These 3 things are just things that we said already,"},{"Start":"01:51.945 ","End":"01:57.050","Text":"that the sequence appears to be bounded between 0 and 1."},{"Start":"01:57.050 ","End":"01:59.080","Text":"We\u0027ll actually prove that."},{"Start":"01:59.080 ","End":"02:01.910","Text":"The terms in the odd positions appear to"},{"Start":"02:01.910 ","End":"02:05.465","Text":"be an increasing sub-sequence and we\u0027ll prove that."},{"Start":"02:05.465 ","End":"02:11.300","Text":"The even positioned terms appear to be a decreasing subsequence and we\u0027ll prove that."},{"Start":"02:11.300 ","End":"02:13.865","Text":"Once we\u0027ve proved these,"},{"Start":"02:13.865 ","End":"02:18.115","Text":"then we can deduce for the odd terms."},{"Start":"02:18.115 ","End":"02:21.710","Text":"Monotonically increasing and bounded from above means"},{"Start":"02:21.710 ","End":"02:28.295","Text":"convergent and monotonically decreasing and bounded from below also means convergent,"},{"Start":"02:28.295 ","End":"02:32.155","Text":"so these 2 are both convergent."},{"Start":"02:32.155 ","End":"02:38.080","Text":"Monotonically increasing and bounded above or monotonically decreasing and bounded below."},{"Start":"02:38.080 ","End":"02:43.360","Text":"The last thing is that we\u0027ll show that both of these"},{"Start":"02:43.360 ","End":"02:48.480","Text":"converge to the same limit L and after that,"},{"Start":"02:48.480 ","End":"02:51.120","Text":"we\u0027ll also compute that limit."},{"Start":"02:51.120 ","End":"02:55.360","Text":"Since we have 2 subsequences that cover the original sequence,"},{"Start":"02:55.360 ","End":"02:58.330","Text":"because it\u0027s just the odds and the evens, of course, they cover,"},{"Start":"02:58.330 ","End":"03:01.240","Text":"so it follows there is a theorem about converging and"},{"Start":"03:01.240 ","End":"03:04.600","Text":"covering subsequences that the original sequence will also"},{"Start":"03:04.600 ","End":"03:12.090","Text":"converge to that same limit as these 2 subsequences."},{"Start":"03:12.090 ","End":"03:15.690","Text":"Now, we\u0027ll prove Part 3, the boundedness."},{"Start":"03:15.690 ","End":"03:21.965","Text":"In fact, we\u0027ll prove by induction that it\u0027s bounded between 0 and 1."},{"Start":"03:21.965 ","End":"03:24.315","Text":"For n equals 1,"},{"Start":"03:24.315 ","End":"03:28.290","Text":"that\u0027s clear because a_1 is 0,"},{"Start":"03:28.290 ","End":"03:30.465","Text":"so it is in this range."},{"Start":"03:30.465 ","End":"03:34.760","Text":"Now the induction stage from n to n plus 1 is that,"},{"Start":"03:34.760 ","End":"03:36.460","Text":"if it\u0027s true for n,"},{"Start":"03:36.460 ","End":"03:38.515","Text":"then it\u0027s true for n plus 1."},{"Start":"03:38.515 ","End":"03:41.461","Text":"We assume this and prove this."},{"Start":"03:41.461 ","End":"03:48.990","Text":"So a_n plus 1 is 1/1 plus a_n from the recursive definition."},{"Start":"03:48.990 ","End":"03:54.585","Text":"Because a_n is bigger or equals to 0,"},{"Start":"03:54.585 ","End":"03:58.800","Text":"if I decrease the denominator possibly then"},{"Start":"03:58.800 ","End":"04:03.080","Text":"I can only increase the fraction and this is equals to 1."},{"Start":"04:03.080 ","End":"04:06.760","Text":"We also have to show that it\u0027s bigger or equals to 0,"},{"Start":"04:06.760 ","End":"04:10.840","Text":"so a_n plus 1 again is equals to 1/1 plus a_n."},{"Start":"04:10.840 ","End":"04:12.980","Text":"This time, I\u0027m using the other part,"},{"Start":"04:12.980 ","End":"04:16.700","Text":"that a_n is less than or equal to"},{"Start":"04:16.700 ","End":"04:21.260","Text":"1 and because this is less than or equal to this and it\u0027s on the denominator,"},{"Start":"04:21.260 ","End":"04:25.625","Text":"then it\u0027s bigger or equal to here and this is a half."},{"Start":"04:25.625 ","End":"04:29.660","Text":"In any event, it\u0027s bigger than 0."},{"Start":"04:33.530 ","End":"04:37.270","Text":"That proves the boundedness."},{"Start":"04:37.270 ","End":"04:39.890","Text":"Now, we\u0027ll prove the monotonicity,"},{"Start":"04:39.890 ","End":"04:42.230","Text":"which is Part 1 and 2."},{"Start":"04:42.230 ","End":"04:48.590","Text":"Just going to derive a useful inequality that in general,"},{"Start":"04:48.590 ","End":"04:53.239","Text":"a_k plus 2 is 1/1 plus a_k plus"},{"Start":"04:53.239 ","End":"04:58.160","Text":"1 from the recursive definition will then equals k plus 1."},{"Start":"04:58.160 ","End":"04:59.990","Text":"Now, a_k plus 1,"},{"Start":"04:59.990 ","End":"05:04.080","Text":"I can also replace by its recursive definition,"},{"Start":"05:04.080 ","End":"05:07.150","Text":"is 1/1 plus a_k."},{"Start":"05:07.150 ","End":"05:09.945","Text":"This is equal to this."},{"Start":"05:09.945 ","End":"05:16.520","Text":"It gives me a formula for jumping 2 indices at once from a_k to a_k plus 2,"},{"Start":"05:16.520 ","End":"05:18.469","Text":"this will be useful."},{"Start":"05:18.469 ","End":"05:23.300","Text":"Part 1 is to show that the odds are increasing."},{"Start":"05:23.300 ","End":"05:26.525","Text":"The odd place terms,"},{"Start":"05:26.525 ","End":"05:28.220","Text":"2n minus 1,1,"},{"Start":"05:28.220 ","End":"05:30.005","Text":"the first, the third, the fifth,"},{"Start":"05:30.005 ","End":"05:34.470","Text":"which means that a_2n plus 1,"},{"Start":"05:34.470 ","End":"05:40.065","Text":"which is replacing n by n plus 1 is bigger or equal to a_2n minus 1."},{"Start":"05:40.065 ","End":"05:44.435","Text":"Actually, I\u0027m going to even show that it\u0027s strictly greater than."},{"Start":"05:44.435 ","End":"05:47.710","Text":"Proof by induction."},{"Start":"05:47.710 ","End":"05:50.040","Text":"For n equals 1,"},{"Start":"05:50.040 ","End":"05:53.325","Text":"we get that a_3,"},{"Start":"05:53.325 ","End":"05:55.515","Text":"which is a 1/2,"},{"Start":"05:55.515 ","End":"05:59.325","Text":"is bigger than 0, which is a_1."},{"Start":"05:59.325 ","End":"06:02.835","Text":"For n equals 1.5 is bigger than 0, that\u0027s clear."},{"Start":"06:02.835 ","End":"06:07.170","Text":"In general, to go from n to n plus 1,"},{"Start":"06:07.170 ","End":"06:10.070","Text":"we need to show that if this is true,"},{"Start":"06:10.070 ","End":"06:12.470","Text":"then it\u0027s true for the following n,"},{"Start":"06:12.470 ","End":"06:16.280","Text":"which means if you replace n by n plus 1, you get this."},{"Start":"06:16.280 ","End":"06:19.610","Text":"This is what we have to show, assuming this."},{"Start":"06:19.610 ","End":"06:22.580","Text":"We\u0027ll use this, I\u0027ll just put it in a box."},{"Start":"06:22.580 ","End":"06:24.565","Text":"This equals this."},{"Start":"06:24.565 ","End":"06:28.650","Text":"See if I let k equal 2n plus 1 here,"},{"Start":"06:28.650 ","End":"06:36.490","Text":"then we\u0027ll get that a_2n plus 3 is 1/1 plus 1/1 plus a_2n plus 1."},{"Start":"06:36.490 ","End":"06:38.015","Text":"Here, I\u0027m going to use"},{"Start":"06:38.015 ","End":"06:44.825","Text":"this induction hypothesis that a_2n plus 1 is bigger than a_2n minus 1."},{"Start":"06:44.825 ","End":"06:51.950","Text":"I claim that this implies that this is bigger than this."},{"Start":"06:51.950 ","End":"06:57.545","Text":"Basically, I have 2 reversal of direction of inequality."},{"Start":"06:57.545 ","End":"06:59.725","Text":"Best to maybe give an example,"},{"Start":"06:59.725 ","End":"07:04.905","Text":"if I put a_2n plus 1 is 7 and a_2n minus 1 is 4,"},{"Start":"07:04.905 ","End":"07:07.155","Text":"which is bigger, this or this?"},{"Start":"07:07.155 ","End":"07:09.630","Text":"Well, 7 is bigger than 4,"},{"Start":"07:09.630 ","End":"07:15.555","Text":"so 1 over this is less than 1 over this and also want to add 1."},{"Start":"07:15.555 ","End":"07:17.720","Text":"But then I put 1 over again,"},{"Start":"07:17.720 ","End":"07:22.715","Text":"so it\u0027s back to bigger than or equal to these 2 changes of direction."},{"Start":"07:22.715 ","End":"07:29.230","Text":"That shows this part and this by the formula,"},{"Start":"07:29.230 ","End":"07:36.390","Text":"with k being 2n minus 1 here is a_2n plus 1."},{"Start":"07:36.390 ","End":"07:38.730","Text":"I\u0027ll just subtract 2 from 2n plus 3,"},{"Start":"07:38.730 ","End":"07:41.835","Text":"I get 2 n plus 1."},{"Start":"07:41.835 ","End":"07:49.290","Text":"Now, we\u0027ll prove Part 2 that the even place members a_2n,"},{"Start":"07:49.290 ","End":"07:52.740","Text":"that\u0027s a decreasing subsequence."},{"Start":"07:52.740 ","End":"07:57.420","Text":"In fact, strictly decreasing and we\u0027ll prove it by induction."},{"Start":"07:57.420 ","End":"07:59.925","Text":"When n equals 1,"},{"Start":"07:59.925 ","End":"08:03.870","Text":"we just basically get that 2/3 is less than 1,"},{"Start":"08:03.870 ","End":"08:09.280","Text":"we\u0027ve already written out the first few terms and a_2 is bigger than a_4."},{"Start":"08:09.280 ","End":"08:11.890","Text":"Now, the induction case,"},{"Start":"08:11.890 ","End":"08:15.490","Text":"we\u0027re assuming that it\u0027s true for n and we\u0027ll"},{"Start":"08:15.490 ","End":"08:19.270","Text":"use this to prove that it\u0027s true for n plus 1,"},{"Start":"08:19.270 ","End":"08:23.210","Text":"this is what we get when we replace n by n plus 1."},{"Start":"08:24.290 ","End":"08:30.130","Text":"We\u0027ll recall this formula we just proved that gets from"},{"Start":"08:30.130 ","End":"08:36.050","Text":"a_k to a_k plus 2 and if we let k equals 2n plus 2 here,"},{"Start":"08:36.050 ","End":"08:41.760","Text":"then we get 2n plus 4 in terms of 2n plus 2, and like so."},{"Start":"08:41.760 ","End":"08:43.995","Text":"From the induction hypothesis,"},{"Start":"08:43.995 ","End":"08:48.460","Text":"a_2n plus 2 is less than a_2n."},{"Start":"08:48.800 ","End":"08:55.430","Text":"So we get an inequality from this that this is also less than this."},{"Start":"08:55.430 ","End":"09:01.680","Text":"There are actually 2 reversals of sine because we have 1 over twice,"},{"Start":"09:01.680 ","End":"09:05.915","Text":"it flips to greater than and then it flips back to less than."},{"Start":"09:05.915 ","End":"09:10.985","Text":"You can check which of these 2 is bigger for example, a numerical example."},{"Start":"09:10.985 ","End":"09:19.320","Text":"Anyway, we\u0027ve shown that a_2n plus 4 is less than a_2n plus 2,"},{"Start":"09:19.320 ","End":"09:24.130","Text":"which is with k equals 2n here."},{"Start":"09:24.130 ","End":"09:29.255","Text":"That\u0027s the induction part that we wanted to show."},{"Start":"09:29.255 ","End":"09:33.425","Text":"Now, we come to Part 5. We\u0027re getting there."},{"Start":"09:33.425 ","End":"09:36.170","Text":"Recall this formula that we proved?"},{"Start":"09:36.170 ","End":"09:41.750","Text":"Let\u0027s apply it first of all to the even members."},{"Start":"09:41.750 ","End":"09:48.150","Text":"Let k equals 2n and we get this formula."},{"Start":"09:49.610 ","End":"09:52.580","Text":"For the odd places,"},{"Start":"09:52.580 ","End":"09:54.470","Text":"we got this formula."},{"Start":"09:54.470 ","End":"09:58.160","Text":"If you let k equals 2n minus 1, we get this."},{"Start":"09:58.160 ","End":"10:01.705","Text":"In each case, if we let L be the limit."},{"Start":"10:01.705 ","End":"10:08.480","Text":"Here, I\u0027ll let L be the limit of a_2n and a_2n and a_2n plus 2 have the same limit."},{"Start":"10:08.480 ","End":"10:12.470","Text":"It\u0027s just offset by 1, same limit."},{"Start":"10:12.470 ","End":"10:16.410","Text":"Similarly here, 2n minus 1 or 2n plus"},{"Start":"10:16.410 ","End":"10:22.310","Text":"1 is the same sequence just with a missing term in one of them."},{"Start":"10:22.310 ","End":"10:28.770","Text":"They both satisfy both the limit of a_2n and the limit of a_2n minus 1."},{"Start":"10:28.770 ","End":"10:32.400","Text":"If I call them L, they\u0027ll satisfy this equation."},{"Start":"10:32.400 ","End":"10:36.020","Text":"This equation, if you simplify it,"},{"Start":"10:36.020 ","End":"10:40.910","Text":"the denominator becomes 2 plus L/1 plus L and then 1 over that,"},{"Start":"10:40.910 ","End":"10:43.700","Text":"so we\u0027ve got L equals this."},{"Start":"10:43.700 ","End":"10:48.580","Text":"If we rearrange it algebraically to a quadratic equation,"},{"Start":"10:48.580 ","End":"10:50.005","Text":"which is this,"},{"Start":"10:50.005 ","End":"10:54.635","Text":"this gives us 2 solutions, the quadratic equation."},{"Start":"10:54.635 ","End":"10:59.150","Text":"But our sequence is non-negative,"},{"Start":"10:59.150 ","End":"11:08.900","Text":"so the negative is ruled out and this is the limit of each of the 2 subsequences,"},{"Start":"11:08.900 ","End":"11:10.595","Text":"the odds and the evens."},{"Start":"11:10.595 ","End":"11:14.734","Text":"Now, there is a theorem that we\u0027ve used before."},{"Start":"11:14.734 ","End":"11:20.000","Text":"We\u0027ve shown that the sequence has 2 subsequences and they cover,"},{"Start":"11:20.000 ","End":"11:23.240","Text":"because one is the odds and one is the evens so together it\u0027s everything and"},{"Start":"11:23.240 ","End":"11:27.725","Text":"they both or each converge to the same thing, which is this."},{"Start":"11:27.725 ","End":"11:34.290","Text":"It follows that the original sequence also converges on to this same thing."},{"Start":"11:34.290 ","End":"11:38.530","Text":"Finally, we are done."}],"ID":31212}],"Thumbnail":null,"ID":294567},{"Name":"Proving Divergence of a Sequence Using 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37s","ChapterTopicVideoID":30481,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.510 ","End":"00:05.820","Text":"In this tutorial clip, we\u0027ll talkabout disproving limits of sequences as"},{"Start":"00:05.820 ","End":"00:08.220","Text":"opposed to what we\u0027ve been doing so far."},{"Start":"00:08.220 ","End":"00:12.480","Text":"Mainly is proving limits ofsequences, but the disproving will"},{"Start":"00:12.480 ","End":"00:15.840","Text":"be from the definition of the limit."},{"Start":"00:16.170 ","End":"00:20.850","Text":"I\u0027m talking about what we sometimes callthe epsilon and definition of a limit."},{"Start":"00:21.270 ","End":"00:25.270","Text":"So let me explain the contextis that we have a sequence, a."},{"Start":"00:26.025 ","End":"00:30.525","Text":"An infinite sequence, usuallyfrom one to infinity, and we have"},{"Start":"00:30.525 ","End":"00:33.465","Text":"some real number L, L for limit."},{"Start":"00:33.495 ","End":"00:37.245","Text":"Then to prove that the limit of an is L."},{"Start":"00:37.515 ","End":"00:41.595","Text":"Often we just write an tends toL and we prove the following."},{"Start":"00:41.894 ","End":"00:45.015","Text":"And this is the definition of the limit."},{"Start":"00:45.735 ","End":"00:50.855","Text":"And by the way, this symbol is such that,and it\u0027s often omitted all together."},{"Start":"00:51.644 ","End":"00:57.405","Text":"In words, what this says is, forall epsilon bigger than zero, there"},{"Start":"00:57.405 ","End":"01:04.664","Text":"exists n nor such that for all Nbigger than n nor the absolute value"},{"Start":"01:04.664 ","End":"01:08.414","Text":"of a N minus L is less than Epsilon."},{"Start":"01:08.745 ","End":"01:14.195","Text":"And I\u0027m assuming, I didn\u0027t say it,that the N and the N nor in the"},{"Start":"01:14.200 ","End":"01:16.095","Text":"definition are natural numbers."},{"Start":"01:17.295 ","End":"01:20.535","Text":"That was just a review ofhow we prove the following."},{"Start":"01:20.955 ","End":"01:26.715","Text":"Now we\u0027re going to learn how to prove thenegation of this, that the limit of an is"},{"Start":"01:26.775 ","End":"01:31.395","Text":"not equal to L or an does not tend to L."},{"Start":"01:31.845 ","End":"01:33.255","Text":"Then we prove the following."},{"Start":"01:34.005 ","End":"01:39.345","Text":"Which is the logical negation ofwhat it says here in case you\u0027re"},{"Start":"01:39.345 ","End":"01:41.715","Text":"wondering how we got to this."},{"Start":"01:41.745 ","End":"01:43.125","Text":"You could just accept it as is."},{"Start":"01:43.125 ","End":"01:48.585","Text":"But if you\u0027re wondering, the way we doit is we change for all to their exists."},{"Start":"01:48.615 ","End":"01:51.074","Text":"Their exists becomes for all."},{"Start":"01:51.225 ","End":"01:53.205","Text":"For all becomes their exists."},{"Start":"01:53.475 ","End":"02:00.565","Text":"That the end, we negate the statement so,Say that this is less than Epsilon is the"},{"Start":"02:00.765 ","End":"02:04.125","Text":"opposite of bigger or equal to epsilon."},{"Start":"02:04.130 ","End":"02:08.984","Text":"So that\u0027s how we get the negationof a logical statement in words."},{"Start":"02:08.984 ","End":"02:14.355","Text":"What we have to show is that thereexists in Epsilon bigger than zero."},{"Start":"02:14.655 ","End":"02:20.445","Text":"Such that for all n nor there existsan N bigger or equal to n nor such"},{"Start":"02:20.445 ","End":"02:25.774","Text":"that the absolute value of a N minusL is bigger or equal to epsilon."},{"Start":"02:26.055 ","End":"02:30.614","Text":"And all you have to do is rememberthis or use this when you\u0027re gonna"},{"Start":"02:30.620 ","End":"02:33.255","Text":"prove non-existence of a limit."},{"Start":"02:33.555 ","End":"02:36.495","Text":"Now a remark though, youprobably know this when we."},{"Start":"02:37.110 ","End":"02:39.720","Text":"The limit of AAN is not equal to L."},{"Start":"02:39.720 ","End":"02:40.980","Text":"It can be one of two things."},{"Start":"02:41.010 ","End":"02:45.090","Text":"It either means that the limitof AAN doesn\u0027t exist at all, or"},{"Start":"02:45.090 ","End":"02:50.970","Text":"it exists and is equal to somelittle L, which is not equal to L."},{"Start":"02:51.060 ","End":"02:51.540","Text":"Okay."},{"Start":"02:51.540 ","End":"02:53.550","Text":"That\u0027s it for the tutorial."},{"Start":"02:53.610 ","End":"02:55.890","Text":"Most of the learningcomes through examples."},{"Start":"02:56.400 ","End":"03:01.050","Text":"I\u0027ll start with an example here,and following the tutorial,"},{"Start":"03:01.260 ","End":"03:03.240","Text":"there\u0027ll be other examples."},{"Start":"03:03.720 ","End":"03:09.570","Text":"The task here is to disprove theclaim that the limit as n goes"},{"Start":"03:09.570 ","End":"03:11.700","Text":"to infinity of one over N is one."},{"Start":"03:11.850 ","End":"03:15.239","Text":"Using the definition, if we didn\u0027thave to use the definition, it would"},{"Start":"03:15.244 ","End":"03:19.410","Text":"be easy because we can say, ah,yeah, the limit of one over N is"},{"Start":"03:19.410 ","End":"03:21.690","Text":"zero, and zero is not equal to one."},{"Start":"03:21.869 ","End":"03:24.299","Text":"But the trick is hereto use the definition."},{"Start":"03:24.390 ","End":"03:26.880","Text":"This definition, I mean, or in words."},{"Start":"03:27.870 ","End":"03:29.790","Text":"The tricky part is to choose."},{"Start":"03:29.820 ","End":"03:32.670","Text":"The epsilon gets easier with experience."},{"Start":"03:32.700 ","End":"03:37.000","Text":"Here, we\u0027ll choose epsilon equalsa half, although any epsilon"},{"Start":"03:37.005 ","End":"03:38.880","Text":"between zero and one will do."},{"Start":"03:39.090 ","End":"03:43.920","Text":"The thinking is that we know thatone over end goes to zero and the"},{"Start":"03:43.920 ","End":"03:46.500","Text":"distance from zero to one is one."},{"Start":"03:46.505 ","End":"03:50.160","Text":"So we wanna choose someepsilon that\u0027s less than one."},{"Start":"03:50.310 ","End":"03:56.340","Text":"Once we\u0027ve chosen Epsilon, And all we haveto do is prove this remainder that for"},{"Start":"03:56.340 ","End":"04:02.170","Text":"all n nor there exists and big equal toend nor such that a N minus L in absolute"},{"Start":"04:02.174 ","End":"04:04.560","Text":"value is bigger or equal to epsilon."},{"Start":"04:04.739 ","End":"04:08.609","Text":"We know what a and l and epsilon are,so this translates to proving that"},{"Start":"04:08.609 ","End":"04:13.200","Text":"for all end nor there exists and nbigger equal to end, nor such that"},{"Start":"04:13.200 ","End":"04:17.849","Text":"what one over and minus one in absolutevalue is bigger or equal to a half."},{"Start":"04:18.149 ","End":"04:21.539","Text":"We can get rid of theabsolute value as follows."},{"Start":"04:22.125 ","End":"04:26.475","Text":"One over N minus one is less thanor equal to zero, cuz one over"},{"Start":"04:26.475 ","End":"04:28.875","Text":"N is less than or equal to one."},{"Start":"04:28.995 ","End":"04:34.425","Text":"So when we put the absolute value, wenegate it, we can reverse the order of the"},{"Start":"04:34.430 ","End":"04:38.295","Text":"term so it becomes one minus one over N."},{"Start":"04:38.895 ","End":"04:45.015","Text":"Then put that in here and weget that one minus one over N"},{"Start":"04:45.135 ","End":"04:47.085","Text":"is bigger or equal to a half."},{"Start":"04:47.145 ","End":"04:48.315","Text":"And this is the same."},{"Start":"04:49.260 ","End":"04:53.580","Text":"One half bigger or equal to one overN cuz just bring the half to the"},{"Start":"04:53.580 ","End":"04:55.409","Text":"left and one over end to the right."},{"Start":"04:56.219 ","End":"04:59.940","Text":"Now we take the reciprocals, butthen we have to change the order."},{"Start":"05:00.390 ","End":"05:04.169","Text":"So this is equivalent toN bigger or equal to two."},{"Start":"05:04.260 ","End":"05:08.280","Text":"So this holds when N is bigger orequal to two, but the requirement is"},{"Start":"05:08.280 ","End":"05:10.469","Text":"to find N bigger or equal to end not."},{"Start":"05:10.469 ","End":"05:11.729","Text":"So we can satisfy both of."},{"Start":"05:12.630 ","End":"05:17.099","Text":"By choosing N to be themaximum of two and N not."},{"Start":"05:17.219 ","End":"05:22.049","Text":"And then N is both big or equal toN, not and bigger or equal to two."},{"Start":"05:22.109 ","End":"05:26.729","Text":"And we also have that thisinequality holds as required."},{"Start":"05:26.789 ","End":"05:29.400","Text":"And that concludes the example."},{"Start":"05:29.489 ","End":"05:34.140","Text":"Like I said, uh, there are moreexamples following this tutorial."},{"Start":"05:34.229 ","End":"05:34.859","Text":"Okay."},{"Start":"05:35.130 ","End":"05:36.510","Text":"That concludes this clip."}],"ID":32617},{"Watched":false,"Name":"Exercise 1","Duration":"3m 25s","ChapterTopicVideoID":26321,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to prove that the limit as n goes to"},{"Start":"00:03.480 ","End":"00:07.185","Text":"infinity of 4n plus 1 plus 2 is not equal to 3."},{"Start":"00:07.185 ","End":"00:11.325","Text":"We have to use the definition of a limit with the Epsilon."},{"Start":"00:11.325 ","End":"00:13.515","Text":"This is a bit ambiguous."},{"Start":"00:13.515 ","End":"00:16.710","Text":"This could mean that the limit exists and is not equal to 3,"},{"Start":"00:16.710 ","End":"00:20.250","Text":"or it could mean that the limit doesn\u0027t exist at all, either will count."},{"Start":"00:20.250 ","End":"00:24.225","Text":"For the solution in general,"},{"Start":"00:24.225 ","End":"00:28.275","Text":"when we show that the limit as n goes to infinity of an is not equal to this L,"},{"Start":"00:28.275 ","End":"00:32.790","Text":"we have to show that there exists an Epsilon bigger than 0,"},{"Start":"00:32.790 ","End":"00:37.020","Text":"such that for all n_0 natural number,"},{"Start":"00:37.020 ","End":"00:40.550","Text":"there exists an n bigger or equal to n_0 such"},{"Start":"00:40.550 ","End":"00:44.510","Text":"that the absolute value of a_n minus L big or equal to Epsilon."},{"Start":"00:44.510 ","End":"00:46.655","Text":"Where did this come from?"},{"Start":"00:46.655 ","End":"00:49.715","Text":"Well, we start from the definition of the limit does"},{"Start":"00:49.715 ","End":"00:54.515","Text":"exist and that means that for all Epsilon bigger than 0,"},{"Start":"00:54.515 ","End":"00:57.110","Text":"there exists n_0 in N,"},{"Start":"00:57.110 ","End":"01:00.890","Text":"such that for all n bigger than n_0,"},{"Start":"01:00.890 ","End":"01:03.800","Text":"absolute value of a_n minus L less than Epsilon."},{"Start":"01:03.800 ","End":"01:05.960","Text":"We don\u0027t need to include words like,"},{"Start":"01:05.960 ","End":"01:09.350","Text":"such that, just write them all contiguously."},{"Start":"01:09.350 ","End":"01:11.360","Text":"Now to negate that,"},{"Start":"01:11.360 ","End":"01:13.820","Text":"you have to reverse the quantifiers,"},{"Start":"01:13.820 ","End":"01:15.215","Text":"wherever you see for all,"},{"Start":"01:15.215 ","End":"01:16.415","Text":"you put there exists."},{"Start":"01:16.415 ","End":"01:18.140","Text":"Here there exists, for all."},{"Start":"01:18.140 ","End":"01:20.365","Text":"Here for all, here there exists."},{"Start":"01:20.365 ","End":"01:23.760","Text":"Finally, you take the statement and negate it."},{"Start":"01:23.760 ","End":"01:25.305","Text":"Instead of less than Epsilon,"},{"Start":"01:25.305 ","End":"01:28.230","Text":"we have bigger or equal to Epsilon."},{"Start":"01:28.390 ","End":"01:30.785","Text":"That gives us this,"},{"Start":"01:30.785 ","End":"01:33.365","Text":"and what does it mean in our case?"},{"Start":"01:33.365 ","End":"01:37.580","Text":"This is the part we replace absolute value of 4n plus 1 plus"},{"Start":"01:37.580 ","End":"01:41.540","Text":"2 minus 3 bigger than or equal to Epsilon."},{"Start":"01:41.540 ","End":"01:44.840","Text":"Let\u0027s compute absolute value of an minus L,"},{"Start":"01:44.840 ","End":"01:48.400","Text":"which is this simplified algebraically."},{"Start":"01:48.400 ","End":"01:50.420","Text":"I don\u0027t think it needs any comment,"},{"Start":"01:50.420 ","End":"01:53.570","Text":"from here we put a common denominator,"},{"Start":"01:53.570 ","End":"01:55.985","Text":"collect some terms together,"},{"Start":"01:55.985 ","End":"01:57.769","Text":"put the n plus 2 separately."},{"Start":"01:57.769 ","End":"02:03.075","Text":"The reason for that is you want this to be a 1 minus 7 plus 2."},{"Start":"02:03.075 ","End":"02:05.430","Text":"We could throw out the absolute value,"},{"Start":"02:05.430 ","End":"02:09.865","Text":"if we knew that 7 plus 2 was less than or equal to 1."},{"Start":"02:09.865 ","End":"02:12.750","Text":"Because then this will be something non-negative,"},{"Start":"02:12.750 ","End":"02:14.780","Text":"so we can throw out the absolute value,"},{"Start":"02:14.780 ","End":"02:18.625","Text":"and we\u0027d like for this to be bigger or equal to Epsilon."},{"Start":"02:18.625 ","End":"02:21.755","Text":"Let\u0027s choose Epsilon equals 1/2."},{"Start":"02:21.755 ","End":"02:24.560","Text":"It could be anything 0 and 1, I suppose."},{"Start":"02:24.560 ","End":"02:25.820","Text":"A half is nice."},{"Start":"02:25.820 ","End":"02:29.285","Text":"We want this to be bigger or equal to Epsilon,"},{"Start":"02:29.285 ","End":"02:33.710","Text":"which is 1/2 and just rearranging a bit,"},{"Start":"02:33.710 ","End":"02:37.640","Text":"this comes out to 7 plus 2 less than or equal to 1/2,"},{"Start":"02:37.640 ","End":"02:41.455","Text":"which is also less than or equal to 1."},{"Start":"02:41.455 ","End":"02:45.935","Text":"This reduces to n bigger or equal to 12."},{"Start":"02:45.935 ","End":"02:48.485","Text":"Now suppose we\u0027re given n naught,"},{"Start":"02:48.485 ","End":"02:50.495","Text":"we have to find the n for it."},{"Start":"02:50.495 ","End":"02:58.355","Text":"I say, let n be maximum of 12 and n_0 and that way n is bigger or equal to n_0,"},{"Start":"02:58.355 ","End":"03:00.350","Text":"which is what\u0027s required."},{"Start":"03:00.350 ","End":"03:03.990","Text":"Also n is bigger or equal to 12 and"},{"Start":"03:03.990 ","End":"03:08.025","Text":"so what\u0027s above will be true that"},{"Start":"03:08.025 ","End":"03:12.610","Text":"a_n minus L absolute value will be bigger or equal to Epsilon."},{"Start":"03:12.610 ","End":"03:15.085","Text":"Summarizing, we found the 2 things."},{"Start":"03:15.085 ","End":"03:22.670","Text":"We found Epsilon and also we found the formula that given n_0 tells us what n should be,"},{"Start":"03:22.670 ","End":"03:26.370","Text":"and we\u0027re done with this exercise."}],"ID":27228},{"Watched":false,"Name":"Exercise 2","Duration":"3m 48s","ChapterTopicVideoID":26322,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to prove that the limit is n goes to"},{"Start":"00:04.020 ","End":"00:08.580","Text":"infinity of n plus 10/4 n plus 2 is not equal to 1/2."},{"Start":"00:08.580 ","End":"00:14.670","Text":"And we do this using the definition of a limit using the Epsilon n thing."},{"Start":"00:14.670 ","End":"00:16.770","Text":"I\u0027ll remind you that in general,"},{"Start":"00:16.770 ","End":"00:21.930","Text":"to show that the sequence doesn\u0027t converge to a limit L,"},{"Start":"00:21.930 ","End":"00:29.310","Text":"what we have to show is there exists an Epsilon such that for all n naught,"},{"Start":"00:29.310 ","End":"00:33.870","Text":"there is some n bigger or equal to n naught for"},{"Start":"00:33.870 ","End":"00:39.810","Text":"which absolute value of a_n minus L is bigger or equal to Epsilon."},{"Start":"00:39.810 ","End":"00:42.225","Text":"The n here is the n here."},{"Start":"00:42.225 ","End":"00:45.215","Text":"In case you\u0027re wondering where this came from."},{"Start":"00:45.215 ","End":"00:53.360","Text":"Just comes from the logical negation of the definition of convergence to a limit"},{"Start":"00:53.360 ","End":"00:55.910","Text":"of a sequence and we just reverse"},{"Start":"00:55.910 ","End":"01:01.850","Text":"the quantifiers for all becomes there exists and vice versa."},{"Start":"01:01.850 ","End":"01:07.190","Text":"The statement less than Epsilon becomes bigger or equal to Epsilon."},{"Start":"01:07.190 ","End":"01:09.275","Text":"And for our particular case,"},{"Start":"01:09.275 ","End":"01:12.080","Text":"what we have to show essentially the same as this,"},{"Start":"01:12.080 ","End":"01:16.910","Text":"except that we can replace the sequence a_n and the limit L. We know what they are."},{"Start":"01:16.910 ","End":"01:20.170","Text":"So a_n is n plus 10/4 n plus 2,"},{"Start":"01:20.170 ","End":"01:21.925","Text":"and L is a 1/2."},{"Start":"01:21.925 ","End":"01:23.809","Text":"Let\u0027s compute this difference,"},{"Start":"01:23.809 ","End":"01:26.420","Text":"which is a_n minus L absolute value."},{"Start":"01:26.420 ","End":"01:29.465","Text":"Believe algebra put a common denominator,"},{"Start":"01:29.465 ","End":"01:30.710","Text":"4n plus 2,"},{"Start":"01:30.710 ","End":"01:35.160","Text":"which means multiplying here top and bottom by 2n plus 1."},{"Start":"01:35.260 ","End":"01:38.735","Text":"Then we can organize it."},{"Start":"01:38.735 ","End":"01:40.670","Text":"We have 9 minus n,"},{"Start":"01:40.670 ","End":"01:43.850","Text":"but we can reverse it because the absolute value of a"},{"Start":"01:43.850 ","End":"01:47.705","Text":"negative is the same as the absolute value of the positive,"},{"Start":"01:47.705 ","End":"01:53.360","Text":"and take a 1/4 out and multiply the numerator here by 4,"},{"Start":"01:53.360 ","End":"01:57.200","Text":"want the same coefficient event top and bottom,"},{"Start":"01:57.200 ","End":"02:00.185","Text":"and this is 4n plus 2 minus 38."},{"Start":"02:00.185 ","End":"02:04.120","Text":"Now, this over this is one so we have this."},{"Start":"02:04.120 ","End":"02:06.740","Text":"Then we can throw the absolute value of y,"},{"Start":"02:06.740 ","End":"02:09.665","Text":"provided that this is less than one."},{"Start":"02:09.665 ","End":"02:13.250","Text":"We want this to be bigger or equal to Epsilon."},{"Start":"02:13.250 ","End":"02:14.600","Text":"Well, we haven\u0027t given Epsilon yet."},{"Start":"02:14.600 ","End":"02:19.380","Text":"Let\u0027s see what Epsilon could be or should be wanted strictly less than 1."},{"Start":"02:19.380 ","End":"02:22.560","Text":"Otherwise, this is going to be 0 and Epsilon will be 0."},{"Start":"02:22.560 ","End":"02:23.990","Text":"Now I need Epsilon to be positive,"},{"Start":"02:23.990 ","End":"02:26.090","Text":"so we take it strictly less than 1."},{"Start":"02:26.090 ","End":"02:29.719","Text":"If you solve this inequality 2n plus 1 is bigger than 19."},{"Start":"02:29.719 ","End":"02:31.850","Text":"So n is bigger than 9,"},{"Start":"02:31.850 ","End":"02:34.010","Text":"means it\u0027s at least 10."},{"Start":"02:34.010 ","End":"02:36.185","Text":"If n equals 10,"},{"Start":"02:36.185 ","End":"02:39.875","Text":"what we get is that this becomes,"},{"Start":"02:39.875 ","End":"02:41.496","Text":"let\u0027s see, 19/21,"},{"Start":"02:41.496 ","End":"02:46.625","Text":"1 minus this is 2/21,"},{"Start":"02:46.625 ","End":"02:50.480","Text":"divided by 4 is 1/42."},{"Start":"02:50.480 ","End":"02:55.625","Text":"So that means the absolute value of a_n minus l is equal to this."},{"Start":"02:55.625 ","End":"03:01.430","Text":"Note that this function is an increasing function of n. Why?"},{"Start":"03:01.430 ","End":"03:04.055","Text":"Because the denominator increases with n,"},{"Start":"03:04.055 ","End":"03:10.130","Text":"so 19 over it decreases and 1 minus that makes it increasing again."},{"Start":"03:10.130 ","End":"03:14.155","Text":"So this, when n is bigger or equal to 10,"},{"Start":"03:14.155 ","End":"03:17.695","Text":"is bigger or equal to 1/42."},{"Start":"03:17.695 ","End":"03:19.585","Text":"That will be our Epsilon."},{"Start":"03:19.585 ","End":"03:24.065","Text":"So, yeah, so all we have to do is specify how we get n from n naught."},{"Start":"03:24.065 ","End":"03:29.090","Text":"I say let n be the maximum of n naught and 10."},{"Start":"03:29.090 ","End":"03:30.290","Text":"Because we need two things,"},{"Start":"03:30.290 ","End":"03:33.020","Text":"we need n bigger or equal to n naught, the requirement,"},{"Start":"03:33.020 ","End":"03:38.085","Text":"and we also want n bigger or equal to 10 so that this will hold."},{"Start":"03:38.085 ","End":"03:42.080","Text":"The inequality we want absolute value of n minus L is bigger or equal to"},{"Start":"03:42.080 ","End":"03:46.220","Text":"Epsilon when n is bigger or equal to n naught."},{"Start":"03:46.220 ","End":"03:48.870","Text":"We are done."}],"ID":27229},{"Watched":false,"Name":"Exercise 3","Duration":"2m 52s","ChapterTopicVideoID":26323,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this exercise we have to prove that it\u0027s not true that"},{"Start":"00:05.100 ","End":"00:10.545","Text":"the limit as n goes to infinity of this quotient is 1."},{"Start":"00:10.545 ","End":"00:16.605","Text":"It could be that the limit doesn\u0027t exist or it could exist and not equal 1 either way."},{"Start":"00:16.605 ","End":"00:21.430","Text":"Anyway, we\u0027re going to do this using definition of a limit with the Epsilon."},{"Start":"00:21.430 ","End":"00:27.680","Text":"Let\u0027s begin. In general when we show that something is not a limit,"},{"Start":"00:27.680 ","End":"00:31.025","Text":"we negate the definition."},{"Start":"00:31.025 ","End":"00:36.050","Text":"Let me show you what I mean. To say that the limit of a_n is L and"},{"Start":"00:36.050 ","End":"00:41.275","Text":"this is the formal definition with the rule and there exists and so on."},{"Start":"00:41.275 ","End":"00:43.185","Text":"If you negate this,"},{"Start":"00:43.185 ","End":"00:45.420","Text":"you get this statement."},{"Start":"00:45.420 ","End":"00:47.790","Text":"It\u0027s adapted to our case,"},{"Start":"00:47.790 ","End":"00:51.870","Text":"we just have to replace a_n and L by what they are."},{"Start":"00:51.870 ","End":"00:56.055","Text":"This will be a_n and L will be 1."},{"Start":"00:56.055 ","End":"00:59.580","Text":"Just the same as this, but spelled out."},{"Start":"00:59.580 ","End":"01:01.455","Text":"There\u0027s really 2 parts to this."},{"Start":"01:01.455 ","End":"01:05.010","Text":"One of them is to find the Epsilon that\u0027s bigger than 0."},{"Start":"01:05.010 ","End":"01:08.120","Text":"It exists, so we have to say what it is."},{"Start":"01:08.120 ","End":"01:15.340","Text":"Secondly this part can be expressed as a rule that given n_0,"},{"Start":"01:15.340 ","End":"01:17.920","Text":"we can say what n is,"},{"Start":"01:17.920 ","End":"01:22.730","Text":"that\u0027s bigger or equal to n_0 but the rule to get from n_0 those are 2 things."},{"Start":"01:22.730 ","End":"01:27.070","Text":"But let\u0027s start by simplifying this using a bit of algebra."},{"Start":"01:27.070 ","End":"01:31.803","Text":"Common denominator of 2n^2 plus 2 will give us this,"},{"Start":"01:31.803 ","End":"01:37.030","Text":"and then simplify the top and then we just have to play around with it."},{"Start":"01:37.030 ","End":"01:40.610","Text":"For example, I could get rid of this n. Well,"},{"Start":"01:40.610 ","End":"01:44.030","Text":"before that it should really say that I can drop the absolute value."},{"Start":"01:44.030 ","End":"01:49.160","Text":"Denominator is positive, this is positive and n minus 1 is non-negative."},{"Start":"01:49.160 ","End":"01:51.560","Text":"Denominator is positive,"},{"Start":"01:51.560 ","End":"01:57.455","Text":"so we can only reduce it if we make n zero and we can write"},{"Start":"01:57.455 ","End":"02:03.470","Text":"this as 2n^2d plus 2 minus 3 to get this the same as this."},{"Start":"02:03.470 ","End":"02:09.030","Text":"That\u0027s equal to 1 minus 3 over 2n^2 plus 2."},{"Start":"02:09.030 ","End":"02:13.665","Text":"This is bigger or equal to 1 minus 3/4"},{"Start":"02:13.665 ","End":"02:19.815","Text":"because 2n^2 plus 2 is bigger or equal to 2 plus 2 which is 4."},{"Start":"02:19.815 ","End":"02:22.635","Text":"This is bigger or equal to 1/4."},{"Start":"02:22.635 ","End":"02:26.420","Text":"We want this to be bigger or equal to Epsilon,"},{"Start":"02:26.420 ","End":"02:31.740","Text":"so just choose Epsilon equals 1/4 and we get this for any"},{"Start":"02:31.740 ","End":"02:37.550","Text":"n. All we have to do is let n equals n_0 for example or something bigger,"},{"Start":"02:37.550 ","End":"02:39.725","Text":"and then n will be bigger or equal to n_0."},{"Start":"02:39.725 ","End":"02:43.790","Text":"This condition, this will be bigger or equal"},{"Start":"02:43.790 ","End":"02:48.110","Text":"to 1/4 because there\u0027s no assumptions made on n here."},{"Start":"02:48.110 ","End":"02:52.890","Text":"That fulfills the requirement and we are done."}],"ID":27230},{"Watched":false,"Name":"Exercise 4","Duration":"3m 31s","ChapterTopicVideoID":26324,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.790","Text":"In this exercise, we have to prove that the limit of this expression is not equal to 1,"},{"Start":"00:07.790 ","End":"00:10.755","Text":"using the definition of a limit."},{"Start":"00:10.755 ","End":"00:13.410","Text":"To say that this limit is not equal to 1,"},{"Start":"00:13.410 ","End":"00:18.525","Text":"means that either it exists and is not equal to 1 or it doesn\u0027t exist."},{"Start":"00:18.525 ","End":"00:27.149","Text":"The negation of saying that the limit exists is this statement here."},{"Start":"00:27.160 ","End":"00:33.200","Text":"I\u0027ll remind you to say that something is the limit is this statement and"},{"Start":"00:33.200 ","End":"00:39.230","Text":"the logical negation of this is this and that\u0027s what we have here."},{"Start":"00:39.230 ","End":"00:43.610","Text":"That\u0027s in general and in our case we know what a_n is."},{"Start":"00:43.610 ","End":"00:49.070","Text":"We\u0027re letting a_n equal this general term and L is 1."},{"Start":"00:49.070 ","End":"00:51.635","Text":"The same as this except that we have to show that"},{"Start":"00:51.635 ","End":"00:55.380","Text":"this minus 1 bigger or equal to Epsilon."},{"Start":"00:55.460 ","End":"01:03.906","Text":"We want to do a bit of algebra to work on this and make a common denominator."},{"Start":"01:03.906 ","End":"01:08.690","Text":"Collect like terms in the numerator and"},{"Start":"01:08.690 ","End":"01:14.375","Text":"then we can play with it to get a bit like the denominator,"},{"Start":"01:14.375 ","End":"01:19.260","Text":"so that this is 1 and then something."},{"Start":"01:20.570 ","End":"01:26.060","Text":"Then what we can do is say that,"},{"Start":"01:26.060 ","End":"01:31.110","Text":"well, this part here is less than or equal to 1/2."},{"Start":"01:31.300 ","End":"01:36.665","Text":"Suppose that this is less than or equal to 1/2,"},{"Start":"01:36.665 ","End":"01:39.380","Text":"any number between 0 and 1 really,"},{"Start":"01:39.380 ","End":"01:41.985","Text":"but the 1/2 is nicely in the middle."},{"Start":"01:41.985 ","End":"01:43.880","Text":"If this is true,"},{"Start":"01:43.880 ","End":"01:47.525","Text":"then what we have here is bigger or equal to 1 minus 1/2,"},{"Start":"01:47.525 ","End":"01:50.740","Text":"which is the 1/2, and that will be our Epsilon."},{"Start":"01:50.740 ","End":"01:57.300","Text":"Now we just have to work on this condition to see when this is less than or equal to 1/2."},{"Start":"01:58.970 ","End":"02:03.630","Text":"If we cross multiply and throw out the 2 here and here,"},{"Start":"02:03.630 ","End":"02:09.820","Text":"we get that 3 times n plus 1 is less than or equal to n^2 plus 1,"},{"Start":"02:09.820 ","End":"02:13.405","Text":"collected as a quadratic expression,"},{"Start":"02:13.405 ","End":"02:16.900","Text":"bigger or equal to 0."},{"Start":"02:17.210 ","End":"02:23.765","Text":"When we have a parabola that\u0027s facing up,"},{"Start":"02:23.765 ","End":"02:28.460","Text":"that\u0027s the right way up parabola,"},{"Start":"02:28.460 ","End":"02:32.900","Text":"then when you have the 2 roots n will either be less than or"},{"Start":"02:32.900 ","End":"02:37.620","Text":"equal to the smaller one or greater or equal to the larger of the 2 roots."},{"Start":"02:37.620 ","End":"02:43.445","Text":"Larger of the 2 roots is 3 plus square root of 17 over 2 using the formula."},{"Start":"02:43.445 ","End":"02:45.290","Text":"So if n is bigger or equal to this,"},{"Start":"02:45.290 ","End":"02:48.370","Text":"then this will be bigger or equal to 0."},{"Start":"02:48.370 ","End":"02:51.960","Text":"Square root of 17 is 4 point something,"},{"Start":"02:51.960 ","End":"02:55.060","Text":"so this will be 7 point something over 2."},{"Start":"02:55.060 ","End":"02:57.050","Text":"Then you ready fence bigger or equal to that?"},{"Start":"02:57.050 ","End":"03:01.750","Text":"It\u0027s got to be bigger or equal to 8/2, which is 4."},{"Start":"03:01.750 ","End":"03:04.545","Text":"Choose Epsilon equals 1/2,"},{"Start":"03:04.545 ","End":"03:11.858","Text":"that\u0027s the 1/2 from here and n is the maximum of n_0, and 4."},{"Start":"03:11.858 ","End":"03:13.910","Text":"It got to be bigger or equal to 4 and"},{"Start":"03:13.910 ","End":"03:17.074","Text":"the requirement is that n is bigger or equal to n_naught."},{"Start":"03:17.074 ","End":"03:20.540","Text":"Under this condition, we got this inequality."},{"Start":"03:20.540 ","End":"03:22.160","Text":"If you go back and summarize it,"},{"Start":"03:22.160 ","End":"03:25.325","Text":"we got that this is bigger or equal to 1/2."},{"Start":"03:25.325 ","End":"03:28.610","Text":"In other words, a_n minus L bigger or equal to Epsilon."},{"Start":"03:28.610 ","End":"03:31.500","Text":"That concludes this exercise."}],"ID":27231},{"Watched":false,"Name":"Exercise 5","Duration":"3m 41s","ChapterTopicVideoID":26325,"CourseChapterTopicPlaylistID":254169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.350 ","End":"00:08.160","Text":"In this exercise, we have to prove that this limit is not 4,"},{"Start":"00:08.160 ","End":"00:09.510","Text":"meaning either it exists,"},{"Start":"00:09.510 ","End":"00:11.775","Text":"and is not equal to 4 or it doesn\u0027t exist."},{"Start":"00:11.775 ","End":"00:14.655","Text":"Actually in this case, the limit doesn\u0027t exist."},{"Start":"00:14.655 ","End":"00:16.890","Text":"Anyway, we\u0027re going to use the definition of a limit,"},{"Start":"00:16.890 ","End":"00:18.555","Text":"the one with the Epsilon."},{"Start":"00:18.555 ","End":"00:21.690","Text":"This remark, if you didn\u0027t have the minus 1 to the n here,"},{"Start":"00:21.690 ","End":"00:22.935","Text":"it would have a limit."},{"Start":"00:22.935 ","End":"00:24.510","Text":"The limit would be 4."},{"Start":"00:24.510 ","End":"00:27.510","Text":"The leading coefficient when they have equal degree,"},{"Start":"00:27.510 ","End":"00:29.490","Text":"it\u0027s 4/1 is 4."},{"Start":"00:29.490 ","End":"00:31.380","Text":"Because of this, it oscillates,"},{"Start":"00:31.380 ","End":"00:33.300","Text":"it\u0027s near minus 4,"},{"Start":"00:33.300 ","End":"00:36.030","Text":"it\u0027s near 4, near minus 4 near 4,"},{"Start":"00:36.030 ","End":"00:38.940","Text":"but doesn\u0027t actually have a limit."},{"Start":"00:38.940 ","End":"00:43.774","Text":"In general, we know that to show that something is not the limit,"},{"Start":"00:43.774 ","End":"00:48.080","Text":"we negate logically the statement that it is a limit."},{"Start":"00:48.080 ","End":"00:49.430","Text":"We\u0027ve done this previously,"},{"Start":"00:49.430 ","End":"00:53.845","Text":"so this is the statement that we have to show in general."},{"Start":"00:53.845 ","End":"00:56.730","Text":"For this exercise, we know what a_n, and L are."},{"Start":"00:56.730 ","End":"00:58.980","Text":"A_n we take it as this,"},{"Start":"00:58.980 ","End":"01:01.095","Text":"and L will be 4."},{"Start":"01:01.095 ","End":"01:05.690","Text":"We have to show that there exists an Epsilon such that for all n_0,"},{"Start":"01:05.690 ","End":"01:09.935","Text":"there exists an n such that this is bigger or equal to Epsilon."},{"Start":"01:09.935 ","End":"01:12.110","Text":"There\u0027s basically two things we have to find here."},{"Start":"01:12.110 ","End":"01:16.430","Text":"We have to find this Epsilon that exists."},{"Start":"01:16.430 ","End":"01:23.220","Text":"But we also have to find a rule that given n_0 tells us what n is."},{"Start":"01:23.220 ","End":"01:25.160","Text":"It\u0027s like to go from here to here."},{"Start":"01:25.160 ","End":"01:28.280","Text":"Notice that if n is odd,"},{"Start":"01:28.280 ","End":"01:31.040","Text":"then this is always minus 1."},{"Start":"01:31.040 ","End":"01:32.810","Text":"If n is large,"},{"Start":"01:32.810 ","End":"01:35.540","Text":"then this will be near minus 4."},{"Start":"01:35.540 ","End":"01:38.440","Text":"In any event, it will be far away from 4,"},{"Start":"01:38.440 ","End":"01:42.560","Text":"and certainly negative from a certain n onwards,"},{"Start":"01:42.560 ","End":"01:43.745","Text":"probably for all n,"},{"Start":"01:43.745 ","End":"01:45.305","Text":"this will be negative."},{"Start":"01:45.305 ","End":"01:47.000","Text":"That if n is odd,"},{"Start":"01:47.000 ","End":"01:52.105","Text":"we can just replace it with minus 1."},{"Start":"01:52.105 ","End":"01:57.190","Text":"I can write 4 as 4n plus 8 over n plus 2."},{"Start":"01:57.190 ","End":"02:03.215","Text":"If I turn this into this after compensate by putting a plus 9 here."},{"Start":"02:03.215 ","End":"02:06.650","Text":"Writing the 1 that\u0027s minus 8 plus 9."},{"Start":"02:06.650 ","End":"02:10.325","Text":"I want it this way because then I can combine these two."},{"Start":"02:10.325 ","End":"02:14.215","Text":"That will give a minus 4 and minus 4 is minus 8."},{"Start":"02:14.215 ","End":"02:18.710","Text":"Can switch them around provided that what\u0027s inside is negative."},{"Start":"02:18.710 ","End":"02:24.310","Text":"Actually need the condition that 9 over n plus 2 is less than or equal to 8,"},{"Start":"02:24.310 ","End":"02:29.420","Text":"and it always is for any n. Think about it, it\u0027s fairly obvious."},{"Start":"02:29.420 ","End":"02:34.000","Text":"This we want to be bigger or equal to Epsilon."},{"Start":"02:34.000 ","End":"02:39.570","Text":"Choose Epsilon equals 7 could be anything between 0 and 8,"},{"Start":"02:39.570 ","End":"02:43.305","Text":"but I\u0027m just thinking 8 minus a bit 7."},{"Start":"02:43.305 ","End":"02:52.130","Text":"Then we get the condition that 8 minus 9 over 2 bigger or equal to Epsilon, which is 7."},{"Start":"02:52.130 ","End":"02:58.365","Text":"This comes out 9 over n plus 2 less than or equal to 1."},{"Start":"02:58.365 ","End":"03:02.530","Text":"N is bigger or equal to 7."},{"Start":"03:04.940 ","End":"03:09.605","Text":"First of all, n is odd because all this is something that condition that n is odd,"},{"Start":"03:09.605 ","End":"03:12.320","Text":"and n is bigger or equal to 7,"},{"Start":"03:12.320 ","End":"03:14.750","Text":"and n is bigger or equal to n_0."},{"Start":"03:14.750 ","End":"03:17.550","Text":"Just put the maximum here."},{"Start":"03:17.840 ","End":"03:23.090","Text":"If that\u0027s true, then n is bigger or equal to n_0 from here."},{"Start":"03:23.090 ","End":"03:26.330","Text":"The absolute value of a_n minus L is bigger or"},{"Start":"03:26.330 ","End":"03:30.185","Text":"equal to Epsilon because this is bigger or equal to this,"},{"Start":"03:30.185 ","End":"03:31.670","Text":"because n is odd,"},{"Start":"03:31.670 ","End":"03:33.995","Text":"and bigger or equal to 7."},{"Start":"03:33.995 ","End":"03:36.170","Text":"This 7 is not the same as this 7,"},{"Start":"03:36.170 ","End":"03:38.420","Text":"it\u0027s just coincidence that they are equal."},{"Start":"03:38.420 ","End":"03:41.790","Text":"That concludes this exercise."}],"ID":27232}],"Thumbnail":null,"ID":254169},{"Name":"Cauchy Criterion","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Cauchy Convergence Test","Duration":"11m 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real function on D,"},{"Start":"00:26.685 ","End":"00:30.420","Text":"and D contains a specific point a."},{"Start":"00:30.420 ","End":"00:33.214","Text":"Now, for this definition,"},{"Start":"00:33.214 ","End":"00:36.275","Text":"f doesn\u0027t actually have to be defined at a."},{"Start":"00:36.275 ","End":"00:39.455","Text":"That sounds strange, but we\u0027ll return to that point in a moment."},{"Start":"00:39.455 ","End":"00:43.520","Text":"Let\u0027s start with the Cauchy definition as a review,"},{"Start":"00:43.520 ","End":"00:46.010","Text":"and then we\u0027ll go to Heine\u0027s definition."},{"Start":"00:46.010 ","End":"00:49.765","Text":"Cauchy definition is also called the Epsilon Delta definition."},{"Start":"00:49.765 ","End":"00:52.460","Text":"Heine\u0027s definition involves sequences."},{"Start":"00:52.460 ","End":"00:59.015","Text":"We say that the limit of f(x) as x tends to a is L,"},{"Start":"00:59.015 ","End":"01:00.620","Text":"L is some number,"},{"Start":"01:00.620 ","End":"01:09.109","Text":"and the notation is limit as x goes to a f(x) equals L. If for any Epsilon bigger than 0,"},{"Start":"01:09.109 ","End":"01:11.605","Text":"there exists Delta bigger than 0,"},{"Start":"01:11.605 ","End":"01:15.905","Text":"such that for all x in the domain,"},{"Start":"01:15.905 ","End":"01:20.570","Text":"if absolute value of x minus a is between 0 and Delta,"},{"Start":"01:20.570 ","End":"01:24.680","Text":"notice this excludes x equals a, this bigger than 0,"},{"Start":"01:24.680 ","End":"01:29.630","Text":"then f(x) minus L in absolute value is less than Epsilon."},{"Start":"01:29.630 ","End":"01:31.730","Text":"This is familiar definition."},{"Start":"01:31.730 ","End":"01:34.100","Text":"You should have learned this already."},{"Start":"01:34.100 ","End":"01:37.895","Text":"Now, note that when we say x minus"},{"Start":"01:37.895 ","End":"01:41.900","Text":"a between 0 and Delta is a punctured neighborhood of a,"},{"Start":"01:41.900 ","End":"01:43.775","Text":"it\u0027s a circle without the center."},{"Start":"01:43.775 ","End":"01:45.895","Text":"We\u0027ve excluded the point a itself."},{"Start":"01:45.895 ","End":"01:48.920","Text":"The reason for that is that the limit of f(x) as"},{"Start":"01:48.920 ","End":"01:54.425","Text":"x goes to a or x is near a and approaches a,"},{"Start":"01:54.425 ","End":"01:58.745","Text":"shouldn\u0027t be affected by the value of f at point a itself,"},{"Start":"01:58.745 ","End":"02:00.695","Text":"which could even be undefined."},{"Start":"02:00.695 ","End":"02:02.495","Text":"When x goes to a,"},{"Start":"02:02.495 ","End":"02:05.300","Text":"it goes to a, but not equal to a."},{"Start":"02:05.300 ","End":"02:10.525","Text":"Now, the Heine definition of a limit using sequences."},{"Start":"02:10.525 ","End":"02:14.695","Text":"We say that the limit of f(x) as x tends to a is L,"},{"Start":"02:14.695 ","End":"02:17.840","Text":"denoted the same way as here,"},{"Start":"02:17.840 ","End":"02:19.355","Text":"but definition is different."},{"Start":"02:19.355 ","End":"02:23.315","Text":"If for any sequence x_n in D,"},{"Start":"02:23.315 ","End":"02:27.440","Text":"but excluding the point a with limit a,"},{"Start":"02:27.440 ","End":"02:30.790","Text":"the limit of f(x_n) is L. Now,"},{"Start":"02:30.790 ","End":"02:33.560","Text":"here and here,"},{"Start":"02:33.560 ","End":"02:35.480","Text":"we\u0027re talking about sequence limit."},{"Start":"02:35.480 ","End":"02:36.890","Text":"Here we have a sequence,"},{"Start":"02:36.890 ","End":"02:39.110","Text":"x_n is a sequence,"},{"Start":"02:39.110 ","End":"02:41.150","Text":"and we know what the limit of a sequence is."},{"Start":"02:41.150 ","End":"02:44.125","Text":"Similarly, f(x_n) is a sequence."},{"Start":"02:44.125 ","End":"02:46.820","Text":"That\u0027s why I said sequence limit,"},{"Start":"02:46.820 ","End":"02:48.290","Text":"which we already know."},{"Start":"02:48.290 ","End":"02:50.510","Text":"I\u0027m not defining limit in terms of itself."},{"Start":"02:50.510 ","End":"02:54.130","Text":"I\u0027m defining limit of a function in terms of limit of a sequence."},{"Start":"02:54.130 ","End":"03:00.095","Text":"Just as before, when we had this punctured neighborhood D minus a,"},{"Start":"03:00.095 ","End":"03:04.810","Text":"from here is a punctured neighborhood of a or a punctured domain."},{"Start":"03:04.810 ","End":"03:08.160","Text":"It has a hole where x is a."},{"Start":"03:08.160 ","End":"03:10.910","Text":"This sentence, we could say it again."},{"Start":"03:10.910 ","End":"03:16.729","Text":"The limit as x goes to a shouldn\u0027t be affected by the value of f at a itself,"},{"Start":"03:16.729 ","End":"03:18.920","Text":"which might even be undefined."},{"Start":"03:18.920 ","End":"03:20.750","Text":"I want to continue on this point,"},{"Start":"03:20.750 ","End":"03:26.540","Text":"what would happen if we used non-punctured neighborhoods without the whole with a?"},{"Start":"03:26.540 ","End":"03:28.070","Text":"I\u0027ll give you an example."},{"Start":"03:28.070 ","End":"03:33.290","Text":"Let\u0027s say f(x) is equal to 1 at x=0,"},{"Start":"03:33.290 ","End":"03:35.420","Text":"and everywhere else is 0."},{"Start":"03:35.420 ","End":"03:39.245","Text":"It\u0027s 0 except that the single-point where x is 0."},{"Start":"03:39.245 ","End":"03:42.560","Text":"The limit as x goes to 0 of f(x),"},{"Start":"03:42.560 ","End":"03:44.540","Text":"we want it to be 0."},{"Start":"03:44.540 ","End":"03:48.070","Text":"If it\u0027s 0 everywhere except at 0,"},{"Start":"03:48.070 ","End":"03:53.560","Text":"then we\u0027d expect the limit to be 0 but it wouldn\u0027t come out 0 if we change"},{"Start":"03:53.560 ","End":"03:59.480","Text":"the definition and remove the word punctured because if it\u0027s not punctured,"},{"Start":"03:59.480 ","End":"04:02.945","Text":"we can let x=0 in the definition."},{"Start":"04:02.945 ","End":"04:07.490","Text":"That\u0027s allowed because x minus 0 is less than Delta."},{"Start":"04:07.490 ","End":"04:10.880","Text":"We\u0027ve removed the condition that this has to be bigger than 0."},{"Start":"04:10.880 ","End":"04:16.040","Text":"Actually, I\u0027m using the Cauchy definition not the Heine."},{"Start":"04:16.040 ","End":"04:20.835","Text":"X minus 0 is less than Delta,"},{"Start":"04:20.835 ","End":"04:29.450","Text":"but f(x) minus L turns out to be not less than Epsilon because it\u0027s equal to 1."},{"Start":"04:29.450 ","End":"04:31.700","Text":"The limit desired is 0,"},{"Start":"04:31.700 ","End":"04:34.455","Text":"f(x) is f(0), which is 1,"},{"Start":"04:34.455 ","End":"04:35.880","Text":"comes out to be 1."},{"Start":"04:35.880 ","End":"04:40.170","Text":"If we take any Epsilon is 1 or less then, it won\u0027t work."},{"Start":"04:40.170 ","End":"04:44.405","Text":"That\u0027s why not only don\u0027t we need the value"},{"Start":"04:44.405 ","End":"04:50.030","Text":"at 0 or at a as we don\u0027t want it because we will get the wrong result."},{"Start":"04:50.030 ","End":"04:53.780","Text":"Now there\u0027s a theorem that both definitions of limit,"},{"Start":"04:53.780 ","End":"04:56.653","Text":"the Cauchy definition and the Heine definition,"},{"Start":"04:56.653 ","End":"05:01.250","Text":"are equivalent but we\u0027ll prove it in a separate clip."},{"Start":"05:01.250 ","End":"05:03.950","Text":"Now an example,"},{"Start":"05:03.950 ","End":"05:08.660","Text":"we often use Heine\u0027s definition to show non-existence of a limit."},{"Start":"05:08.660 ","End":"05:10.340","Text":"This is what this example is."},{"Start":"05:10.340 ","End":"05:15.165","Text":"Let f(x) equals cosine of 1/x for positive x."},{"Start":"05:15.165 ","End":"05:20.910","Text":"We have to show that the limit as x goes to 0 of f(x) doesn\u0027t exist."},{"Start":"05:20.910 ","End":"05:23.645","Text":"I\u0027ll change this to not equals."},{"Start":"05:23.645 ","End":"05:27.485","Text":"We\u0027ll solve it using Heine\u0027s definition of a limit,"},{"Start":"05:27.485 ","End":"05:33.590","Text":"and afterwards we\u0027ll solve it using Cauchy\u0027s definition, just for comparison."},{"Start":"05:33.590 ","End":"05:36.470","Text":"By contradiction, suppose the limit does exist."},{"Start":"05:36.470 ","End":"05:40.625","Text":"If it exists, it\u0027s equal to sum L. Choose a sequence"},{"Start":"05:40.625 ","End":"05:46.450","Text":"x_n=1 Pi so certainly x_n goes to 0."},{"Start":"05:46.450 ","End":"05:50.835","Text":"Then, by Heine\u0027s definition of a limit, x_n goes to 0,"},{"Start":"05:50.835 ","End":"05:57.779","Text":"f(x_n) goes to L. That\u0027s what this means in Heine\u0027s definition."},{"Start":"05:57.779 ","End":"06:05.400","Text":"Now, f(x_n) is cosine of 1 over this, which is nPi."},{"Start":"06:05.400 ","End":"06:08.985","Text":"Cosine(nPi) is (-1)^n."},{"Start":"06:08.985 ","End":"06:16.775","Text":"Minus 1^n gives us the sequence minus 1,1 minus 1,1 alternating and this has no limit,"},{"Start":"06:16.775 ","End":"06:19.130","Text":"and that\u0027s a contradiction."},{"Start":"06:19.130 ","End":"06:23.770","Text":"Now, let\u0027s do it using Cauchy\u0027s definition of a limit."},{"Start":"06:23.770 ","End":"06:30.125","Text":"Again by contradiction, suppose that f does have a limit and that limit is L,"},{"Start":"06:30.125 ","End":"06:32.965","Text":"then for every Epsilon there exists Delta."},{"Start":"06:32.965 ","End":"06:39.680","Text":"Choose Epsilon equals 1 and it has a corresponding Delta such that if x minus a,"},{"Start":"06:39.680 ","End":"06:41.345","Text":"in this case x minus 0,"},{"Start":"06:41.345 ","End":"06:44.990","Text":"absolute value is less than Delta, bigger than 0,"},{"Start":"06:44.990 ","End":"06:46.775","Text":"meaning that x is not equal to 0,"},{"Start":"06:46.775 ","End":"06:51.830","Text":"then absolute value of f(x) minus L is less than Epsilon, which is 1."},{"Start":"06:51.830 ","End":"06:56.765","Text":"Choose some n bigger than 1/2 Pi Delta."},{"Start":"06:56.765 ","End":"06:59.030","Text":"Got to this from reverse engineering."},{"Start":"06:59.030 ","End":"07:01.010","Text":"If you go to the end, you see what you need to come back."},{"Start":"07:01.010 ","End":"07:04.535","Text":"You see that you need n to be bigger than 1/2 Pi Delta."},{"Start":"07:04.535 ","End":"07:07.250","Text":"Then we\u0027ll pick 2 x\u0027s,"},{"Start":"07:07.250 ","End":"07:08.600","Text":"x_1 and x_2,"},{"Start":"07:08.600 ","End":"07:15.295","Text":"x_1 will be 1/2nPi and x_2 will be 1/2n plus 1 Pi."},{"Start":"07:15.295 ","End":"07:16.775","Text":"I forgot to write it,"},{"Start":"07:16.775 ","End":"07:18.290","Text":"but each of these x_1,"},{"Start":"07:18.290 ","End":"07:20.120","Text":"x_2 is less than Delta."},{"Start":"07:20.120 ","End":"07:24.590","Text":"For example, if you multiply both sides by 2Pi,"},{"Start":"07:24.590 ","End":"07:28.055","Text":"we get 2Pin is bigger than 1 over Delta,"},{"Start":"07:28.055 ","End":"07:34.027","Text":"and then take the reciprocal and invert the sign we have 1/2Pi n is less than Delta."},{"Start":"07:34.027 ","End":"07:38.790","Text":"Similarly, 1/2n plus 1Pi is less than Delta."},{"Start":"07:38.790 ","End":"07:41.070","Text":"Each of these is less than Delta."},{"Start":"07:41.070 ","End":"07:46.830","Text":"f(x_1) is equal to cosine 2nPi is 1,"},{"Start":"07:46.830 ","End":"07:50.175","Text":"and f(x_2) is cosine 2n plus 1Pi,"},{"Start":"07:50.175 ","End":"07:51.654","Text":"which is minus 1."},{"Start":"07:51.654 ","End":"07:53.716","Text":"Cosine of uneven number times Pi is 1,"},{"Start":"07:53.716 ","End":"07:56.315","Text":"cosine of an odd number times Pi is minus 1."},{"Start":"07:56.315 ","End":"08:02.645","Text":"These both satisfy this because absolute value of x minus 0 is just x when x is positive."},{"Start":"08:02.645 ","End":"08:07.715","Text":"We already said that each of these is less than Delta."},{"Start":"08:07.715 ","End":"08:11.380","Text":"If you plug in x_1 or x_2 here, this is true."},{"Start":"08:11.380 ","End":"08:19.260","Text":"By this implication, f(x_1) minus L is less than 1 and f(x_2) minus L is less than 1."},{"Start":"08:19.260 ","End":"08:24.410","Text":"You want to replace x by x_1 here and here and wants you replace it by x_2 here and here."},{"Start":"08:24.410 ","End":"08:26.450","Text":"Now we\u0027re going to use the triangle inequality."},{"Start":"08:26.450 ","End":"08:28.070","Text":"The absolute value of f(x_1)"},{"Start":"08:28.070 ","End":"08:31.640","Text":"minus f(x_2) is less than or equal to the absolute value of this plus"},{"Start":"08:31.640 ","End":"08:37.855","Text":"absolute value of this because the sum of these 2 is just f(x_1) minus f(x_2)."},{"Start":"08:37.855 ","End":"08:39.735","Text":"Now, this is less than 1,"},{"Start":"08:39.735 ","End":"08:41.430","Text":"and this is less than 1,"},{"Start":"08:41.430 ","End":"08:44.305","Text":"so together this is less than 2."},{"Start":"08:44.305 ","End":"08:49.055","Text":"On the other hand, we\u0027ve already computed f(x_1) and f(x_2) and"},{"Start":"08:49.055 ","End":"08:55.085","Text":"f(x_1) minus f(x_2) in absolute value is 1 minus minus 1, which is 2."},{"Start":"08:55.085 ","End":"08:56.974","Text":"On the 1 hand,"},{"Start":"08:56.974 ","End":"09:00.995","Text":"the expression is less than 2 and on the other hand,"},{"Start":"09:00.995 ","End":"09:02.930","Text":"it\u0027s equal to 2."},{"Start":"09:02.930 ","End":"09:07.055","Text":"That can\u0027t be, that gives us a contradiction."},{"Start":"09:07.055 ","End":"09:11.370","Text":"That\u0027s the end of this proof and the clip."}],"ID":31282},{"Watched":false,"Name":"Cauchy and Heine Limits are the Same - Proof","Duration":"4m 27s","ChapterTopicVideoID":29643,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this clip, we\u0027ll prove that both definitions of a limit are equivalent,"},{"Start":"00:05.010 ","End":"00:08.730","Text":"meaning Cauchy\u0027s definition and Heine\u0027s definition."},{"Start":"00:08.730 ","End":"00:12.705","Text":"To show they\u0027re equivalent we\u0027ll show that each one implies the other."},{"Start":"00:12.705 ","End":"00:16.575","Text":"We\u0027ll start with Cauchy\u0027s limit implies Heine limit."},{"Start":"00:16.575 ","End":"00:19.920","Text":"Let f be defined on a domain D,"},{"Start":"00:19.920 ","End":"00:24.435","Text":"which is usually an open interval and it contains a."},{"Start":"00:24.435 ","End":"00:30.360","Text":"Now suppose that the limit exists according to Cauchy\u0027s we\u0027ll prove."},{"Start":"00:30.360 ","End":"00:31.940","Text":"Well, it\u0027s written the same,"},{"Start":"00:31.940 ","End":"00:36.990","Text":"but we\u0027ll understand the second time meaning terms of Heine."},{"Start":"00:37.340 ","End":"00:45.605","Text":"What we have to show for Heine is that if we have a sequence x_n in D excluding a,"},{"Start":"00:45.605 ","End":"00:50.685","Text":"and x_n tends to a as a sequence then f(x_n),"},{"Start":"00:50.685 ","End":"00:53.190","Text":"tends to L as a sequence."},{"Start":"00:53.190 ","End":"01:00.590","Text":"To say that this tends to this as a sequence means that given Epsilon bigger than 0,"},{"Start":"01:00.590 ","End":"01:03.860","Text":"there exists a natural number n,"},{"Start":"01:03.860 ","End":"01:06.280","Text":"such that for all little n,"},{"Start":"01:06.280 ","End":"01:08.445","Text":"from N onwards,"},{"Start":"01:08.445 ","End":"01:12.520","Text":"f(x_n) is close to L to within Epsilon."},{"Start":"01:12.520 ","End":"01:15.065","Text":"Now, by Cauchy\u0027s definition,"},{"Start":"01:15.065 ","End":"01:17.045","Text":"which we know is satisfied,"},{"Start":"01:17.045 ","End":"01:19.580","Text":"we can find a Delta for this Epsilon."},{"Start":"01:19.580 ","End":"01:26.293","Text":"There exists a Delta such that if x is close to a within Delta,"},{"Start":"01:26.293 ","End":"01:27.725","Text":"but not equal to a,"},{"Start":"01:27.725 ","End":"01:31.550","Text":"then f(x) is close to L within Epsilon."},{"Start":"01:31.550 ","End":"01:35.645","Text":"Now the idea is to somehow replace this x by x_n,"},{"Start":"01:35.645 ","End":"01:37.190","Text":"which is what we want."},{"Start":"01:37.190 ","End":"01:40.295","Text":"Since x_n tends to a,"},{"Start":"01:40.295 ","End":"01:43.520","Text":"again, by definition of sequential limit,"},{"Start":"01:43.520 ","End":"01:51.330","Text":"there exists a big N such that x_n minus a is less than Delta,"},{"Start":"01:51.330 ","End":"01:53.925","Text":"Delta is taking the role of Epsilon here,"},{"Start":"01:53.925 ","End":"01:57.155","Text":"whenever n is N or more,"},{"Start":"01:57.155 ","End":"02:02.585","Text":"we can add the bigger than 0 here because x_n is not equal to a,"},{"Start":"02:02.585 ","End":"02:05.585","Text":"because we took it from the punctured domain."},{"Start":"02:05.585 ","End":"02:12.145","Text":"Now, if n is bigger or equal to N x_n minus a is less than Delta,"},{"Start":"02:12.145 ","End":"02:15.860","Text":"which implies that f(x_n) minus L is less than"},{"Start":"02:15.860 ","End":"02:20.110","Text":"Epsilon because x_n satisfies this condition."},{"Start":"02:20.110 ","End":"02:24.125","Text":"This is what we require and that\u0027s 1.5 of the proof."},{"Start":"02:24.125 ","End":"02:29.575","Text":"Next, we\u0027re going to show that the Heine limit implies the Cauchy limit."},{"Start":"02:29.575 ","End":"02:32.105","Text":"We\u0027ll do it by contradiction."},{"Start":"02:32.105 ","End":"02:39.755","Text":"Suppose it\u0027s not true that the limit of f(x) is L as x goes to a."},{"Start":"02:39.755 ","End":"02:43.865","Text":"Basically we\u0027re using the contrapositive instead of showing Heine implies Cauchy."},{"Start":"02:43.865 ","End":"02:47.285","Text":"We\u0027re showing that not Cauchy implies not Heine,"},{"Start":"02:47.285 ","End":"02:50.420","Text":"which is really what proof by contradiction is."},{"Start":"02:50.420 ","End":"02:53.735","Text":"Not Cauchy means that for some Epsilon,"},{"Start":"02:53.735 ","End":"02:55.609","Text":"there is no good Delta."},{"Start":"02:55.609 ","End":"02:59.105","Text":"In particular, 1 over n is not a good Delta."},{"Start":"02:59.105 ","End":"03:02.720","Text":"Which means that instead of for all x_n there exists"},{"Start":"03:02.720 ","End":"03:06.725","Text":"an x_n such that what we want is not true,"},{"Start":"03:06.725 ","End":"03:13.560","Text":"meaning that x_n minus a absolute value is less than this Delta."},{"Start":"03:13.850 ","End":"03:18.380","Text":"Yet, f(x_n) minus L is not close within Epsilon."},{"Start":"03:18.380 ","End":"03:20.300","Text":"It\u0027s bigger or equal to Epsilon."},{"Start":"03:20.300 ","End":"03:22.430","Text":"For each n, 1, 2, 3,"},{"Start":"03:22.430 ","End":"03:25.750","Text":"etc., there exists an x_n."},{"Start":"03:25.750 ","End":"03:32.480","Text":"This gives us the existence of a sequence x_n in D excluding a,"},{"Start":"03:32.480 ","End":"03:35.670","Text":"and the sequences in D minus a,"},{"Start":"03:35.670 ","End":"03:38.770","Text":"because x_n belongs to D at x_n"},{"Start":"03:38.770 ","End":"03:42.650","Text":"minus a in absolute value bigger than 0 to x is not equal to a,"},{"Start":"03:42.650 ","End":"03:46.770","Text":"such that x_n tends to a."},{"Start":"03:46.770 ","End":"03:48.905","Text":"The reason x_n tends to a,"},{"Start":"03:48.905 ","End":"03:54.700","Text":"I didn\u0027t spell out the details is if x_n minus a is less than 1 over n,"},{"Start":"03:54.700 ","End":"03:57.090","Text":"then, of course, x_n goes to a."},{"Start":"03:57.090 ","End":"04:04.760","Text":"However, f(x_n) does not tend to L Because f of x_n is always far away,"},{"Start":"04:04.760 ","End":"04:09.920","Text":"at least Epsilon from L. This contradicts the fact that"},{"Start":"04:09.920 ","End":"04:15.440","Text":"the limit as x goes to a of f(x) is equal to L using Heine limit."},{"Start":"04:15.440 ","End":"04:19.918","Text":"Just to repeat, we found a sequence of x_n that goes to a,"},{"Start":"04:19.918 ","End":"04:27.480","Text":"and still f(x_n) does not go to L. That concludes the proof."}],"ID":31283},{"Watched":false,"Name":"Cauchy and Heine Continuity","Duration":"6m 57s","ChapterTopicVideoID":29644,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.660","Text":"In this clip, we\u0027ll define continuity according to Heine,"},{"Start":"00:06.660 ","End":"00:12.615","Text":"assuming you\u0027ve already learned about continuity and that would be the Cauchy definition."},{"Start":"00:12.615 ","End":"00:17.550","Text":"We\u0027ll talk about continuity at a point because continuity"},{"Start":"00:17.550 ","End":"00:22.826","Text":"on a whole interval just means continuous at every point in the interval for example."},{"Start":"00:22.826 ","End":"00:25.410","Text":"Let\u0027s start with general definition."},{"Start":"00:25.410 ","End":"00:28.140","Text":"That\u0027s neither Cauchy nor Heine."},{"Start":"00:28.140 ","End":"00:31.340","Text":"If f is defined on a domain D,"},{"Start":"00:31.340 ","End":"00:35.000","Text":"usually an interval containing the point a,"},{"Start":"00:35.000 ","End":"00:43.695","Text":"we say that f is continuous at a if there exists the limit as x goes to a of f(x)."},{"Start":"00:43.695 ","End":"00:45.140","Text":"No it only does it exist,"},{"Start":"00:45.140 ","End":"00:47.890","Text":"but it exactly equals f(a)."},{"Start":"00:47.890 ","End":"00:53.000","Text":"But we\u0027ve defined continuity in terms of the concept of a limit."},{"Start":"00:53.000 ","End":"00:57.265","Text":"Now, limit could be defined according to Cauchy or Heine and"},{"Start":"00:57.265 ","End":"01:02.150","Text":"if we interpret this in terms of the Cauchy or the Heine definition,"},{"Start":"01:02.150 ","End":"01:05.120","Text":"we\u0027ll get the definition of continuity according to"},{"Start":"01:05.120 ","End":"01:09.470","Text":"Cauchy and definition of continuity according to Heine."},{"Start":"01:09.470 ","End":"01:13.760","Text":"Remember, we don\u0027t use punctured neighborhoods to define continuity."},{"Start":"01:13.760 ","End":"01:18.980","Text":"We need f to be defined at a because we\u0027re finding a relationship between"},{"Start":"01:18.980 ","End":"01:24.685","Text":"the value of the function at a and the values of f(x) when x is near a."},{"Start":"01:24.685 ","End":"01:26.250","Text":"The Cauchy definition is"},{"Start":"01:26.250 ","End":"01:30.885","Text":"the Epsilon Delta definition and the Heine definition uses sequences."},{"Start":"01:30.885 ","End":"01:35.145","Text":"We said that f is continuous at a if for any Epsilon,"},{"Start":"01:35.145 ","End":"01:43.175","Text":"there exists a Delta such that for any x in D whose distance to a is less than Delta,"},{"Start":"01:43.175 ","End":"01:48.475","Text":"the distance from f(x) to f(a) is less than Epsilon."},{"Start":"01:48.475 ","End":"01:52.320","Text":"Here, we do not need the bigger than 0."},{"Start":"01:52.320 ","End":"01:55.515","Text":"Also, L is replaced by f(a)."},{"Start":"01:55.515 ","End":"01:59.900","Text":"The similar adaptation in the Heine definition,"},{"Start":"01:59.900 ","End":"02:01.130","Text":"we\u0027re just jumping ahead."},{"Start":"02:01.130 ","End":"02:04.880","Text":"We see that we don\u0027t puncture the neighborhood."},{"Start":"02:04.880 ","End":"02:08.845","Text":"The limit is not just some L, but it\u0027s f(a)."},{"Start":"02:08.845 ","End":"02:13.285","Text":"F is continuous at a if for any sequence x_n,"},{"Start":"02:13.285 ","End":"02:19.650","Text":"whose limit is a the sequence f(x_n) has the limit f(a)."},{"Start":"02:19.650 ","End":"02:22.820","Text":"The two definitions are equivalent because we\u0027ve already"},{"Start":"02:22.820 ","End":"02:26.935","Text":"proved that the two definitions of the limit are equivalent."},{"Start":"02:26.935 ","End":"02:30.140","Text":"That\u0027s just what I\u0027m saying here, its continuity is based on limit."},{"Start":"02:30.140 ","End":"02:33.965","Text":"Since Cauchy limit is the same as Heine limit,"},{"Start":"02:33.965 ","End":"02:37.070","Text":"then the continuity is the same for both."},{"Start":"02:37.070 ","End":"02:39.155","Text":"Now let\u0027s do a couple of examples."},{"Start":"02:39.155 ","End":"02:44.810","Text":"In the first example, we will prove that x^2 is continuous at any point x=a."},{"Start":"02:44.810 ","End":"02:49.405","Text":"We\u0027ll do it in two ways: using Heine and then using Cauchy."},{"Start":"02:49.405 ","End":"02:53.630","Text":"In Heine, let\u0027s suppose x_n is a sequence that tends to a."},{"Start":"02:53.630 ","End":"02:57.085","Text":"We need to show that f(x_n) tends to f(a.)"},{"Start":"02:57.085 ","End":"03:01.090","Text":"Now the words that x_n^ 2 goes to a^2."},{"Start":"03:01.090 ","End":"03:03.770","Text":"Well, we can use the product rule for a limit that"},{"Start":"03:03.770 ","End":"03:06.485","Text":"if we multiply 2 sequences with a limit,"},{"Start":"03:06.485 ","End":"03:10.080","Text":"the limit of the product is the product of the limits."},{"Start":"03:10.160 ","End":"03:16.270","Text":"The limit of x_n^2 is the limit of x_n^2."},{"Start":"03:16.270 ","End":"03:19.460","Text":"That proves that very simple."},{"Start":"03:19.460 ","End":"03:21.800","Text":"Cauchy is not so simple,"},{"Start":"03:21.800 ","End":"03:23.785","Text":"but not that difficult."},{"Start":"03:23.785 ","End":"03:29.180","Text":"That Epsilon bigger than 0 be given and choose Delta to be"},{"Start":"03:29.180 ","End":"03:34.670","Text":"the minimum of 1 and Epsilon over 1 plus 2 absolute value of a."},{"Start":"03:34.670 ","End":"03:36.470","Text":"You get this by reverse engineering."},{"Start":"03:36.470 ","End":"03:38.210","Text":"If you left it blank and work forward,"},{"Start":"03:38.210 ","End":"03:40.840","Text":"you could figure out what it needs to be and then work back."},{"Start":"03:40.840 ","End":"03:46.715","Text":"We need to show that if absolute value of x minus a is less than Delta,"},{"Start":"03:46.715 ","End":"03:51.650","Text":"then the absolute value of f(x) minus f(a)is less than Epsilon."},{"Start":"03:51.650 ","End":"03:55.895","Text":"A bit of algebra, absolute value of x^2 minus a^2"},{"Start":"03:55.895 ","End":"04:00.650","Text":"is absolute value of x minus a times x plus a."},{"Start":"04:00.650 ","End":"04:02.630","Text":"The x plus a,"},{"Start":"04:02.630 ","End":"04:06.830","Text":"we can write as x minus a plus 2a."},{"Start":"04:06.830 ","End":"04:12.215","Text":"Now we know that x minus a is less than Delta."},{"Start":"04:12.215 ","End":"04:16.075","Text":"Note also that Delta is less than or equal to 1."},{"Start":"04:16.075 ","End":"04:19.140","Text":"That\u0027s why we put this min here."},{"Start":"04:19.140 ","End":"04:22.225","Text":"Here, we\u0027ll replace x minus a by Delta,"},{"Start":"04:22.225 ","End":"04:24.515","Text":"and here we replace it by 1."},{"Start":"04:24.515 ","End":"04:26.495","Text":"We get this inequality."},{"Start":"04:26.495 ","End":"04:32.270","Text":"Delta is less than or equal to this times this,"},{"Start":"04:32.270 ","End":"04:35.585","Text":"and that gives us Epsilon as required."},{"Start":"04:35.585 ","End":"04:40.550","Text":"Second example, we have to show that this function called the"},{"Start":"04:40.550 ","End":"04:45.385","Text":"Heavyside function is not continuous at x=0."},{"Start":"04:45.385 ","End":"04:54.310","Text":"The function is defined to be 1 when x is non-negative and 0 when x is negative."},{"Start":"04:54.310 ","End":"04:59.360","Text":"Here\u0027s the picture, starts at 0,"},{"Start":"04:59.360 ","End":"05:02.225","Text":"and then at 0 it jumps to 1."},{"Start":"05:02.225 ","End":"05:06.655","Text":"At 0 itself, we define it to be equal to 1."},{"Start":"05:06.655 ","End":"05:10.430","Text":"First, we\u0027ll prove non-continuity by Heine."},{"Start":"05:10.430 ","End":"05:12.290","Text":"Let\u0027s take the sequence x,"},{"Start":"05:12.290 ","End":"05:19.060","Text":"n to be minus 1 over n. It\u0027s negative but approaches 0."},{"Start":"05:19.060 ","End":"05:27.180","Text":"Continuity according to Heine means that h(x_n) tends to h(0)."},{"Start":"05:27.180 ","End":"05:29.390","Text":"But h(x_n) is 0,"},{"Start":"05:29.390 ","End":"05:33.955","Text":"at any negative number it\u0027s 0 and h(0)= 1."},{"Start":"05:33.955 ","End":"05:37.650","Text":"A sequence of 0s does not tend to 1."},{"Start":"05:37.650 ","End":"05:42.360","Text":"This is not true and so it\u0027s not continuous."},{"Start":"05:42.360 ","End":"05:45.105","Text":"Now, by Cauchy,"},{"Start":"05:45.105 ","End":"05:50.230","Text":"let Epsilon=1, you could choose less than 1."},{"Start":"05:50.230 ","End":"05:51.740","Text":"Do a proof by contradiction."},{"Start":"05:51.740 ","End":"05:57.930","Text":"Suppose that h is continuous at x=0."},{"Start":"05:57.930 ","End":"06:02.345","Text":"There exists a Delta bigger than 0 corresponding to this Epsilon,"},{"Start":"06:02.345 ","End":"06:07.760","Text":"meaning that h(x)minus h(0) absolute is less than Epsilon."},{"Start":"06:07.760 ","End":"06:12.940","Text":"Whenever x minus 0 absolute is less than Delta."},{"Start":"06:12.940 ","End":"06:18.530","Text":"Now, choose the point x to be minus Delta over 2."},{"Start":"06:18.530 ","End":"06:27.890","Text":"It certainly satisfies this inequality because x minus 0 absolute value is Delta over 2,"},{"Start":"06:27.890 ","End":"06:30.310","Text":"which is less than Delta."},{"Start":"06:30.310 ","End":"06:33.700","Text":"On the other hand, this expression,"},{"Start":"06:33.700 ","End":"06:38.555","Text":"h(x) minus h(0) is 0 here,"},{"Start":"06:38.555 ","End":"06:42.770","Text":"because h of negative is 0 and h(0) is 1."},{"Start":"06:42.770 ","End":"06:47.285","Text":"This comes out to be 1 which is not less than Epsilon,"},{"Start":"06:47.285 ","End":"06:49.010","Text":"1 is not less than 1."},{"Start":"06:49.010 ","End":"06:50.860","Text":"That\u0027s a contradiction."},{"Start":"06:50.860 ","End":"06:53.775","Text":"That\u0027s the proof by Cauchy,"},{"Start":"06:53.775 ","End":"06:58.540","Text":"and that completes the examples and this clip."}],"ID":31284},{"Watched":false,"Name":"Nonexistence of Limit via Heine","Duration":"5m 41s","ChapterTopicVideoID":29645,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"This clip is about 1 of the applications of sequences."},{"Start":"00:04.890 ","End":"00:08.775","Text":"They can be used in certain cases to"},{"Start":"00:08.775 ","End":"00:12.690","Text":"show that a function doesn\u0027t have a limit at some point."},{"Start":"00:12.690 ","End":"00:19.810","Text":"This is based on a theorem due to Heine, this picture here."},{"Start":"00:19.990 ","End":"00:23.125","Text":"What Heine says, is that,"},{"Start":"00:23.125 ","End":"00:26.565","Text":"if the limit of a function f,"},{"Start":"00:26.565 ","End":"00:30.555","Text":"when x goes to a is L,"},{"Start":"00:30.555 ","End":"00:36.780","Text":"then basically we can let x go to a, along a sequence."},{"Start":"00:36.780 ","End":"00:40.410","Text":"In other words, we can take a sequence x_n,"},{"Start":"00:40.410 ","End":"00:42.825","Text":"who\u0027s limit is a,"},{"Start":"00:42.825 ","End":"00:47.045","Text":"and then we\u0027d expect that if we take f at each x_n,"},{"Start":"00:47.045 ","End":"00:51.410","Text":"that it would approach L. It actually works the other way round too,"},{"Start":"00:51.410 ","End":"00:58.625","Text":"if every sequence that tends to a satisfies this,"},{"Start":"00:58.625 ","End":"01:01.280","Text":"then the limit of the function exists,"},{"Start":"01:01.280 ","End":"01:03.865","Text":"we\u0027re going to use it in this direction."},{"Start":"01:03.865 ","End":"01:07.975","Text":"Now this gives rise to 2 main techniques,"},{"Start":"01:07.975 ","End":"01:11.495","Text":"to prove the nonexistence of a limit of a function."},{"Start":"01:11.495 ","End":"01:15.585","Text":"1 involves taking 2 sequences with a different limit,"},{"Start":"01:15.585 ","End":"01:18.695","Text":"and 1 involves taking a sequence which has no limits."},{"Start":"01:18.695 ","End":"01:21.680","Text":"The first we\u0027ll call it Method 1."},{"Start":"01:21.680 ","End":"01:24.155","Text":"If there are 2 sequences,"},{"Start":"01:24.155 ","End":"01:25.655","Text":"call them x_n and y_n,"},{"Start":"01:25.655 ","End":"01:29.030","Text":"then each of them tends to a."},{"Start":"01:29.030 ","End":"01:33.530","Text":"But if we apply f to each of the sequences,"},{"Start":"01:33.530 ","End":"01:36.200","Text":"the limits are not the same,"},{"Start":"01:36.200 ","End":"01:39.544","Text":"then this limit does not exist."},{"Start":"01:39.544 ","End":"01:42.470","Text":"Because if it existed it would be some L,"},{"Start":"01:42.470 ","End":"01:43.850","Text":"then L would have to be this,"},{"Start":"01:43.850 ","End":"01:47.640","Text":"and L would have to be this, but they\u0027re not equal."},{"Start":"01:47.810 ","End":"01:50.400","Text":"I\u0027ll give an example of Method 1,"},{"Start":"01:50.400 ","End":"01:52.755","Text":"and then we\u0027ll talk about Method 2."},{"Start":"01:52.755 ","End":"01:57.540","Text":"We want to prove that when x tends to 0,"},{"Start":"01:57.540 ","End":"02:07.060","Text":"the function f(x) is 4^1/x doesn\u0027t exist using this Heine\u0027s theorem Method 1."},{"Start":"02:07.970 ","End":"02:12.300","Text":"We\u0027ll choose 2 sequences,"},{"Start":"02:12.300 ","End":"02:15.530","Text":"x_n will be 1,"},{"Start":"02:15.530 ","End":"02:18.080","Text":"that\u0027s certainly goes to 0,"},{"Start":"02:18.080 ","End":"02:21.725","Text":"and y_n will be minus 1/ n,"},{"Start":"02:21.725 ","End":"02:24.260","Text":"and it also goes to 0."},{"Start":"02:24.260 ","End":"02:26.735","Text":"Now let\u0027s apply f,"},{"Start":"02:26.735 ","End":"02:30.180","Text":"whereas f (x) is just 4^1/x."},{"Start":"02:30.180 ","End":"02:33.040","Text":"Apply this to each of the sequences,"},{"Start":"02:33.040 ","End":"02:35.945","Text":"or substitute the sequence in the function,"},{"Start":"02:35.945 ","End":"02:43.910","Text":"then we get f( x_n) is 4^1/1,"},{"Start":"02:43.910 ","End":"02:45.500","Text":"which is 4^n,"},{"Start":"02:45.500 ","End":"02:47.770","Text":"and that goes to infinity."},{"Start":"02:47.770 ","End":"02:49.650","Text":"On the other hand,"},{"Start":"02:49.650 ","End":"02:51.540","Text":"if we plug in,"},{"Start":"02:51.540 ","End":"02:54.765","Text":"should be f(y_n),"},{"Start":"02:54.765 ","End":"02:59.000","Text":"and that comes out 4^minus n,"},{"Start":"02:59.000 ","End":"03:02.315","Text":"which goes to 0 as n goes to infinity."},{"Start":"03:02.315 ","End":"03:05.720","Text":"Certainly 0 is not equal to infinity,"},{"Start":"03:05.720 ","End":"03:09.995","Text":"so we have 2 sequences going to 0,"},{"Start":"03:09.995 ","End":"03:13.205","Text":"but f of them goes to different things,"},{"Start":"03:13.205 ","End":"03:17.285","Text":"so the function f does not have a limit at 0."},{"Start":"03:17.285 ","End":"03:21.255","Text":"Method 2, is to find a sequence,"},{"Start":"03:21.255 ","End":"03:27.270","Text":"such that limit of x_n is a,"},{"Start":"03:27.270 ","End":"03:32.400","Text":"but the limit of f(x_n) doesn\u0027t exist at all,"},{"Start":"03:32.400 ","End":"03:37.610","Text":"never mind that it doesn\u0027t equal L. If The limit doesn\u0027t exist altogether then"},{"Start":"03:37.610 ","End":"03:43.820","Text":"you know certainly it\u0027s not equal to L. Let\u0027s take an example of that,"},{"Start":"03:43.820 ","End":"03:46.700","Text":"in fact will take the same example,"},{"Start":"03:46.700 ","End":"03:52.925","Text":"but we\u0027ll do it differently is to use Method 2 with a single sequence."},{"Start":"03:52.925 ","End":"03:59.400","Text":"We\u0027ll take the sequence x_n is an alternating sequence,"},{"Start":"03:59.400 ","End":"04:01.770","Text":"we can see from the minus 1^n,"},{"Start":"04:01.770 ","End":"04:05.460","Text":"minus 1^n goes to 0."},{"Start":"04:05.460 ","End":"04:08.355","Text":"It\u0027s minus 1 plus a 1/2,"},{"Start":"04:08.355 ","End":"04:12.195","Text":"minus a 1/3 plus a 1/4 and so on, and it goes to 0."},{"Start":"04:12.195 ","End":"04:19.905","Text":"But if I substitute in the function f(x) is 4^1/x,"},{"Start":"04:19.905 ","End":"04:24.570","Text":"what we get, and we just figure it."},{"Start":"04:24.570 ","End":"04:30.160","Text":"Let\u0027s see f(x_n) is 1 this expression."},{"Start":"04:30.800 ","End":"04:35.390","Text":"What we get is n over minus 1^n."},{"Start":"04:35.390 ","End":"04:38.081","Text":"You can actually bring minus 1^n,"},{"Start":"04:38.081 ","End":"04:40.690","Text":"to the numerator,"},{"Start":"04:40.690 ","End":"04:46.460","Text":"because if n is even this is plus and it\u0027s safe to bring to the numerator."},{"Start":"04:46.460 ","End":"04:48.275","Text":"This is what we get."},{"Start":"04:48.275 ","End":"04:50.150","Text":"This minus 1^n,"},{"Start":"04:50.150 ","End":"04:53.940","Text":"it alternates between plus and minus 1,"},{"Start":"04:54.310 ","End":"04:58.370","Text":"which indicates that it\u0027s not going to have a limit."},{"Start":"04:58.370 ","End":"05:00.140","Text":"If we write it out,"},{"Start":"05:00.140 ","End":"05:04.835","Text":"it becomes even more clear because we have 4^minus 4^2,"},{"Start":"05:04.835 ","End":"05:07.860","Text":"4^minus 3, 4^4th."},{"Start":"05:07.860 ","End":"05:09.710","Text":"If you look at the red ones,"},{"Start":"05:09.710 ","End":"05:13.190","Text":"they\u0027re going to 0, 1/4,"},{"Start":"05:13.190 ","End":"05:19.950","Text":"1/64, 1/1024,"},{"Start":"05:19.950 ","End":"05:23.760","Text":"and this going to infinity, 16,"},{"Start":"05:23.760 ","End":"05:27.210","Text":"256, it just jumps all over the place,"},{"Start":"05:27.210 ","End":"05:29.355","Text":"obviously it doesn\u0027t converge."},{"Start":"05:29.355 ","End":"05:31.667","Text":"The result follows that,"},{"Start":"05:31.667 ","End":"05:34.565","Text":"this function does not have a limit,"},{"Start":"05:34.565 ","End":"05:38.540","Text":"this 4^1/x when x goes to 0."},{"Start":"05:38.540 ","End":"05:41.790","Text":"We\u0027re done with this clip."}],"ID":31285},{"Watched":false,"Name":"Exercise 1","Duration":"3m 25s","ChapterTopicVideoID":29646,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this exercise, we have to prove that the limit as x goes to"},{"Start":"00:05.190 ","End":"00:10.770","Text":"4 of x minus 4 over absolute value of x minus 4 does not exist."},{"Start":"00:10.770 ","End":"00:12.960","Text":"I\u0027m going to show it in 2 different ways using"},{"Start":"00:12.960 ","End":"00:17.040","Text":"Heine\u0027s definition and using one-sided limits."},{"Start":"00:17.040 ","End":"00:22.300","Text":"Let\u0027s denote this function as f (x)."},{"Start":"00:23.480 ","End":"00:27.345","Text":"What Heine said is that if the limit exists,"},{"Start":"00:27.345 ","End":"00:34.395","Text":"then all sequences that tend to 4 f of that sequence will tend to that limit."},{"Start":"00:34.395 ","End":"00:39.900","Text":"What it implies is that if I can find 2 different sequences that tend to 4,"},{"Start":"00:39.900 ","End":"00:44.495","Text":"but f of them is a different answer,"},{"Start":"00:44.495 ","End":"00:47.269","Text":"then there\u0027s no limit."},{"Start":"00:47.269 ","End":"00:53.235","Text":"What I\u0027m going to do is choose the following 2 sequences."},{"Start":"00:53.235 ","End":"01:00.660","Text":"First of all, we\u0027ll let x_n be 4 plus 1 over n. The other sequence will be y_n,"},{"Start":"01:00.660 ","End":"01:04.070","Text":"which will be 4 minus 1 over n. Now,"},{"Start":"01:04.070 ","End":"01:06.655","Text":"when n goes to infinity,"},{"Start":"01:06.655 ","End":"01:10.365","Text":"1 over n goes to 0 here and here."},{"Start":"01:10.365 ","End":"01:12.540","Text":"So here we have 4 plus 0,"},{"Start":"01:12.540 ","End":"01:14.555","Text":"and here we have 4 minus 0."},{"Start":"01:14.555 ","End":"01:16.810","Text":"In other words, in both cases we have 4,"},{"Start":"01:16.810 ","End":"01:18.754","Text":"so we have 2 different sequences."},{"Start":"01:18.754 ","End":"01:22.790","Text":"They are different, and they go to 4."},{"Start":"01:22.790 ","End":"01:28.940","Text":"Now, if we substitute each of these sequences in the function f(x),"},{"Start":"01:28.940 ","End":"01:30.405","Text":"which is this,"},{"Start":"01:30.405 ","End":"01:32.355","Text":"if we substitute x_n,"},{"Start":"01:32.355 ","End":"01:35.306","Text":"then we get 4 plus 1 over n minus 4,"},{"Start":"01:35.306 ","End":"01:37.790","Text":"and then the same thing with absolute value."},{"Start":"01:37.790 ","End":"01:43.025","Text":"Now 4 plus 1 over n minus 4 is 1 over n. Since 1 over n is positive,"},{"Start":"01:43.025 ","End":"01:45.215","Text":"we can throw out the absolute value."},{"Start":"01:45.215 ","End":"01:48.350","Text":"This is 1, so the limit is 1."},{"Start":"01:48.350 ","End":"01:51.515","Text":"Now if we substitute the other one,"},{"Start":"01:51.515 ","End":"01:56.775","Text":"the y_n, then we have 4 minus 1 over n instead."},{"Start":"01:56.775 ","End":"02:01.205","Text":"At this point, we get minus 1 over n divided by the absolute value,"},{"Start":"02:01.205 ","End":"02:07.280","Text":"but the absolute value of minus 1 over n is plus 1 over n. So this quotient is minus 1,"},{"Start":"02:07.280 ","End":"02:09.314","Text":"and therefore tends to minus 1."},{"Start":"02:09.314 ","End":"02:12.275","Text":"So here, 2 sequences,"},{"Start":"02:12.275 ","End":"02:16.505","Text":"both tending to the same 4,"},{"Start":"02:16.505 ","End":"02:20.180","Text":"but f of them doesn\u0027t go to the same thing."},{"Start":"02:20.180 ","End":"02:27.530","Text":"So by Heine, there is no limit of f at x=4."},{"Start":"02:27.530 ","End":"02:33.230","Text":"Now we\u0027ll do it the other way using one-sided limits."},{"Start":"02:33.230 ","End":"02:35.765","Text":"f(x) is this."},{"Start":"02:35.765 ","End":"02:39.440","Text":"Now we\u0027ll write it as a piecewise defined function because"},{"Start":"02:39.440 ","End":"02:46.130","Text":"the absolute value depends on if x minus 4 is non-negative or negative."},{"Start":"02:46.130 ","End":"02:49.045","Text":"In other words, if x is bigger or equal to 4,"},{"Start":"02:49.045 ","End":"02:50.790","Text":"then this is non-negative."},{"Start":"02:50.790 ","End":"02:52.770","Text":"So I\u0027ll throw the absolute value away."},{"Start":"02:52.770 ","End":"02:54.200","Text":"If x is less than 4,"},{"Start":"02:54.200 ","End":"02:56.006","Text":"then denominator is negative,"},{"Start":"02:56.006 ","End":"02:57.650","Text":"so we put a minus here."},{"Start":"02:57.650 ","End":"02:59.885","Text":"This is what f is equal to."},{"Start":"02:59.885 ","End":"03:03.170","Text":"In other words, if x is bigger or equal to 4,"},{"Start":"03:03.170 ","End":"03:05.810","Text":"this is just the same as this, so the quotient is 1."},{"Start":"03:05.810 ","End":"03:07.040","Text":"If x is less than 4,"},{"Start":"03:07.040 ","End":"03:10.280","Text":"we have a minus, so minus 1."},{"Start":"03:10.280 ","End":"03:12.845","Text":"If x goes to 4 from the right,"},{"Start":"03:12.845 ","End":"03:15.680","Text":"then I\u0027m using this definition, and I\u0027ll get 1."},{"Start":"03:15.680 ","End":"03:17.270","Text":"If our x goes to 4 from the left,"},{"Start":"03:17.270 ","End":"03:19.925","Text":"I\u0027ll be using this, and I\u0027ll get minus 1."},{"Start":"03:19.925 ","End":"03:22.460","Text":"Since the 2 one-sided limits are not equal,"},{"Start":"03:22.460 ","End":"03:25.890","Text":"there is no limit. We\u0027re done."}],"ID":31286},{"Watched":false,"Name":"Exercise 2","Duration":"2m 45s","ChapterTopicVideoID":29647,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.494","Text":"In this exercise, we have to use Heine\u0027s definition of a limit of a function to prove"},{"Start":"00:06.494 ","End":"00:14.025","Text":"that the limit as x goes to infinity of this expression does not exist."},{"Start":"00:14.025 ","End":"00:16.410","Text":"I\u0027ll call this f(x)."},{"Start":"00:16.410 ","End":"00:21.240","Text":"The way we use Heine\u0027s definition to prove nonexistence of"},{"Start":"00:21.240 ","End":"00:26.640","Text":"a limit is defined 2 separate sequences,"},{"Start":"00:26.640 ","End":"00:32.535","Text":"each of them tending to infinity or whatever it is that it was here."},{"Start":"00:32.535 ","End":"00:37.920","Text":"Then to plug both of the sequences into f and show that we"},{"Start":"00:37.920 ","End":"00:43.675","Text":"get different results for the limit. Well, I\u0027ll show you."},{"Start":"00:43.675 ","End":"00:47.210","Text":"What I\u0027m going to do is take the following 2 sequences."},{"Start":"00:47.210 ","End":"00:53.030","Text":"First of all, we\u0027ll take x_n as being multiples of 2Pi,"},{"Start":"00:53.030 ","End":"00:55.430","Text":"4Pi, 6Pi, and so on."},{"Start":"00:55.430 ","End":"00:58.850","Text":"Certainly, that goes to infinity."},{"Start":"00:58.850 ","End":"01:02.930","Text":"The other limit I\u0027ll choose will be y_n,"},{"Start":"01:02.930 ","End":"01:07.820","Text":"which will also be in jumps of 2Pi but we\u0027ll start with 2.5Pi,"},{"Start":"01:07.820 ","End":"01:09.215","Text":"4.5, 6.5,"},{"Start":"01:09.215 ","End":"01:10.750","Text":"8.5, and so on."},{"Start":"01:10.750 ","End":"01:12.775","Text":"2n plus 1/2 Pi,"},{"Start":"01:12.775 ","End":"01:14.450","Text":"it also goes to infinity."},{"Start":"01:14.450 ","End":"01:25.025","Text":"Now we\u0027ll plug each of x_n and then y_n into the function f. If we plug x_n,"},{"Start":"01:25.025 ","End":"01:30.110","Text":"we get x_n is 2nPi in general."},{"Start":"01:30.110 ","End":"01:34.855","Text":"So 2nPi plus 4 over cosine nPi plus 10."},{"Start":"01:34.855 ","End":"01:39.815","Text":"Now, the sine of multiples of 2Pi is 0,"},{"Start":"01:39.815 ","End":"01:45.590","Text":"it\u0027s like sine of 0, and cosine of 0 or multiples of 2Pi is 1."},{"Start":"01:45.590 ","End":"01:51.820","Text":"What we get is 4/11 for a limit of f(x_n)."},{"Start":"01:51.820 ","End":"01:55.110","Text":"Now, let\u0027s take the limit of f(y_n)."},{"Start":"01:55.250 ","End":"02:03.150","Text":"f(y_n), we plug into the same function 2n plus 1/2 Pi."},{"Start":"02:06.740 ","End":"02:10.005","Text":"This is going to be the same as sine of 1/2 Pi,"},{"Start":"02:10.005 ","End":"02:12.480","Text":"and this will be like cosine of 1/2 Pi."},{"Start":"02:12.480 ","End":"02:15.270","Text":"Sine of 90 degrees is 1,"},{"Start":"02:15.270 ","End":"02:17.295","Text":"cosine of 90 degrees is 0,"},{"Start":"02:17.295 ","End":"02:20.700","Text":"90 is 1/2 Pi here. We get what?"},{"Start":"02:20.700 ","End":"02:24.640","Text":"5/10, which you could simplify to 1/2."},{"Start":"02:24.640 ","End":"02:27.791","Text":"In any event, this is not the same as this."},{"Start":"02:27.791 ","End":"02:32.915","Text":"We found 2 different limits for f of a sequence,"},{"Start":"02:32.915 ","End":"02:36.755","Text":"and both the sequences went to the same infinity,"},{"Start":"02:36.755 ","End":"02:39.170","Text":"so by Heine,"},{"Start":"02:39.170 ","End":"02:45.480","Text":"the limit of f does not exist. We\u0027re done."}],"ID":31287},{"Watched":false,"Name":"Exercise 3","Duration":"2m 18s","ChapterTopicVideoID":29648,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this exercise, we have to use"},{"Start":"00:02.490 ","End":"00:06.090","Text":"Heine\u0027s definition of a limit of a function to prove that"},{"Start":"00:06.090 ","End":"00:11.729","Text":"the limit as x goes to 0 of sine of 1/x does not exist."},{"Start":"00:11.729 ","End":"00:13.350","Text":"Let\u0027s give this a label,"},{"Start":"00:13.350 ","End":"00:16.260","Text":"let\u0027s call this f(x)."},{"Start":"00:16.260 ","End":"00:21.840","Text":"Now, Heine\u0027s definition implies that if we can find 2 different sequences,"},{"Start":"00:21.840 ","End":"00:23.640","Text":"say x_n and y_n,"},{"Start":"00:23.640 ","End":"00:25.470","Text":"both going to 0,"},{"Start":"00:25.470 ","End":"00:27.930","Text":"but when I apply f to them,"},{"Start":"00:27.930 ","End":"00:29.790","Text":"they go to different limits,"},{"Start":"00:29.790 ","End":"00:33.165","Text":"then it will prove that this limit doesn\u0027t exist."},{"Start":"00:33.165 ","End":"00:35.730","Text":"Let\u0027s choose 2 sequences."},{"Start":"00:35.730 ","End":"00:42.030","Text":"Let\u0027s choose x_n to be 1 over multiples of 2Pi;"},{"Start":"00:42.030 ","End":"00:43.800","Text":"2Pi, 4Pi,"},{"Start":"00:43.800 ","End":"00:46.125","Text":"6Pi, 8Pi, in general 2nPi."},{"Start":"00:46.125 ","End":"00:51.790","Text":"This will of course go to 0 because the denominator goes to infinity."},{"Start":"00:51.790 ","End":"00:55.020","Text":"When n goes to infinity, 2nPi goes to infinity."},{"Start":"00:55.020 ","End":"00:59.615","Text":"The other sequence will also go to 0."},{"Start":"00:59.615 ","End":"01:02.535","Text":"Instead of multiples of 2Pi,"},{"Start":"01:02.535 ","End":"01:06.660","Text":"we\u0027ll have 2.5Pi and then jump by 2Pi each time,"},{"Start":"01:06.660 ","End":"01:09.285","Text":"4.5 6.5, 8.5Pi,"},{"Start":"01:09.285 ","End":"01:13.095","Text":"and so on, also goes to 0."},{"Start":"01:13.095 ","End":"01:20.705","Text":"Now, we\u0027re going to apply the function f to each of these."},{"Start":"01:20.705 ","End":"01:22.445","Text":"If we apply f to x_n,"},{"Start":"01:22.445 ","End":"01:26.790","Text":"we get sine 1 over 1 over,"},{"Start":"01:27.800 ","End":"01:30.795","Text":"means it\u0027s just 2nPi."},{"Start":"01:30.795 ","End":"01:35.460","Text":"Sine of multiples of 2Pi is sine 0 is 0."},{"Start":"01:35.460 ","End":"01:38.324","Text":"Constant sequence goes to 0."},{"Start":"01:38.324 ","End":"01:42.810","Text":"If I apply to y_n,"},{"Start":"01:42.810 ","End":"01:48.930","Text":"then 1 over 1 over just brings us to the top,"},{"Start":"01:48.930 ","End":"01:52.530","Text":"and multiples of 2nPi don\u0027t matter."},{"Start":"01:52.530 ","End":"01:56.760","Text":"It\u0027s like sine of 1/2Pi or sine of 90 degrees is 1,"},{"Start":"01:56.760 ","End":"02:01.960","Text":"and a constant also goes to 1."},{"Start":"02:02.660 ","End":"02:05.570","Text":"We have 2 different sequences,"},{"Start":"02:05.570 ","End":"02:06.920","Text":"both tending to 0,"},{"Start":"02:06.920 ","End":"02:09.005","Text":"but after I apply f to them,"},{"Start":"02:09.005 ","End":"02:11.070","Text":"then we get different limits."},{"Start":"02:11.070 ","End":"02:15.975","Text":"By Heine, this limit does not exist,"},{"Start":"02:15.975 ","End":"02:18.640","Text":"and so we\u0027re done."}],"ID":31288},{"Watched":false,"Name":"Exercise 4","Duration":"3m 38s","ChapterTopicVideoID":29649,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.530","Text":"In this exercise, we have to use"},{"Start":"00:03.530 ","End":"00:08.040","Text":"Heine\u0027s definition of a limit of a function to prove that the limit as"},{"Start":"00:08.040 ","End":"00:15.495","Text":"x goes to infinity of e to the x minus the floor function of x does not exist."},{"Start":"00:15.495 ","End":"00:21.155","Text":"I\u0027ll explain. The floor function means to round down to a whole number."},{"Start":"00:21.155 ","End":"00:28.185","Text":"For example, the floor function of 6.8 is 6."},{"Start":"00:28.185 ","End":"00:35.375","Text":"But the floor function of the whole number like 9 is 9 itself."},{"Start":"00:35.375 ","End":"00:39.005","Text":"Sometimes, it\u0027s written with square brackets,"},{"Start":"00:39.005 ","End":"00:41.720","Text":"so you could write it like that."},{"Start":"00:41.720 ","End":"00:47.950","Text":"Like the square brackets of 5.9 is 5."},{"Start":"00:47.950 ","End":"00:52.730","Text":"Heine\u0027s definition implies that if I take"},{"Start":"00:52.730 ","End":"00:59.390","Text":"2 different sequences that both go to infinity and then I apply,"},{"Start":"00:59.390 ","End":"01:02.495","Text":"let\u0027s call this function f(x)."},{"Start":"01:02.495 ","End":"01:05.705","Text":"If we apply f to both sequences,"},{"Start":"01:05.705 ","End":"01:10.189","Text":"we should get the same limit and if we don\u0027t,"},{"Start":"01:10.189 ","End":"01:12.750","Text":"then it means the limit doesn\u0027t exist."},{"Start":"01:12.750 ","End":"01:16.870","Text":"I\u0027ll choose 1 sequence. Let\u0027s see."},{"Start":"01:16.870 ","End":"01:22.170","Text":"We\u0027ll choose x_n to be n for all n,"},{"Start":"01:22.170 ","End":"01:25.035","Text":"1, 2, 3, 4, 5, etc."},{"Start":"01:25.035 ","End":"01:30.125","Text":"I\u0027ll choose y_n to be n plus 1/2."},{"Start":"01:30.125 ","End":"01:33.365","Text":"Now, both of these clearly tend to infinity."},{"Start":"01:33.365 ","End":"01:35.795","Text":"1, 2, 3, 4, 5, n goes to infinity."},{"Start":"01:35.795 ","End":"01:37.130","Text":"When n goes to infinity,"},{"Start":"01:37.130 ","End":"01:40.474","Text":"n plus 1/2 goes to infinity plus 1/2,"},{"Start":"01:40.474 ","End":"01:42.555","Text":"yeah, also infinity."},{"Start":"01:42.555 ","End":"01:48.180","Text":"Now, let\u0027s apply f to both x and y and"},{"Start":"01:48.180 ","End":"01:54.020","Text":"f(x) is e to the x minus the round down function of x."},{"Start":"01:54.020 ","End":"01:56.870","Text":"What we get is as follows."},{"Start":"01:56.870 ","End":"02:03.590","Text":"f(x_n) is e^n minus floor function"},{"Start":"02:03.590 ","End":"02:08.720","Text":"of n. But n minus the floor function of n,"},{"Start":"02:08.720 ","End":"02:11.645","Text":"where n is a whole number is always 0."},{"Start":"02:11.645 ","End":"02:18.890","Text":"Just like 9 minus floor function of 9 is just rounds it down to a whole number,"},{"Start":"02:18.890 ","End":"02:20.165","Text":"but it is a whole number,"},{"Start":"02:20.165 ","End":"02:22.590","Text":"leaves it as it is."},{"Start":"02:22.790 ","End":"02:28.605","Text":"This is just e^0 is 1."},{"Start":"02:28.605 ","End":"02:33.465","Text":"If we apply f to y_n,"},{"Start":"02:33.465 ","End":"02:38.870","Text":"then we get n plus 1/2 minus round down or"},{"Start":"02:38.870 ","End":"02:44.225","Text":"floor function of n plus 1/2 and that would just be n itself."},{"Start":"02:44.225 ","End":"02:46.160","Text":"For example, if n is 17,"},{"Start":"02:46.160 ","End":"02:54.755","Text":"then here we have 17 plus 1/2 minus floor function of 17.5."},{"Start":"02:54.755 ","End":"02:57.980","Text":"This is 17, so this is 1/2."},{"Start":"02:57.980 ","End":"03:04.175","Text":"In general, n plus 1/2 minus round down of n plus 1/2,"},{"Start":"03:04.175 ","End":"03:12.260","Text":"will equal 1/2 because this is just n. This is 1/2."},{"Start":"03:12.260 ","End":"03:14.840","Text":"It\u0027s a constant sequence,"},{"Start":"03:14.840 ","End":"03:19.500","Text":"so it tends to e^1/2 as n goes to infinity."},{"Start":"03:19.580 ","End":"03:22.365","Text":"These 2 are different,"},{"Start":"03:22.365 ","End":"03:25.290","Text":"e^1/2 is not equal to 1."},{"Start":"03:25.290 ","End":"03:26.790","Text":"This is the square root of e,"},{"Start":"03:26.790 ","End":"03:29.290","Text":"it\u0027s more than 1."},{"Start":"03:30.050 ","End":"03:34.750","Text":"Because we got different limits by Heine,"},{"Start":"03:34.750 ","End":"03:39.810","Text":"this limit doesn\u0027t exist, and we\u0027re done."}],"ID":31289},{"Watched":false,"Name":"Exercise 5","Duration":"2m 44s","ChapterTopicVideoID":29651,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this exercise, we have a function from"},{"Start":"00:03.810 ","End":"00:07.695","Text":"the reals to the reals and we have a real number x naught."},{"Start":"00:07.695 ","End":"00:14.145","Text":"Now, suppose that the limit as x goes to x naught of f(x) exists."},{"Start":"00:14.145 ","End":"00:18.885","Text":"We have to show that the limit as x goes to 0 of"},{"Start":"00:18.885 ","End":"00:24.885","Text":"f(x) plus x naught is the same limit as this."},{"Start":"00:24.885 ","End":"00:26.220","Text":"If this limit exists,"},{"Start":"00:26.220 ","End":"00:27.780","Text":"we can give it a name,"},{"Start":"00:27.780 ","End":"00:31.260","Text":"call it capital L. Now what we\u0027re going to do,"},{"Start":"00:31.260 ","End":"00:34.440","Text":"we have to show that this limit exists and is equal to this."},{"Start":"00:34.440 ","End":"00:37.275","Text":"We\u0027ll use the sequential form of a limit."},{"Start":"00:37.275 ","End":"00:40.230","Text":"We\u0027ll show that for any sequence x_n,"},{"Start":"00:40.230 ","End":"00:44.570","Text":"such that x_n goes to 0."},{"Start":"00:44.570 ","End":"00:49.895","Text":"Let me just pause here. Let write just the right arrow here with sequences,"},{"Start":"00:49.895 ","End":"00:54.100","Text":"it\u0027s a shorthand for saying the limit as n goes to infinity."},{"Start":"00:54.100 ","End":"00:57.980","Text":"Just wanted to state that in case it wasn\u0027t obvious."},{"Start":"00:57.980 ","End":"01:03.515","Text":"We show that for any sequence that tends to 0, x_n,"},{"Start":"01:03.515 ","End":"01:06.830","Text":"we have f(x)_n plus x naught goes to"},{"Start":"01:06.830 ","End":"01:11.015","Text":"L. That will be the sequence definition of this limit."},{"Start":"01:11.015 ","End":"01:16.385","Text":"You replace x by x_n and you let x_n go to whatever x goes to."},{"Start":"01:16.385 ","End":"01:19.850","Text":"Now, there\u0027s an addition of limits."},{"Start":"01:19.850 ","End":"01:24.170","Text":"If you add 2 sequences term wise,"},{"Start":"01:24.170 ","End":"01:27.095","Text":"the limit of the sum is the sum of the limits."},{"Start":"01:27.095 ","End":"01:31.820","Text":"X naught can be seen as a constant sequence, if you like."},{"Start":"01:31.820 ","End":"01:36.225","Text":"X_n plus x naught goes to,"},{"Start":"01:36.225 ","End":"01:39.390","Text":"x_n goes to 0, x_0 goes to x_0,"},{"Start":"01:39.390 ","End":"01:41.775","Text":"so it\u0027s just x_0."},{"Start":"01:41.775 ","End":"01:46.885","Text":"This x_n plus x naught goes to x_0."},{"Start":"01:46.885 ","End":"01:51.290","Text":"Now, since this limit is equal to L,"},{"Start":"01:51.290 ","End":"01:52.805","Text":"that\u0027s the original limit."},{"Start":"01:52.805 ","End":"01:59.134","Text":"If we have some other sequence y_n that goes to x naught,"},{"Start":"01:59.134 ","End":"02:03.830","Text":"then f(y_n) goes to the limit"},{"Start":"02:03.830 ","End":"02:10.495","Text":"L. That\u0027s sort of the sequential definition of this limit just using letter y_n."},{"Start":"02:10.495 ","End":"02:15.720","Text":"Now, if we take y_n to equal x_n plus x naught,"},{"Start":"02:15.720 ","End":"02:21.240","Text":"we get that f(x _n) plus x naught goes to"},{"Start":"02:21.240 ","End":"02:27.300","Text":"L. This is true because y_n goes to x naught."},{"Start":"02:27.300 ","End":"02:30.885","Text":"Y_n is this and it goes to x naught."},{"Start":"02:30.885 ","End":"02:32.890","Text":"If y_n goes to x naught,"},{"Start":"02:32.890 ","End":"02:34.530","Text":"f(y_n) goes to L,"},{"Start":"02:34.530 ","End":"02:36.825","Text":"but y_n it happens to be this sum,"},{"Start":"02:36.825 ","End":"02:39.650","Text":"and that\u0027s what we had to prove."},{"Start":"02:39.650 ","End":"02:41.420","Text":"It\u0027s what we said here we\u0027re looking for,"},{"Start":"02:41.420 ","End":"02:44.940","Text":"so we\u0027ve done it and that\u0027s it."}],"ID":31290},{"Watched":false,"Name":"Exercise 6","Duration":"1m 54s","ChapterTopicVideoID":29652,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.475","Text":"In this exercise, f(x) is the absolute value function."},{"Start":"00:05.475 ","End":"00:09.375","Text":"We have to show that it\u0027s continuous everywhere."},{"Start":"00:09.375 ","End":"00:13.800","Text":"Now what we have to do in terms of sequences is to"},{"Start":"00:13.800 ","End":"00:18.825","Text":"show that if a sequence x_n converges to some x_naught,"},{"Start":"00:18.825 ","End":"00:26.775","Text":"then we can apply f to it and say that f(x_n) tends to f of x_naught."},{"Start":"00:26.775 ","End":"00:30.060","Text":"Now, we\u0027ll interpret this in terms of Epsilon."},{"Start":"00:30.060 ","End":"00:34.095","Text":"What it means is that given Epsilon bigger than 0,"},{"Start":"00:34.095 ","End":"00:42.615","Text":"there is some N such that whenever n is bigger than N,"},{"Start":"00:42.615 ","End":"00:48.450","Text":"then the distance from x_n to x_naught is less than Epsilon."},{"Start":"00:48.820 ","End":"00:54.860","Text":"Now, one of the variations of the triangle inequality allows us to write this that"},{"Start":"00:54.860 ","End":"00:57.590","Text":"the absolute value of the difference is bigger or equal"},{"Start":"00:57.590 ","End":"01:01.220","Text":"to the difference of the absolute values."},{"Start":"01:01.220 ","End":"01:04.220","Text":"Note that here,"},{"Start":"01:04.220 ","End":"01:05.390","Text":"we have f(x _n),"},{"Start":"01:05.390 ","End":"01:08.330","Text":"and here we have f of x_naught."},{"Start":"01:08.330 ","End":"01:15.365","Text":"What we have is that for all n bigger or equal to N,"},{"Start":"01:15.365 ","End":"01:23.330","Text":"this difference, f(x_n) minus f of x_naught is less than or equal to x_n minus x_naught."},{"Start":"01:23.330 ","End":"01:26.030","Text":"That\u0027s from here, just reversing it."},{"Start":"01:26.030 ","End":"01:30.745","Text":"This is less than Epsilon, from here."},{"Start":"01:30.745 ","End":"01:33.935","Text":"If you look at the colored part,"},{"Start":"01:33.935 ","End":"01:37.685","Text":"this difference is less than Epsilon."},{"Start":"01:37.685 ","End":"01:41.269","Text":"That\u0027s basically what we need for showing this convergence."},{"Start":"01:41.269 ","End":"01:47.090","Text":"In other words, we\u0027re using the same n from the x_n goes to"},{"Start":"01:47.090 ","End":"01:54.990","Text":"x_naught to show that f(x_n) goes to f of x_naught. That\u0027s it."}],"ID":31291},{"Watched":false,"Name":"Exercise 7","Duration":"2m 10s","ChapterTopicVideoID":29653,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.225","Text":"In this exercise, we have a function f which is defined on the closed interval 0, pi,"},{"Start":"00:06.225 ","End":"00:11.655","Text":"and it\u0027s defined split wise for x_naught equal to 0,"},{"Start":"00:11.655 ","End":"00:14.280","Text":"f of x is equal to this expression."},{"Start":"00:14.280 ","End":"00:19.200","Text":"Notice that it has a 1/x in it and it\u0027s not defined at x equals 0,"},{"Start":"00:19.200 ","End":"00:24.435","Text":"so we define separately at 0 to be 0,"},{"Start":"00:24.435 ","End":"00:27.480","Text":"and the question is, is it continuous?"},{"Start":"00:27.480 ","End":"00:31.575","Text":"Well, certainly, continuous when x is not 0."},{"Start":"00:31.575 ","End":"00:33.640","Text":"The question is what happens at 0?"},{"Start":"00:33.640 ","End":"00:38.010","Text":"I\u0027ll tell you, it\u0027s not continuous at 0."},{"Start":"00:38.010 ","End":"00:40.270","Text":"How do we show this?"},{"Start":"00:40.270 ","End":"00:46.235","Text":"The easiest way is to find a sequence that goes to 0."},{"Start":"00:46.235 ","End":"00:48.860","Text":"In fact, this will be our sequence,"},{"Start":"00:48.860 ","End":"00:53.810","Text":"x_n is 1/2 n Pi."},{"Start":"00:53.810 ","End":"00:57.260","Text":"You\u0027ll see why this is useful,"},{"Start":"00:57.260 ","End":"01:02.000","Text":"and the claim is that if you apply f to it,"},{"Start":"01:02.000 ","End":"01:05.375","Text":"f of x_n doesn\u0027t go to f of 0,"},{"Start":"01:05.375 ","End":"01:08.240","Text":"and that will prove that\u0027s not continuous."},{"Start":"01:08.240 ","End":"01:11.135","Text":"Let\u0027s see what is f of x_n,"},{"Start":"01:11.135 ","End":"01:16.520","Text":"x is 1/2 Pi n. That\u0027s this,"},{"Start":"01:16.520 ","End":"01:24.495","Text":"1/x is just 2Pi n. That\u0027s sine of 2Pi n over this,"},{"Start":"01:24.495 ","End":"01:27.450","Text":"and here we have the 2Pi n here,"},{"Start":"01:27.450 ","End":"01:30.925","Text":"cosine of 2Pi n. Now,"},{"Start":"01:30.925 ","End":"01:37.605","Text":"the sine is 0 on multiples of Pi or 2Pi."},{"Start":"01:37.605 ","End":"01:44.815","Text":"This part is 0. The cosine of a multiple of 2Pi is equal to 1,"},{"Start":"01:44.815 ","End":"01:50.905","Text":"so what we\u0027re left with is the minus 2Pi n. Certainly,"},{"Start":"01:50.905 ","End":"01:52.585","Text":"when n goes to infinity,"},{"Start":"01:52.585 ","End":"01:58.885","Text":"minus 2Pi n doesn\u0027t go to 0, which is f of 0."},{"Start":"01:58.885 ","End":"02:02.925","Text":"In fact, it goes to minus infinity."},{"Start":"02:02.925 ","End":"02:06.870","Text":"Anyway, this was the thing we had to show,"},{"Start":"02:06.870 ","End":"02:08.255","Text":"and we\u0027ve shown it,"},{"Start":"02:08.255 ","End":"02:10.710","Text":"and we are done."}],"ID":31292},{"Watched":false,"Name":"Exercise 8","Duration":"2m 15s","ChapterTopicVideoID":29654,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this exercise, we have 2 functions,"},{"Start":"00:02.730 ","End":"00:06.630","Text":"f and g and they\u0027re both continuous and they"},{"Start":"00:06.630 ","End":"00:10.995","Text":"have this property that when you have any 2 points,"},{"Start":"00:10.995 ","End":"00:13.035","Text":"x_1 less than x_2,"},{"Start":"00:13.035 ","End":"00:16.290","Text":"at some point in between x_1 and x_2,"},{"Start":"00:16.290 ","End":"00:19.299","Text":"where f and g are equal."},{"Start":"00:19.299 ","End":"00:24.200","Text":"You could rephrase this by saying that on any open interval that\u0027s the x_1,"},{"Start":"00:24.200 ","End":"00:28.905","Text":"x_2, there\u0027s the point where the functions are equal."},{"Start":"00:28.905 ","End":"00:34.325","Text":"Our task is to show that f equals g everywhere."},{"Start":"00:34.325 ","End":"00:38.284","Text":"Now as is often the case in this question,"},{"Start":"00:38.284 ","End":"00:41.900","Text":"it\u0027s useful to look at the difference,"},{"Start":"00:41.900 ","End":"00:43.640","Text":"that of saying f equals g,"},{"Start":"00:43.640 ","End":"00:45.410","Text":"the difference is zero."},{"Start":"00:45.410 ","End":"00:51.830","Text":"We\u0027re going to show that h is zero by showing that any point at 0 for any x_0."},{"Start":"00:51.830 ","End":"00:55.380","Text":"Now given x_0,"},{"Start":"00:56.300 ","End":"01:01.185","Text":"we can take x_1 to be x_0,"},{"Start":"01:01.185 ","End":"01:07.665","Text":"and we can take x_2 to be x_0 plus 1 over n,"},{"Start":"01:07.665 ","End":"01:11.930","Text":"and we are guaranteed the existence of some x_3,"},{"Start":"01:11.930 ","End":"01:15.950","Text":"and we\u0027ll call it x_n because it depends on n,"},{"Start":"01:15.950 ","End":"01:25.080","Text":"such that f(x_n)=g(x_n) or in other words, h(x_n)=0."},{"Start":"01:25.080 ","End":"01:31.155","Text":"Now note that this sequence x_0 plus 1 over n tends to x_0."},{"Start":"01:31.155 ","End":"01:37.560","Text":"What we have is that x_n is sandwiched between x_0 and something that tends to x_0."},{"Start":"01:37.560 ","End":"01:42.985","Text":"We apply the sandwich theorem and x_n tends to x_0."},{"Start":"01:42.985 ","End":"01:46.625","Text":"Now, h is continuous."},{"Start":"01:46.625 ","End":"01:51.005","Text":"Difference of 2 continuous functions is continuous."},{"Start":"01:51.005 ","End":"01:53.240","Text":"If x_n goes to x_0,"},{"Start":"01:53.240 ","End":"01:57.790","Text":"we have to have h(x_n) goes to h(x_0)."},{"Start":"01:57.790 ","End":"02:02.145","Text":"But h(x_n) is zero from here."},{"Start":"02:02.145 ","End":"02:04.965","Text":"It\u0027s only limit could be zero."},{"Start":"02:04.965 ","End":"02:07.215","Text":"h(x_0) is 0."},{"Start":"02:07.215 ","End":"02:10.080","Text":"Since x_0 was arbitrary,"},{"Start":"02:10.080 ","End":"02:15.370","Text":"h is the zero function and we are done."}],"ID":31293},{"Watched":false,"Name":"Exercise 9","Duration":"2m 3s","ChapterTopicVideoID":29655,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, we have a function f which is"},{"Start":"00:04.800 ","End":"00:09.720","Text":"defined to be 0 when x is rational and 1,"},{"Start":"00:09.720 ","End":"00:11.760","Text":"when x is irrational."},{"Start":"00:11.760 ","End":"00:17.670","Text":"What we have to determine is which points f is continuous"},{"Start":"00:17.670 ","End":"00:24.765","Text":"at and it turns out that f is not continuous anywhere."},{"Start":"00:24.765 ","End":"00:29.370","Text":"Let\u0027s take some point x naught,"},{"Start":"00:29.370 ","End":"00:32.885","Text":"we\u0027ll show that f is not continuous at x naught."},{"Start":"00:32.885 ","End":"00:36.290","Text":"Now, for each n,"},{"Start":"00:36.290 ","End":"00:40.655","Text":"we can find a rational number, call it x_n,"},{"Start":"00:40.655 ","End":"00:44.020","Text":"and an irrational number y_n,"},{"Start":"00:44.020 ","End":"00:48.060","Text":"such that x_n is between x naught and x naught plus 1 over n,"},{"Start":"00:48.060 ","End":"00:51.475","Text":"and also y_n is in the same interval."},{"Start":"00:51.475 ","End":"00:57.024","Text":"In any interval, you can find a rational and an irrational."},{"Start":"00:57.024 ","End":"01:00.275","Text":"That\u0027s 1 of the properties of the real numbers."},{"Start":"01:00.275 ","End":"01:04.050","Text":"Now, both x_n and y_n,"},{"Start":"01:04.050 ","End":"01:06.855","Text":"they both tend to x naught."},{"Start":"01:06.855 ","End":"01:09.920","Text":"They\u0027re just sandwiched between this and this,"},{"Start":"01:09.920 ","End":"01:12.095","Text":"and this goes to x naught."},{"Start":"01:12.095 ","End":"01:14.975","Text":"The sandwich theorem, we get both of these."},{"Start":"01:14.975 ","End":"01:19.070","Text":"Now, the rest is by contradiction."},{"Start":"01:19.070 ","End":"01:22.805","Text":"Suppose that f is continuous at x naught."},{"Start":"01:22.805 ","End":"01:29.245","Text":"Now if f were continuous at x naught since x_n goes to x naught,"},{"Start":"01:29.245 ","End":"01:35.475","Text":"the limit of f (x_n) would have to be f of x naught but the limit is"},{"Start":"01:35.475 ","End":"01:41.970","Text":"0 because f(x_n) is 0 for each rational x_n."},{"Start":"01:41.970 ","End":"01:45.920","Text":"Similarly, because y_n is irrational,"},{"Start":"01:45.920 ","End":"01:48.940","Text":"the limit would have to equal 1."},{"Start":"01:48.940 ","End":"01:55.350","Text":"That would imply that 0 equals 1 and that\u0027s a contradiction."},{"Start":"01:55.350 ","End":"01:58.550","Text":"The contradiction came from the assumption that f is"},{"Start":"01:58.550 ","End":"02:03.990","Text":"continuous and therefore it\u0027s not continuous. We\u0027re done."}],"ID":31294},{"Watched":false,"Name":"Exercise 10","Duration":"2m 27s","ChapterTopicVideoID":29657,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have a function f defined on"},{"Start":"00:03.360 ","End":"00:06.458","Text":"the reals and it takes positive values,"},{"Start":"00:06.458 ","End":"00:11.745","Text":"and it has the property that it takes sums to products as written here."},{"Start":"00:11.745 ","End":"00:16.905","Text":"Now, suppose that f is continuous at 0,"},{"Start":"00:16.905 ","End":"00:20.325","Text":"we have to show that f is continuous everywhere."},{"Start":"00:20.325 ","End":"00:24.180","Text":"We start out by trying to figure out what is f(0),"},{"Start":"00:24.180 ","End":"00:25.335","Text":"call it a,"},{"Start":"00:25.335 ","End":"00:30.150","Text":"so a is also f(0) plus 0 and by this property,"},{"Start":"00:30.150 ","End":"00:34.350","Text":"it\u0027s equal to f(0)^2, which is a^2."},{"Start":"00:34.350 ","End":"00:35.970","Text":"We have a equals a^2,"},{"Start":"00:35.970 ","End":"00:40.070","Text":"and if we didn\u0027t have this restriction,"},{"Start":"00:40.070 ","End":"00:41.780","Text":"we\u0027d say a is 0 or 1,"},{"Start":"00:41.780 ","End":"00:43.070","Text":"but since a is positive,"},{"Start":"00:43.070 ","End":"00:45.670","Text":"it has to be 1."},{"Start":"00:45.670 ","End":"00:52.070","Text":"Now let\u0027s get another property for f relating to negatives of a number."},{"Start":"00:52.070 ","End":"00:56.840","Text":"If we have any x, then we already saw that"},{"Start":"00:56.840 ","End":"01:03.080","Text":"f(0) equals 1 and 0 is x minus x so by this property,"},{"Start":"01:03.080 ","End":"01:08.150","Text":"this is f(x) times f(-x) so we\u0027ve just discovered a property"},{"Start":"01:08.150 ","End":"01:13.935","Text":"that f(x) is 1 over f(-x) for any x."},{"Start":"01:13.935 ","End":"01:17.330","Text":"Now let\u0027s take any x-naught,"},{"Start":"01:17.330 ","End":"01:21.200","Text":"and we want to show that f is continuous at x naught."},{"Start":"01:21.200 ","End":"01:23.555","Text":"We\u0027ll take a sequence x_n,"},{"Start":"01:23.555 ","End":"01:25.700","Text":"which tends to x naught."},{"Start":"01:25.700 ","End":"01:29.555","Text":"Now because f is continuous at 0,"},{"Start":"01:29.555 ","End":"01:34.115","Text":"and because x_n minus x naught tends to 0,"},{"Start":"01:34.115 ","End":"01:39.020","Text":"we must have that f(x_n minus x naught) goes to f(0)."},{"Start":"01:39.020 ","End":"01:44.270","Text":"f(0) is 1 and f of this,"},{"Start":"01:44.270 ","End":"01:50.660","Text":"we\u0027ll look at it as a sum x_n minus x naught goes to the product so we get that"},{"Start":"01:50.660 ","End":"01:57.640","Text":"f(x_n) goes to 1 over f( minus x naught)."},{"Start":"01:57.640 ","End":"02:03.095","Text":"We take this limit and just divide both sides by the constant f(minus x naught)."},{"Start":"02:03.095 ","End":"02:05.344","Text":"On the other hand,"},{"Start":"02:05.344 ","End":"02:14.509","Text":"we have over here that f(x naught) is 1 over f(minus x naught),"},{"Start":"02:14.509 ","End":"02:18.155","Text":"just replacing x by x naught so we have equality here."},{"Start":"02:18.155 ","End":"02:22.580","Text":"This shows that f(x_n) goes to f(x naught) and"},{"Start":"02:22.580 ","End":"02:27.840","Text":"that concludes the proof that f is continuous at x naught. We\u0027re done."}],"ID":31295},{"Watched":false,"Name":"Exercise 11","Duration":"3m 6s","ChapterTopicVideoID":29658,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have a function f on the reals,"},{"Start":"00:03.660 ","End":"00:05.235","Text":"which is continuous,"},{"Start":"00:05.235 ","End":"00:11.190","Text":"and it has this property that f(x) is the same as f(x^2) for all x."},{"Start":"00:11.190 ","End":"00:14.280","Text":"Our task is to show that f is a constant,"},{"Start":"00:14.280 ","End":"00:18.255","Text":"and in fact that constant will be 1."},{"Start":"00:18.255 ","End":"00:23.130","Text":"Now, let\u0027s start with the positive numbers first."},{"Start":"00:23.130 ","End":"00:25.920","Text":"Suppose x is bigger than 0,"},{"Start":"00:25.920 ","End":"00:30.015","Text":"then x is the square root of x^2."},{"Start":"00:30.015 ","End":"00:37.805","Text":"Applying this formula, we\u0027ve got that f(x) equals f of square root of x."},{"Start":"00:37.805 ","End":"00:41.840","Text":"Now, I can apply the formula again to square root of x,"},{"Start":"00:41.840 ","End":"00:43.805","Text":"and if I do this n times,"},{"Start":"00:43.805 ","End":"00:49.925","Text":"we\u0027ll end up with f(x) equals the 2^n root of x."},{"Start":"00:49.925 ","End":"00:52.717","Text":"If you just take the square root of the square root of the square root n times,"},{"Start":"00:52.717 ","End":"00:56.640","Text":"you\u0027ll get root 2^n."},{"Start":"00:57.340 ","End":"01:03.745","Text":"Now, this root 2^n of x tends to 1,"},{"Start":"01:03.745 ","End":"01:11.425","Text":"you can think of it as x^1 over 2^n and that tends to x^0 which is 1."},{"Start":"01:11.425 ","End":"01:14.045","Text":"F(x), which is equal to this,"},{"Start":"01:14.045 ","End":"01:16.595","Text":"and this tends to f(1),"},{"Start":"01:16.595 ","End":"01:18.410","Text":"that\u0027s from the continuity,"},{"Start":"01:18.410 ","End":"01:25.400","Text":"so f(x) is equal to f(1), I mean,"},{"Start":"01:25.400 ","End":"01:34.650","Text":"this is a constant sequence which is always equal to 1 and so it tends to 1."},{"Start":"01:35.300 ","End":"01:43.880","Text":"We\u0027ve proved that f(x) is equal to f(1) for the positive x."},{"Start":"01:43.880 ","End":"01:47.350","Text":"Now we\u0027ll do it for the negative x."},{"Start":"01:47.350 ","End":"01:50.580","Text":"Now again, x is still positive,"},{"Start":"01:50.580 ","End":"01:58.355","Text":"f(minus x) is going to equal f (minus x^2) and that\u0027s the same as f(x^2),"},{"Start":"01:58.355 ","End":"01:59.870","Text":"which is f(x),"},{"Start":"01:59.870 ","End":"02:03.535","Text":"so f (minus x) equals f(x)."},{"Start":"02:03.535 ","End":"02:05.975","Text":"Also for negative x,"},{"Start":"02:05.975 ","End":"02:09.125","Text":"we have f(x) equals f(1)."},{"Start":"02:09.125 ","End":"02:16.325","Text":"Now all that\u0027s left is the case f(0)."},{"Start":"02:16.325 ","End":"02:24.230","Text":"Now, since f(x) is 1 for all x not 0 and f is continuous at 0,"},{"Start":"02:24.230 ","End":"02:30.260","Text":"we can take a sequence x that tends to 0 but is not equal to 0."},{"Start":"02:30.260 ","End":"02:39.115","Text":"As such, the limit of f(x) will be the limit of f(1)."},{"Start":"02:39.115 ","End":"02:41.855","Text":"That\u0027s a constant sequence, so that\u0027s f(1)."},{"Start":"02:41.855 ","End":"02:45.650","Text":"We have that f(0) equals f(1)."},{"Start":"02:45.650 ","End":"02:51.740","Text":"For all x, we get that f(x) is f(1) for 0,"},{"Start":"02:51.740 ","End":"02:54.973","Text":"for the negatives,"},{"Start":"02:54.973 ","End":"02:57.020","Text":"and for the positives,"},{"Start":"02:57.020 ","End":"02:59.795","Text":"that covers all the real numbers."},{"Start":"02:59.795 ","End":"03:02.015","Text":"We don\u0027t know what this constant is,"},{"Start":"03:02.015 ","End":"03:06.690","Text":"but it\u0027s some constant. Okay, we\u0027re done."}],"ID":31296},{"Watched":false,"Name":"Exercise 12","Duration":"3m 35s","ChapterTopicVideoID":29660,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this exercise, we have a function f from the interval 0,"},{"Start":"00:05.460 ","End":"00:06.600","Text":"1 to the interval a,"},{"Start":"00:06.600 ","End":"00:08.985","Text":"b, closed intervals."},{"Start":"00:08.985 ","End":"00:11.535","Text":"It\u0027s 1-1 and onto,"},{"Start":"00:11.535 ","End":"00:15.105","Text":"we have to show that if f is continuous,"},{"Start":"00:15.105 ","End":"00:22.455","Text":"then so is the inverse of f. We\u0027re going to use the result of a previous exercise"},{"Start":"00:22.455 ","End":"00:30.323","Text":"which states as follows that if we have a sequence x_n and a point x naught,"},{"Start":"00:30.323 ","End":"00:34.400","Text":"and suppose that every subsequence of x_n"},{"Start":"00:34.400 ","End":"00:39.695","Text":"has itself a subsequence converging to x naught."},{"Start":"00:39.695 ","End":"00:44.255","Text":"Then the original sequence x_n converges to x naught."},{"Start":"00:44.255 ","End":"00:47.348","Text":"This subsequence of a subsequence,"},{"Start":"00:47.348 ","End":"00:49.535","Text":"you could call it a sub subsequence."},{"Start":"00:49.535 ","End":"00:52.865","Text":"Anyway, this is the main result we\u0027ll use."},{"Start":"00:52.865 ","End":"00:55.055","Text":"Now let\u0027s see, what do we have to prove?"},{"Start":"00:55.055 ","End":"01:01.160","Text":"We have to prove using sequences that if y_n tends to y naught,"},{"Start":"01:01.160 ","End":"01:07.420","Text":"and I should say that these are in the range in the image."},{"Start":"01:07.420 ","End":"01:11.186","Text":"Then f minus 1,"},{"Start":"01:11.186 ","End":"01:17.405","Text":"inverse of f (y_n) tends to f minus 1 of y naught."},{"Start":"01:17.405 ","End":"01:21.440","Text":"That\u0027s basically what continuity is in terms of sequences."},{"Start":"01:21.440 ","End":"01:25.985","Text":"Now if we label this as x_n and this is x naught,"},{"Start":"01:25.985 ","End":"01:30.740","Text":"we could rephrase this as follows."},{"Start":"01:30.740 ","End":"01:36.825","Text":"What we need to show is that if f(x_n) tends to f of x naught,"},{"Start":"01:36.825 ","End":"01:39.435","Text":"then x_n tends to x naught."},{"Start":"01:39.435 ","End":"01:45.290","Text":"This time the sequence x_n and the point x naught are in 0, 1."},{"Start":"01:45.290 ","End":"01:50.030","Text":"They\u0027re all in where they\u0027re supposed to be in the right domain or range."},{"Start":"01:50.030 ","End":"01:52.510","Text":"This is the thing to show,"},{"Start":"01:52.510 ","End":"01:56.580","Text":"and we show it using this result."},{"Start":"01:56.580 ","End":"02:01.710","Text":"Let\u0027s take a subsequence x_nk( x_n)."},{"Start":"02:01.710 ","End":"02:06.525","Text":"Now it\u0027s bounded in 0, 1."},{"Start":"02:06.525 ","End":"02:11.029","Text":"By the Bolzano-Weierstrass theorem,"},{"Start":"02:11.029 ","End":"02:17.274","Text":"it has a subsequence which converges to some Alpha."},{"Start":"02:17.274 ","End":"02:23.270","Text":"Actually Alpha\u0027s going to be in the interval 0, 1."},{"Start":"02:23.270 ","End":"02:31.940","Text":"Now, this sub subsequence is in particular a sequence and it converges to Alpha."},{"Start":"02:31.940 ","End":"02:34.915","Text":"By the continuity of f,"},{"Start":"02:34.915 ","End":"02:40.070","Text":"f of this sequence tends to f of Alpha."},{"Start":"02:40.070 ","End":"02:44.090","Text":"On the other hand, there\u0027s a theorem or"},{"Start":"02:44.090 ","End":"02:49.070","Text":"proposition that if a sequence converges to something,"},{"Start":"02:49.070 ","End":"02:53.045","Text":"then any subsequence of it converges to the same thing."},{"Start":"02:53.045 ","End":"03:00.635","Text":"The limit of f of this sub subsequence is the same as the limit of the original sequence."},{"Start":"03:00.635 ","End":"03:03.200","Text":"Note that we have this limit."},{"Start":"03:03.200 ","End":"03:06.810","Text":"This is f of x naught."},{"Start":"03:07.550 ","End":"03:11.355","Text":"F of Alpha has got to be equal to f of x naught."},{"Start":"03:11.355 ","End":"03:12.575","Text":"Since f is 1-1,"},{"Start":"03:12.575 ","End":"03:17.190","Text":"Alpha is equal to x naught."},{"Start":"03:17.360 ","End":"03:25.405","Text":"Every subsequence of x_n has a subsequence converging to x naught."},{"Start":"03:25.405 ","End":"03:27.635","Text":"By the result above,"},{"Start":"03:27.635 ","End":"03:31.580","Text":"it means that x_n tends to x naught."},{"Start":"03:31.580 ","End":"03:33.913","Text":"That\u0027s what we had to show,"},{"Start":"03:33.913 ","End":"03:36.390","Text":"and so we\u0027re done."}],"ID":31297},{"Watched":false,"Name":"Exercise 13","Duration":"4m 55s","ChapterTopicVideoID":29661,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.260 ","End":"00:05.655","Text":"In this exercise, we\u0027re given a function f,"},{"Start":"00:05.655 ","End":"00:09.015","Text":"which is continuous and additive,"},{"Start":"00:09.015 ","End":"00:11.550","Text":"meaning what\u0027s written here."},{"Start":"00:11.550 ","End":"00:20.640","Text":"We have to show that f(x)=f(1) times x for all x."},{"Start":"00:21.500 ","End":"00:25.580","Text":"The way we\u0027ll do this is we\u0027ll do it gradually."},{"Start":"00:25.580 ","End":"00:31.400","Text":"First of all, we\u0027ll show it\u0027s true for 0 and the natural numbers,"},{"Start":"00:31.400 ","End":"00:34.520","Text":"and then for integers,"},{"Start":"00:34.520 ","End":"00:37.070","Text":"and then for rational numbers."},{"Start":"00:37.070 ","End":"00:38.990","Text":"Then the final step,"},{"Start":"00:38.990 ","End":"00:43.520","Text":"and here we\u0027ll use the continuity is to show it\u0027s true for the reals,"},{"Start":"00:43.520 ","End":"00:45.860","Text":"given that it\u0027s true for the rationals."},{"Start":"00:45.860 ","End":"00:48.380","Text":"Let\u0027s get started."},{"Start":"00:48.380 ","End":"00:52.130","Text":"F(0) is f(0) plus 0,"},{"Start":"00:52.130 ","End":"00:56.210","Text":"which is f(0) plus f(0) by the additivity."},{"Start":"00:56.210 ","End":"01:00.515","Text":"What we get is that f(0)=0,"},{"Start":"01:00.515 ","End":"01:03.140","Text":"because if a=a plus a,"},{"Start":"01:03.140 ","End":"01:11.249","Text":"then a is 0 and 0 is certainly equal to f(1) times 0,"},{"Start":"01:11.249 ","End":"01:16.685","Text":"whatever f(1) is so that\u0027s the 0 part."},{"Start":"01:16.685 ","End":"01:19.370","Text":"Now, what about the natural numbers?"},{"Start":"01:19.370 ","End":"01:24.285","Text":"Well, f(n) is f(1) plus 1 plus 1, etc,"},{"Start":"01:24.285 ","End":"01:26.930","Text":"which is f(1) plus f(1),"},{"Start":"01:26.930 ","End":"01:30.315","Text":"which is f(1) times n,"},{"Start":"01:30.315 ","End":"01:34.180","Text":"because you just count n times f(1)."},{"Start":"01:36.260 ","End":"01:44.710","Text":"That means that it\u0027s true for natural numbers and next,"},{"Start":"01:44.710 ","End":"01:47.425","Text":"we\u0027ll go to the negative as follows."},{"Start":"01:47.425 ","End":"01:51.700","Text":"f(0) is 0, but 0 is minus x plus x."},{"Start":"01:51.700 ","End":"01:55.645","Text":"By the additivity, it\u0027s f of minus x plus f(x)"},{"Start":"01:55.645 ","End":"02:00.715","Text":"so we can bring the f(x) to the other side."},{"Start":"02:00.715 ","End":"02:06.290","Text":"This plus this is 0, so this is minus this and this is minus this."},{"Start":"02:06.290 ","End":"02:09.730","Text":"If I let x be a natural number,"},{"Start":"02:09.730 ","End":"02:12.505","Text":"then we get f of minus n,"},{"Start":"02:12.505 ","End":"02:14.410","Text":"a negative equals minus f(n),"},{"Start":"02:14.410 ","End":"02:18.340","Text":"which is minus f(1) times n. We can bring the minus inside,"},{"Start":"02:18.340 ","End":"02:23.900","Text":"which is f(1) times minus n. This is true for n bigger than 0,"},{"Start":"02:23.900 ","End":"02:26.090","Text":"which means that minus n is less than 0."},{"Start":"02:26.090 ","End":"02:29.773","Text":"It\u0027s true for the negatives and it\u0027s true positives,"},{"Start":"02:29.773 ","End":"02:31.235","Text":"and true for 0."},{"Start":"02:31.235 ","End":"02:36.320","Text":"For all integers f(m)= f(1) times"},{"Start":"02:36.320 ","End":"02:42.155","Text":"m. Now we want to move on to the rational numbers."},{"Start":"02:42.155 ","End":"02:44.915","Text":"Let\u0027s take a rational q."},{"Start":"02:44.915 ","End":"02:50.025","Text":"We can write q as the quotient of 2 integers."},{"Start":"02:50.025 ","End":"02:56.090","Text":"In fact the denominator we can make positive or else move the minus upstairs."},{"Start":"02:56.090 ","End":"03:00.245","Text":"m is an integer and n is a natural number."},{"Start":"03:00.245 ","End":"03:09.420","Text":"That means we can write m as m over n plus m over n, n terms altogether."},{"Start":"03:09.420 ","End":"03:13.655","Text":"Then we can use the additivity of f,"},{"Start":"03:13.655 ","End":"03:18.095","Text":"say this is f of this plus f of this, again n terms."},{"Start":"03:18.095 ","End":"03:20.930","Text":"Then just by counting there\u0027s n here,"},{"Start":"03:20.930 ","End":"03:26.510","Text":"so it\u0027s n times f(m) over n. That means that f(m) over n,"},{"Start":"03:26.510 ","End":"03:28.940","Text":"if we divide both sides by n,"},{"Start":"03:28.940 ","End":"03:31.175","Text":"is f(m) over n,"},{"Start":"03:31.175 ","End":"03:39.710","Text":"which is f(1 ) times m over n. The numerator comes from the results for integers."},{"Start":"03:39.710 ","End":"03:49.320","Text":"Then that\u0027s just f(1) times m over n so we\u0027ve shown that this is true for rationals."},{"Start":"03:49.320 ","End":"03:52.500","Text":"We can write this as follows,"},{"Start":"03:52.500 ","End":"03:59.030","Text":"that f(q)= f(1)q for all rational q is the m over n here."},{"Start":"03:59.030 ","End":"04:02.930","Text":"Now we want to move from the rationals to the reals."},{"Start":"04:02.930 ","End":"04:05.000","Text":"Let r be a real number."},{"Start":"04:05.000 ","End":"04:09.685","Text":"We can take a sequence of rationals that converge to it."},{"Start":"04:09.685 ","End":"04:13.985","Text":"Then we get that f(r),"},{"Start":"04:13.985 ","End":"04:18.190","Text":"which is f of the limit of q_n,"},{"Start":"04:18.190 ","End":"04:20.565","Text":"is equal to,"},{"Start":"04:20.565 ","End":"04:22.625","Text":"that\u0027s the continuity part."},{"Start":"04:22.625 ","End":"04:24.905","Text":"We can take the limit outside."},{"Start":"04:24.905 ","End":"04:28.097","Text":"Is the limit of f(q_n),"},{"Start":"04:28.097 ","End":"04:33.530","Text":"but f(q_n) we know is f(1) times q_n."},{"Start":"04:33.530 ","End":"04:38.133","Text":"We can bring the constant f(1) to the front."},{"Start":"04:38.133 ","End":"04:41.480","Text":"We\u0027ve got f(1) times the limit of q_n,"},{"Start":"04:41.480 ","End":"04:50.645","Text":"but the limit of q_n is r. We get that this is f(1) times r. That\u0027s what we had to show."},{"Start":"04:50.645 ","End":"04:52.574","Text":"We\u0027ve shown it for all real r,"},{"Start":"04:52.574 ","End":"04:55.680","Text":"and that means that we are done."}],"ID":31298},{"Watched":false,"Name":"Exercise 14","Duration":"8m 35s","ChapterTopicVideoID":29662,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we have an interesting example of a function which is"},{"Start":"00:05.370 ","End":"00:11.085","Text":"continuous at every irrational and discontinuous that every rational."},{"Start":"00:11.085 ","End":"00:14.025","Text":"Here\u0027s the definition of the function,"},{"Start":"00:14.025 ","End":"00:18.945","Text":"and then we have 3 parts to solve this in."},{"Start":"00:18.945 ","End":"00:22.950","Text":"The function is 0 when x is irrational."},{"Start":"00:22.950 ","End":"00:25.770","Text":"Now, if x is rational,"},{"Start":"00:25.770 ","End":"00:31.560","Text":"then we can write it as the quotient of 2 natural numbers,"},{"Start":"00:31.560 ","End":"00:34.350","Text":"and we assume that it is reduced."},{"Start":"00:34.350 ","End":"00:37.890","Text":"In other words, p and q have no common factor."},{"Start":"00:37.890 ","End":"00:39.465","Text":"In part a,"},{"Start":"00:39.465 ","End":"00:41.700","Text":"we have a sequence x_n,"},{"Start":"00:41.700 ","End":"00:44.250","Text":"which converges to some x_0,"},{"Start":"00:44.250 ","End":"00:51.915","Text":"but x_n is not equal to x_0 for any of the x_n\u0027s."},{"Start":"00:51.915 ","End":"00:57.935","Text":"I\u0027m also suppose that we write x_n as p_n/q_n, just as here."},{"Start":"00:57.935 ","End":"01:01.385","Text":"In other words, it\u0027s the reduced form rational number."},{"Start":"01:01.385 ","End":"01:06.680","Text":"What we have to show is that this sequence of denominators,"},{"Start":"01:06.680 ","End":"01:09.371","Text":"the q_n, go to infinity."},{"Start":"01:09.371 ","End":"01:13.070","Text":"We\u0027ll do this by contradiction."},{"Start":"01:13.070 ","End":"01:18.395","Text":"Let\u0027s first of all say what it means for q_n to go to infinity."},{"Start":"01:18.395 ","End":"01:21.650","Text":"We can incondense logical format,"},{"Start":"01:21.650 ","End":"01:23.300","Text":"write it this way."},{"Start":"01:23.300 ","End":"01:26.755","Text":"For any positive number m,"},{"Start":"01:26.755 ","End":"01:31.625","Text":"there\u0027s some natural number n,"},{"Start":"01:31.625 ","End":"01:36.350","Text":"such that whenever little n is bigger or equal to n,"},{"Start":"01:36.350 ","End":"01:41.240","Text":"q_n is bigger than m. Now, by contradiction,"},{"Start":"01:41.240 ","End":"01:44.990","Text":"we\u0027re going to suppose that q_n does not go to infinity,"},{"Start":"01:44.990 ","End":"01:47.225","Text":"and at the end we\u0027ll get a contradiction."},{"Start":"01:47.225 ","End":"01:50.780","Text":"What does it mean? Just the logical inverse of this."},{"Start":"01:50.780 ","End":"01:54.650","Text":"We replace for all with there exists and vice versa,"},{"Start":"01:54.650 ","End":"01:56.000","Text":"and at the end,"},{"Start":"01:56.000 ","End":"02:00.070","Text":"we invert the statement here,"},{"Start":"02:00.070 ","End":"02:07.240","Text":"so we have that there exists an m such that for all n,"},{"Start":"02:07.240 ","End":"02:10.548","Text":"there exists a little n,"},{"Start":"02:10.548 ","End":"02:12.160","Text":"depends on big N,"},{"Start":"02:12.160 ","End":"02:14.560","Text":"which is bigger or equal to N,"},{"Start":"02:14.560 ","End":"02:18.145","Text":"such that q_n is less than or equal to m,"},{"Start":"02:18.145 ","End":"02:21.640","Text":"the opposite of bigger than m. Let\u0027s just rewrite this."},{"Start":"02:21.640 ","End":"02:22.726","Text":"Instead of big N,"},{"Start":"02:22.726 ","End":"02:24.000","Text":"we\u0027ll use little k,"},{"Start":"02:24.000 ","End":"02:26.140","Text":"and instead of function notation,"},{"Start":"02:26.140 ","End":"02:27.970","Text":"N of n,"},{"Start":"02:27.970 ","End":"02:31.125","Text":"we\u0027ll have N_k,"},{"Start":"02:31.125 ","End":"02:38.040","Text":"so we have a sequence q of n_k,"},{"Start":"02:38.040 ","End":"02:44.240","Text":"which is bounded above by m and below by 1,"},{"Start":"02:44.240 ","End":"02:46.645","Text":"because the q_n are natural numbers,"},{"Start":"02:46.645 ","End":"02:52.025","Text":"as something I should have remarked earlier that when we have x as p/q,"},{"Start":"02:52.025 ","End":"02:57.470","Text":"p has to be less than q because the fraction is between 0 and 1,"},{"Start":"02:57.470 ","End":"03:00.520","Text":"it\u0027s \u003c 1, so p \u003c q."},{"Start":"03:00.520 ","End":"03:06.470","Text":"If we take the set of all the elements of the sequence"},{"Start":"03:06.470 ","End":"03:13.235","Text":"whose subscript and k is in the same n_k that we found from here,"},{"Start":"03:13.235 ","End":"03:15.545","Text":"n depends on N, so it depends on k,"},{"Start":"03:15.545 ","End":"03:21.160","Text":"and each such x_n_k is of the form p/q,"},{"Start":"03:21.160 ","End":"03:26.445","Text":"where 1 of the q_n\u0027s in this range,"},{"Start":"03:26.445 ","End":"03:32.900","Text":"and p \u003c q can only be a finite number of these."},{"Start":"03:32.900 ","End":"03:42.590","Text":"There\u0027s only a finite number of natural numbers between 1 and m. For each of these,"},{"Start":"03:42.590 ","End":"03:48.940","Text":"there is only a finite number of numerators that could be between 1 and q_n."},{"Start":"03:48.940 ","End":"03:54.035","Text":"So altogether, we get a finite set of fractions."},{"Start":"03:54.035 ","End":"04:03.635","Text":"Now, the thing is that if we have a convergent sequence and all the elements, terms,"},{"Start":"04:03.635 ","End":"04:11.825","Text":"of the sequence come from a finite set of numbers,"},{"Start":"04:11.825 ","End":"04:15.070","Text":"then it has to be eventually constant."},{"Start":"04:15.070 ","End":"04:17.470","Text":"For example, you could take Epsilon less than"},{"Start":"04:17.470 ","End":"04:22.060","Text":"the smallest difference between a pair in this finite set,"},{"Start":"04:22.060 ","End":"04:24.160","Text":"and at some point onwards,"},{"Start":"04:24.160 ","End":"04:27.790","Text":"the distance between any 2 is less than that,"},{"Start":"04:27.790 ","End":"04:29.560","Text":"so it has to be constant."},{"Start":"04:29.560 ","End":"04:35.350","Text":"x_0 is 1 of the x and k for some k,"},{"Start":"04:35.350 ","End":"04:37.300","Text":"for infinitely many k,"},{"Start":"04:37.300 ","End":"04:39.167","Text":"nut at least for 1 anyway."},{"Start":"04:39.167 ","End":"04:45.475","Text":"That\u0027s a contradiction because we said that here,"},{"Start":"04:45.475 ","End":"04:49.700","Text":"that none of the elements of the sequence is x_0."},{"Start":"04:49.700 ","End":"04:51.945","Text":"That does part a,"},{"Start":"04:51.945 ","End":"04:55.620","Text":"and now on to part b."},{"Start":"04:55.620 ","End":"05:03.600","Text":"Suppose x_0 is irrational and x_n tends to x_0,"},{"Start":"05:03.600 ","End":"05:05.330","Text":"and we can still assume, as above,"},{"Start":"05:05.330 ","End":"05:07.535","Text":"that x_n is not equal to x_0."},{"Start":"05:07.535 ","End":"05:10.180","Text":"Now I\u0027m going to divide into 2 cases."},{"Start":"05:10.180 ","End":"05:11.770","Text":"In case 1,"},{"Start":"05:11.770 ","End":"05:15.710","Text":"the sequence only has a finite number of rationals."},{"Start":"05:15.710 ","End":"05:19.103","Text":"In case 2, we will have an infinite number."},{"Start":"05:19.103 ","End":"05:24.075","Text":"In this case, f of x_n is 0,"},{"Start":"05:24.075 ","End":"05:30.860","Text":"for most n. At some n will be passed the last of these rationals,"},{"Start":"05:30.860 ","End":"05:33.155","Text":"and then we only have irrationals,"},{"Start":"05:33.155 ","End":"05:36.500","Text":"and f an irrational is 0."},{"Start":"05:36.500 ","End":"05:38.800","Text":"The limit is 0."},{"Start":"05:38.800 ","End":"05:40.687","Text":"That\u0027s case 1."},{"Start":"05:40.687 ","End":"05:46.365","Text":"Case 2, x_n has infinitely many rationals,"},{"Start":"05:46.365 ","End":"05:51.050","Text":"and these form a subsequence x_n_k."},{"Start":"05:51.050 ","End":"05:54.610","Text":"It\u0027s just a set of rationals in x_n."},{"Start":"05:54.610 ","End":"06:01.710","Text":"Now, we can write each of these x_n_k as p_k/q_k,"},{"Start":"06:01.710 ","End":"06:03.980","Text":"to reduced fraction,"},{"Start":"06:03.980 ","End":"06:05.615","Text":"the p is less than q."},{"Start":"06:05.615 ","End":"06:11.470","Text":"By part a, q_k goes to infinity,"},{"Start":"06:11.470 ","End":"06:15.673","Text":"and f of x_n_k is 1/q_k,"},{"Start":"06:15.673 ","End":"06:18.995","Text":"you just go back and look at the definition of f,"},{"Start":"06:18.995 ","End":"06:21.365","Text":"and that tends to 0."},{"Start":"06:21.365 ","End":"06:23.770","Text":"Now if you think about it and if you think about"},{"Start":"06:23.770 ","End":"06:28.807","Text":"the difference between f of x_n and f of x_n_k,"},{"Start":"06:28.807 ","End":"06:33.465","Text":"this sequence is like this 1 with all the zeros removed,"},{"Start":"06:33.465 ","End":"06:39.220","Text":"because f of x n is non-zero if and only if x_n is rational."},{"Start":"06:39.220 ","End":"06:45.350","Text":"When we removed the irrationals from x_n,"},{"Start":"06:45.350 ","End":"06:51.640","Text":"it\u0027s like removing zeros from f of x_n."},{"Start":"06:51.640 ","End":"06:53.630","Text":"If this goes to 0,"},{"Start":"06:53.630 ","End":"06:58.235","Text":"if I throw back some 0\u0027s into the sequence,"},{"Start":"06:58.235 ","End":"07:00.755","Text":"it will still tend to 0."},{"Start":"07:00.755 ","End":"07:02.630","Text":"Not going to give a formal proof of that."},{"Start":"07:02.630 ","End":"07:04.675","Text":"It\u0027s intuitively obvious."},{"Start":"07:04.675 ","End":"07:09.945","Text":"What we get is that in both case 1 and case 2,"},{"Start":"07:09.945 ","End":"07:13.200","Text":"you get that f of x_n goes to 0,"},{"Start":"07:13.200 ","End":"07:15.975","Text":"which is f of x_ 0,"},{"Start":"07:15.975 ","End":"07:22.035","Text":"because x_0 is irrational."},{"Start":"07:22.035 ","End":"07:24.860","Text":"f is continuous at x_0."},{"Start":"07:24.860 ","End":"07:28.530","Text":"We showed that every sequence x_ n goes to x_0,"},{"Start":"07:28.530 ","End":"07:30.885","Text":"then f of x_n goes to f of x_0,"},{"Start":"07:30.885 ","End":"07:35.765","Text":"so that\u0027s the irrational case."},{"Start":"07:35.765 ","End":"07:44.330","Text":"Then we have part c where we have to show that it\u0027s not continuous at any rational,"},{"Start":"07:44.330 ","End":"07:46.700","Text":"so suppose x_0 is rational,"},{"Start":"07:46.700 ","End":"07:50.240","Text":"then f of x_0 is not equal to 0."},{"Start":"07:50.240 ","End":"07:52.715","Text":"Choose a sequence x_n,"},{"Start":"07:52.715 ","End":"07:56.315","Text":"where x_n is irrational,"},{"Start":"07:56.315 ","End":"07:58.756","Text":"but at x_n goes to x_0."},{"Start":"07:58.756 ","End":"08:01.070","Text":"Meaning for any real number,"},{"Start":"08:01.070 ","End":"08:03.860","Text":"you could find a sequence of irrationals that tend to it."},{"Start":"08:03.860 ","End":"08:06.560","Text":"You can also find a sequence of rationals that tend to it."},{"Start":"08:06.560 ","End":"08:14.600","Text":"What we get, if we have a sequence of irrationals tending to a rational,"},{"Start":"08:14.600 ","End":"08:20.494","Text":"you get that the limit of f of x_n is the limit of 0\u0027s, which is 0."},{"Start":"08:20.494 ","End":"08:27.249","Text":"But that\u0027s not equal to f of x_0 because that\u0027s not 0 since x_0 is rational."},{"Start":"08:27.249 ","End":"08:30.125","Text":"Because this does not equal to this,"},{"Start":"08:30.125 ","End":"08:33.080","Text":"f is discontinuous at x_0,"},{"Start":"08:33.080 ","End":"08:36.450","Text":"and that proves part c, and we\u0027re done."}],"ID":31299},{"Watched":false,"Name":"Exercise 15","Duration":"3m 25s","ChapterTopicVideoID":29665,"CourseChapterTopicPlaylistID":294580,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"In this exercise, function f is from the reals to the positive reals,"},{"Start":"00:05.880 ","End":"00:09.180","Text":"and it has this property that for all x and y,"},{"Start":"00:09.180 ","End":"00:14.820","Text":"f(x+y) is [fx, fy]^2."},{"Start":"00:14.820 ","End":"00:19.785","Text":"Now, suppose that f is continuous at x=0,"},{"Start":"00:19.785 ","End":"00:23.145","Text":"we have to show it\u0027s continuous everywhere."},{"Start":"00:23.145 ","End":"00:27.570","Text":"The first step is to show that f(0)=1, so here,"},{"Start":"00:27.570 ","End":"00:30.615","Text":"f(0) is f(0+0),"},{"Start":"00:30.615 ","End":"00:32.265","Text":"and then from the formula,"},{"Start":"00:32.265 ","End":"00:33.840","Text":"this is equal to [f(0),"},{"Start":"00:33.840 ","End":"00:36.870","Text":"f(0)]^2, which is f(0)^4."},{"Start":"00:36.870 ","End":"00:41.880","Text":"When you have an equation like x= x^4,"},{"Start":"00:41.880 ","End":"00:44.905","Text":"then x can be 0 or 1,"},{"Start":"00:44.905 ","End":"00:48.709","Text":"but f is positive valued,"},{"Start":"00:48.709 ","End":"00:53.960","Text":"so we have to rule out the 0 that gives us f(0) is 1."},{"Start":"00:53.960 ","End":"00:56.525","Text":"Now, we\u0027ll get another formula."},{"Start":"00:56.525 ","End":"00:58.580","Text":"1 is f(0),"},{"Start":"00:58.580 ","End":"01:01.250","Text":"and that\u0027s f(x-x),"},{"Start":"01:01.250 ","End":"01:03.455","Text":"which from here is f(x)^2,"},{"Start":"01:03.455 ","End":"01:05.450","Text":"f of minus x^2,"},{"Start":"01:05.450 ","End":"01:12.875","Text":"so we have the formula that f of minus x^2 is 1 over f(x)^2."},{"Start":"01:12.875 ","End":"01:15.530","Text":"Now we\u0027re ready to tackle the continuity."},{"Start":"01:15.530 ","End":"01:18.830","Text":"Suppose we have some x_naught and we have a sequence x_n,"},{"Start":"01:18.830 ","End":"01:19.940","Text":"which tends to it,"},{"Start":"01:19.940 ","End":"01:24.305","Text":"we have to show that f(x_n) tends to f(x_naught)."},{"Start":"01:24.305 ","End":"01:27.905","Text":"Now, since x_n goes to x_naught,"},{"Start":"01:27.905 ","End":"01:32.855","Text":"we can subtract and say that x_n minus x_naught goes to 0,"},{"Start":"01:32.855 ","End":"01:39.170","Text":"and that means that f(x_n) minus x_naught goes to f(0) because"},{"Start":"01:39.170 ","End":"01:45.380","Text":"we know that f is continuous at 0. f(0) is equal to 1,"},{"Start":"01:45.380 ","End":"01:53.270","Text":"and f(x_n minus x_0) is equal to this as 2 steps in 1."},{"Start":"01:53.270 ","End":"02:00.605","Text":"We should say that it\u0027s f(x_n)^2 times f of minus x_naught^2 if we\u0027re using this formula."},{"Start":"02:00.605 ","End":"02:05.120","Text":"But f of minus x_naught^2 is 1 over f(x_naught)^2,"},{"Start":"02:05.120 ","End":"02:07.250","Text":"so that\u0027s how we get this."},{"Start":"02:07.250 ","End":"02:13.325","Text":"Next, we can multiply both sides by this constant doesn\u0027t depend on n,"},{"Start":"02:13.325 ","End":"02:19.490","Text":"so f(x_n)^2 goes to f(x_naught)^2."},{"Start":"02:19.490 ","End":"02:22.250","Text":"Now, f is positive value,"},{"Start":"02:22.250 ","End":"02:23.690","Text":"there\u0027s no negatives here,"},{"Start":"02:23.690 ","End":"02:24.940","Text":"so there\u0027s no ambiguity,"},{"Start":"02:24.940 ","End":"02:30.950","Text":"we can take the square root of both and get that f(x_n) goes to f(x_naugh)."},{"Start":"02:30.950 ","End":"02:34.625","Text":"We\u0027re done, but I\u0027d like to show you an alternate solution,"},{"Start":"02:34.625 ","End":"02:36.905","Text":"so here it goes,"},{"Start":"02:36.905 ","End":"02:44.000","Text":"f(0)=f(0+0), which is [f(0),f(0)]^2, which is f(0)^0."},{"Start":"02:44.000 ","End":"02:46.595","Text":"That\u0027s like we had previously,"},{"Start":"02:46.595 ","End":"02:50.945","Text":"so f(0) equals 0 or 1, so f(0)=1."},{"Start":"02:50.945 ","End":"02:52.970","Text":"This is the same as before."},{"Start":"02:52.970 ","End":"02:54.785","Text":"Now here\u0027s where we differ."},{"Start":"02:54.785 ","End":"03:01.100","Text":"f(x) is f(x+0) which is f(x)^2 times f(0)^2 from the formula,"},{"Start":"03:01.100 ","End":"03:03.815","Text":"but we\u0027ve just shown that this is equal to 1,"},{"Start":"03:03.815 ","End":"03:06.905","Text":"so this is just f(x)^2."},{"Start":"03:06.905 ","End":"03:10.535","Text":"Now, if f(x) is equal to f(x)^2,"},{"Start":"03:10.535 ","End":"03:13.775","Text":"then f(x) has to be 0 or 1,"},{"Start":"03:13.775 ","End":"03:16.295","Text":"but once again, since we\u0027re positive valued,"},{"Start":"03:16.295 ","End":"03:18.665","Text":"f(x) has to equal 1."},{"Start":"03:18.665 ","End":"03:20.420","Text":"So f is a constant function,"},{"Start":"03:20.420 ","End":"03:21.680","Text":"is constantly equal to 1,"},{"Start":"03:21.680 ","End":"03:26.400","Text":"and therefore, it\u0027s continuous. That\u0027s it."}],"ID":31300}],"Thumbnail":null,"ID":294580},{"Name":"Advanced Theory Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"10m 10s","ChapterTopicVideoID":29734,"CourseChapterTopicPlaylistID":294604,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":31363},{"Watched":false,"Name":"Exercise 2","Duration":"9m 12s","ChapterTopicVideoID":29733,"CourseChapterTopicPlaylistID":294604,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":31364}],"Thumbnail":null,"ID":294604}]

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1.1

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