Introduction to Power Series
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- Intro and Definition
- Region of Convergence
- Radius of Convergence
- Exercise 1 part a
- Exercise 1 part b
- Exercise 1 part c
- Exercise 1 part d
- Exercise 1 part e
- Exercise 1 part f
- Exercise 2 part a
- Exercise 2 part b
- Exercise 2 part c
- Exercise 2 part d
- Exercise 2 part e
- Exercise 2 part f
- Exercise 3 part a
- Exercise 3 part b
- Exercise 3 part c
- Exercise 3 part d
- Exercise 3 part e
- Exercise 3 part f
- Exercise 3 part g
- Exercise 3 part h
- Exercise 3 part i
- Exercise 3 part j
- Exercise 3 part k
- Exercise 3 part l
- Exercise 4 part a
- Exercise 4 part b
- Exercise 4 part c
- Exercise 4 part d
- Exercise 4 part e

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[{"Name":"Introduction to Power Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro and Definition","Duration":"12m 9s","ChapterTopicVideoID":7927,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/7927.jpeg","UploadDate":"2020-01-16T13:01:31.4600000","DurationForVideoObject":"PT12M9S","Description":null,"MetaTitle":"Intro and Definition: Video + Workbook | Proprep","MetaDescription":"Power Series - Introduction to Power Series. 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/power-series/introduction-to-power-series/vid7958","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"A power series is like an infinite polynomial."},{"Start":"00:03.450 ","End":"00:05.265","Text":"Let me show you what I mean."},{"Start":"00:05.265 ","End":"00:08.445","Text":"We all know what a polynomial is."},{"Start":"00:08.445 ","End":"00:12.840","Text":"Let\u0027s say you want a polynomial of degree n. We could"},{"Start":"00:12.840 ","End":"00:18.100","Text":"write it as the sum i goes from 0 to n,"},{"Start":"00:18.110 ","End":"00:21.450","Text":"a_i, x to the power of i."},{"Start":"00:21.450 ","End":"00:22.920","Text":"That\u0027s a bit abstract."},{"Start":"00:22.920 ","End":"00:25.935","Text":"If you put it in regular notation without Sigma,"},{"Start":"00:25.935 ","End":"00:30.045","Text":"It\u0027s a_0 plus a_1 x,"},{"Start":"00:30.045 ","End":"00:33.930","Text":"x^1, plus a_2 x squared plus,"},{"Start":"00:33.930 ","End":"00:35.550","Text":"and so on and so on,"},{"Start":"00:35.550 ","End":"00:38.970","Text":"up to a_n x^n."},{"Start":"00:38.970 ","End":"00:44.610","Text":"We write it in increasing powers of x, more convenient."},{"Start":"00:44.610 ","End":"00:47.060","Text":"When it\u0027s a polynomial of degree 2,"},{"Start":"00:47.060 ","End":"00:53.580","Text":"we might just write a_x squared plus bx plus c,"},{"Start":"00:53.580 ","End":"00:55.684","Text":"but we run out of letters,"},{"Start":"00:55.684 ","End":"01:01.010","Text":"so we write it from decreasing to increasing power."},{"Start":"01:01.010 ","End":"01:06.885","Text":"This might be written as a_0 plus a_1 x plus a_2 x squared,"},{"Start":"01:06.885 ","End":"01:09.370","Text":"then we generalize it to degree n. Now,"},{"Start":"01:09.370 ","End":"01:14.110","Text":"I want to make one more generalization to change this n to infinity."},{"Start":"01:14.110 ","End":"01:17.930","Text":"A power series, very much like this,"},{"Start":"01:17.930 ","End":"01:22.340","Text":"it\u0027s just that we take from 0 to infinity,"},{"Start":"01:22.340 ","End":"01:24.920","Text":"but I\u0027m going to use the letter n instead of i."},{"Start":"01:24.920 ","End":"01:30.380","Text":"So I\u0027ll say n goes from 0 to infinity, a_n x^n."},{"Start":"01:30.380 ","End":"01:32.585","Text":"If I write it out in expanded form,"},{"Start":"01:32.585 ","End":"01:34.360","Text":"it looks like this,"},{"Start":"01:34.360 ","End":"01:39.185","Text":"a_0 plus a_1 x plus a_2 x squared."},{"Start":"01:39.185 ","End":"01:40.755","Text":"Only it doesn\u0027t stop,"},{"Start":"01:40.755 ","End":"01:43.840","Text":"it keeps on going forever."},{"Start":"01:45.260 ","End":"01:47.855","Text":"Let me give an example."},{"Start":"01:47.855 ","End":"01:49.309","Text":"I\u0027ll give a few examples."},{"Start":"01:49.309 ","End":"01:51.560","Text":"In fact, let\u0027s start with Example 1."},{"Start":"01:51.560 ","End":"01:57.965","Text":"I\u0027ll take the sum from 0 to infinity of 2^n, x^n."},{"Start":"01:57.965 ","End":"01:59.585","Text":"If I write this out,"},{"Start":"01:59.585 ","End":"02:04.765","Text":"this is like 1 plus 2x plus 2 squared,"},{"Start":"02:04.765 ","End":"02:06.580","Text":"which is 4x squared."},{"Start":"02:06.580 ","End":"02:07.650","Text":"I\u0027ll write another term,"},{"Start":"02:07.650 ","End":"02:10.640","Text":"2 cubed is 8x cubed plus,"},{"Start":"02:10.640 ","End":"02:12.525","Text":"and so on, and so on."},{"Start":"02:12.525 ","End":"02:18.080","Text":"Sometimes we could just briefly describe this by just saying what a_n is in this formula."},{"Start":"02:18.080 ","End":"02:21.860","Text":"Here we would say a_n is 2^n,"},{"Start":"02:21.860 ","End":"02:26.495","Text":"and that would describe the series based on this."},{"Start":"02:26.495 ","End":"02:28.910","Text":"Let\u0027s give another example."},{"Start":"02:28.910 ","End":"02:32.765","Text":"This one will be Example 2, of course."},{"Start":"02:32.765 ","End":"02:39.685","Text":"This time I\u0027ll take the sum of 1 over n x^n."},{"Start":"02:39.685 ","End":"02:43.985","Text":"Only I can\u0027t take it from 0 to infinity because when n is 0,"},{"Start":"02:43.985 ","End":"02:45.620","Text":"we\u0027d be dividing by 0,"},{"Start":"02:45.620 ","End":"02:49.040","Text":"so we\u0027ll take it just from 1 to infinity. That\u0027s another thing."},{"Start":"02:49.040 ","End":"02:53.790","Text":"The power series doesn\u0027t necessarily have to start at 0."},{"Start":"02:53.930 ","End":"02:57.540","Text":"This would equal, if n is 1,"},{"Start":"02:57.540 ","End":"02:59.330","Text":"we get x^1 over 1,"},{"Start":"02:59.330 ","End":"03:07.825","Text":"which I would write just as x plus x squared over 2 plus x cubed over 3 plus, and so on."},{"Start":"03:07.825 ","End":"03:13.190","Text":"Here, I could describe this by saying a_n equals 1 over n,"},{"Start":"03:13.190 ","End":"03:17.220","Text":"but I\u0027d also have to make a note that we start from 1."},{"Start":"03:17.740 ","End":"03:21.365","Text":"Let\u0027s take yet another example."},{"Start":"03:21.365 ","End":"03:25.880","Text":"This time I\u0027ll take the sum also from 0 to infinity"},{"Start":"03:25.880 ","End":"03:32.870","Text":"of 2^n over n factorial x^n,"},{"Start":"03:32.870 ","End":"03:39.175","Text":"and this would equal 2^0 over 0 factorial x^0."},{"Start":"03:39.175 ","End":"03:40.695","Text":"I don\u0027t write that."},{"Start":"03:40.695 ","End":"03:44.145","Text":"2 to the 1 over 1 factorial x,"},{"Start":"03:44.145 ","End":"03:48.330","Text":"2 squared over 2 factorial x squared."},{"Start":"03:48.330 ","End":"03:52.065","Text":"I\u0027ll add one more, 2 cubed over 3 factorial x cubed,"},{"Start":"03:52.065 ","End":"03:56.460","Text":"and so on. Yet another example."},{"Start":"03:56.460 ","End":"04:00.650","Text":"This time it\u0027s given in a slightly different form."},{"Start":"04:00.650 ","End":"04:05.235","Text":"Sometimes you have to modify it a bit."},{"Start":"04:05.235 ","End":"04:13.790","Text":"I have minus 2x over n to the power of n. Then I might just have"},{"Start":"04:13.790 ","End":"04:22.980","Text":"to break it open a bit to get minus 2 over n to the power of n,"},{"Start":"04:22.980 ","End":"04:25.910","Text":"and then x^n, just using the rules of exponents,"},{"Start":"04:25.910 ","End":"04:28.235","Text":"taking the x^n outside the brackets."},{"Start":"04:28.235 ","End":"04:30.020","Text":"Okay, that\u0027s enough examples."},{"Start":"04:30.020 ","End":"04:35.130","Text":"Now, the next topic I want to talk about is convergence."},{"Start":"04:35.560 ","End":"04:40.640","Text":"Specifically I\u0027m going to talk about something called region of convergence."},{"Start":"04:40.640 ","End":"04:43.115","Text":"I\u0027m going to illustrate with an example."},{"Start":"04:43.115 ","End":"04:46.175","Text":"Let me keep Example 2,"},{"Start":"04:46.175 ","End":"04:49.100","Text":"and I\u0027ll move it up here."},{"Start":"04:49.100 ","End":"04:54.475","Text":"A power series is like a polynomial, but infinite."},{"Start":"04:54.475 ","End":"04:58.910","Text":"But it\u0027s like a polynomial also in the sense that we"},{"Start":"04:58.910 ","End":"05:03.275","Text":"can substitute different values of x and get a result."},{"Start":"05:03.275 ","End":"05:05.600","Text":"Now, what happens here, for example,"},{"Start":"05:05.600 ","End":"05:08.540","Text":"if I substitute x equals 1,"},{"Start":"05:08.540 ","End":"05:10.820","Text":"if I put x equals 1,"},{"Start":"05:10.820 ","End":"05:12.440","Text":"then I get the sum,"},{"Start":"05:12.440 ","End":"05:15.115","Text":"n equals 1 to infinity."},{"Start":"05:15.115 ","End":"05:22.185","Text":"1^n is just 1, so I get the sum of 1 to the sum of 1 over n. If I write it out,"},{"Start":"05:22.185 ","End":"05:32.060","Text":"it\u0027s 1 over 1 plus 1/2 plus 1/3 plus 1/4 plus, and so on."},{"Start":"05:32.060 ","End":"05:37.115","Text":"This is a specific series, not depending on x."},{"Start":"05:37.115 ","End":"05:40.305","Text":"As a number series which we\u0027ve studied,"},{"Start":"05:40.305 ","End":"05:44.540","Text":"there could be 2 possibilities or actually 3."},{"Start":"05:44.540 ","End":"05:51.100","Text":"I\u0027d like to remind you that the number series can have possibilities."},{"Start":"05:51.100 ","End":"05:54.900","Text":"It could diverge or it could converge."},{"Start":"05:54.900 ","End":"06:02.900","Text":"Actually, the converge, we can split into 2 cases, converge conditionally."},{"Start":"06:03.980 ","End":"06:06.729","Text":"Let us write cond, conditionally,"},{"Start":"06:06.729 ","End":"06:10.250","Text":"or it could converge absolutely."},{"Start":"06:16.070 ","End":"06:19.610","Text":"Question is, what happens in this particular case?"},{"Start":"06:19.610 ","End":"06:24.665","Text":"If x is 1, this is the harmonic series and we know it diverges."},{"Start":"06:24.665 ","End":"06:27.990","Text":"I\u0027ll write diverges."},{"Start":"06:28.540 ","End":"06:30.830","Text":"Let\u0027s take another example."},{"Start":"06:30.830 ","End":"06:34.610","Text":"Suppose I put x equals minus 1,"},{"Start":"06:34.610 ","End":"06:37.910","Text":"then I get the sum from 1 to infinity."},{"Start":"06:37.910 ","End":"06:39.350","Text":"If I put x as minus 1,"},{"Start":"06:39.350 ","End":"06:47.085","Text":"I get minus 1^n over n. If we expand this,"},{"Start":"06:47.085 ","End":"06:58.170","Text":"when n is 1, we get minus 1 plus 1/2 minus a 1/3 plus 1/4, and so on."},{"Start":"06:58.170 ","End":"07:03.470","Text":"Turns out that this converges won\u0027t get into too much detail,"},{"Start":"07:03.470 ","End":"07:05.044","Text":"but if you remember series,"},{"Start":"07:05.044 ","End":"07:07.565","Text":"this is an alternating Leibniz series,"},{"Start":"07:07.565 ","End":"07:10.040","Text":"is plus minus, plus minus, and so on."},{"Start":"07:10.040 ","End":"07:11.560","Text":"If I ignore the sign,"},{"Start":"07:11.560 ","End":"07:14.690","Text":"it\u0027s a series that decreases and goes down to 0."},{"Start":"07:14.690 ","End":"07:16.385","Text":"1, 1/2, 1/3, 1/4,"},{"Start":"07:16.385 ","End":"07:20.270","Text":"keeps getting smaller and smaller and tends to 0."},{"Start":"07:20.270 ","End":"07:24.205","Text":"Let\u0027s take a third example, x equals 0."},{"Start":"07:24.205 ","End":"07:29.785","Text":"When x is 0, we get the sum from 1 to infinity."},{"Start":"07:29.785 ","End":"07:31.700","Text":"0^n is just 0,"},{"Start":"07:31.700 ","End":"07:33.530","Text":"so each term is 0,"},{"Start":"07:33.530 ","End":"07:35.480","Text":"and this definitely converges."},{"Start":"07:35.480 ","End":"07:38.140","Text":"I can even tell you what the sum is."},{"Start":"07:38.140 ","End":"07:42.080","Text":"The sum is 0, but the point is that it converges."},{"Start":"07:42.080 ","End":"07:45.590","Text":"Not only that, but in general,"},{"Start":"07:45.590 ","End":"07:49.520","Text":"the sum for any power series a_n x^n,"},{"Start":"07:49.520 ","End":"07:52.550","Text":"n goes from something,"},{"Start":"07:52.550 ","End":"07:54.140","Text":"could be 0, could be something else,"},{"Start":"07:54.140 ","End":"08:00.820","Text":"to infinity, always converges for x equals 0."},{"Start":"08:00.820 ","End":"08:04.725","Text":"One thing we can count on is that."},{"Start":"08:04.725 ","End":"08:07.920","Text":"Now, what I want to discuss here is that for some values of x,"},{"Start":"08:07.920 ","End":"08:09.710","Text":"we can substitute, it will converge,"},{"Start":"08:09.710 ","End":"08:11.000","Text":"for some it will divert."},{"Start":"08:11.000 ","End":"08:14.060","Text":"If it converges, for some values it will converge absolutely,"},{"Start":"08:14.060 ","End":"08:16.145","Text":"and for some values conditionally."},{"Start":"08:16.145 ","End":"08:19.985","Text":"We want to see other than haphazardly trying each value,"},{"Start":"08:19.985 ","End":"08:24.080","Text":"if I can some way tell from the series for which values of"},{"Start":"08:24.080 ","End":"08:28.760","Text":"x it\u0027s going to converge and for which values of x it\u0027s going to diverge, and so on."},{"Start":"08:28.760 ","End":"08:32.890","Text":"Is there a general rule that we can find out?"},{"Start":"08:32.890 ","End":"08:37.460","Text":"I\u0027d just like to get one more example in before I discuss in general."},{"Start":"08:37.460 ","End":"08:42.920","Text":"Let\u0027s just try x equals 2 and then we get the sum from 1 to infinity"},{"Start":"08:42.920 ","End":"08:51.325","Text":"of 2^n over n. I claim that this diverges."},{"Start":"08:51.325 ","End":"08:54.890","Text":"One of the ways of proving divergence,"},{"Start":"08:54.890 ","End":"08:57.575","Text":"the easiest, if it\u0027s possible,"},{"Start":"08:57.575 ","End":"09:01.115","Text":"is to show that the general term does not tend to 0."},{"Start":"09:01.115 ","End":"09:10.235","Text":"In fact, the limit as n goes to infinity of 2^n over n is infinity."},{"Start":"09:10.235 ","End":"09:12.665","Text":"There\u0027s many ways of showing this."},{"Start":"09:12.665 ","End":"09:19.920","Text":"One way would be to use a function,"},{"Start":"09:19.920 ","End":"09:21.555","Text":"2 to the x over x,"},{"Start":"09:21.555 ","End":"09:22.950","Text":"and then use L\u0027Hospital\u0027s rule."},{"Start":"09:22.950 ","End":"09:25.195","Text":"I don\u0027t want to get into this too much anyway."},{"Start":"09:25.195 ","End":"09:27.365","Text":"It goes to infinity."},{"Start":"09:27.365 ","End":"09:30.424","Text":"It does not go to 0."},{"Start":"09:30.424 ","End":"09:33.620","Text":"If the general term doesn\u0027t tend to 0,"},{"Start":"09:33.620 ","End":"09:35.330","Text":"then the series diverges."},{"Start":"09:35.330 ","End":"09:39.630","Text":"Now we\u0027ve got some values."},{"Start":"09:40.630 ","End":"09:44.495","Text":"Let me now just tell you,"},{"Start":"09:44.495 ","End":"09:48.935","Text":"I\u0027ll give you the answer for which values of x this series converges"},{"Start":"09:48.935 ","End":"09:55.035","Text":"without explanation and then we\u0027ll discuss it in more detail."},{"Start":"09:55.035 ","End":"10:03.450","Text":"I\u0027ll tell you that it converges when x is between 1 and minus 1,"},{"Start":"10:03.450 ","End":"10:08.040","Text":"but it includes the minus 1,"},{"Start":"10:08.040 ","End":"10:10.240","Text":"and does not include the 1."},{"Start":"10:10.240 ","End":"10:13.050","Text":"Never mind how we got to this, that will be later."},{"Start":"10:13.050 ","End":"10:14.540","Text":"I\u0027m just giving you the answer."},{"Start":"10:14.540 ","End":"10:17.765","Text":"It always turns out that it\u0027s in an interval."},{"Start":"10:17.765 ","End":"10:20.390","Text":"In this case, from minus 1 to 1,"},{"Start":"10:20.390 ","End":"10:23.300","Text":"and sometimes including the endpoints, sometimes not."},{"Start":"10:23.300 ","End":"10:30.725","Text":"But the set of values for which it converges is called the range of convergence."},{"Start":"10:30.725 ","End":"10:33.125","Text":"That\u0027s one name for it."},{"Start":"10:33.125 ","End":"10:37.070","Text":"More commonly, now that we"},{"Start":"10:37.070 ","End":"10:41.155","Text":"know that I\u0027m telling you that it\u0027s going to always be continuous as an interval,"},{"Start":"10:41.155 ","End":"10:44.290","Text":"it can\u0027t be like for 3, it will converge;"},{"Start":"10:44.290 ","End":"10:45.805","Text":"for 4 it will diverge,"},{"Start":"10:45.805 ","End":"10:48.185","Text":"then from 5 to 6, certainly converge again."},{"Start":"10:48.185 ","End":"10:53.130","Text":"It\u0027s always a contiguous interval."},{"Start":"10:53.130 ","End":"11:00.250","Text":"So it\u0027s sometimes more commonly called the interval of convergence."},{"Start":"11:00.250 ","End":"11:03.720","Text":"I\u0027ve even seen region of convergence,"},{"Start":"11:03.720 ","End":"11:08.645","Text":"so range, interval, region of convergence."},{"Start":"11:08.645 ","End":"11:14.835","Text":"Now, let me give the general rule, a theorem,"},{"Start":"11:14.835 ","End":"11:18.720","Text":"and then we\u0027ll also see in this example"},{"Start":"11:18.720 ","End":"11:23.315","Text":"how I got to this result which I quoted without showing you why."},{"Start":"11:23.315 ","End":"11:30.335","Text":"The set of values for which the series converges has many names,"},{"Start":"11:30.335 ","End":"11:34.190","Text":"it\u0027s called the region of convergence."},{"Start":"11:34.190 ","End":"11:35.900","Text":"It\u0027s just, as I say,"},{"Start":"11:35.900 ","End":"11:44.345","Text":"the set of values of x for which the series converges."},{"Start":"11:44.345 ","End":"11:46.640","Text":"In this case, we can see it\u0027s an interval."},{"Start":"11:46.640 ","End":"11:49.565","Text":"In fact, it always turns out to be some kind of interval,"},{"Start":"11:49.565 ","End":"11:53.390","Text":"so it\u0027s also called the interval of convergence,"},{"Start":"11:53.390 ","End":"11:57.590","Text":"and sometimes the range of convergence."},{"Start":"11:57.590 ","End":"12:03.215","Text":"All these 3 are possible: region of convergence, interval, range."},{"Start":"12:03.215 ","End":"12:09.809","Text":"Now, I\u0027m going to give a theorem about what we can expect."}],"ID":7958},{"Watched":false,"Name":"Region of Convergence","Duration":"6m 51s","ChapterTopicVideoID":7928,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:05.835","Text":"I kept the example and now I\u0027m coming to the theorem."},{"Start":"00:05.835 ","End":"00:11.715","Text":"The theorem relates to convergence of power series,"},{"Start":"00:11.715 ","End":"00:20.235","Text":"and it says that when we have a series of the general form a_n x^n,"},{"Start":"00:20.235 ","End":"00:23.670","Text":"then there are 3, and only 3,"},{"Start":"00:23.670 ","End":"00:28.395","Text":"possibilities, and I\u0027ll list them in a second."},{"Start":"00:28.395 ","End":"00:33.295","Text":"Those possibilities are, A,"},{"Start":"00:33.295 ","End":"00:35.720","Text":"that the series, this one,"},{"Start":"00:35.720 ","End":"00:41.930","Text":"converges only for x equals 0."},{"Start":"00:41.930 ","End":"00:45.980","Text":"We already mentioned before that the series always converges when x is 0,"},{"Start":"00:45.980 ","End":"00:48.080","Text":"because we just get the sum of 0s."},{"Start":"00:48.080 ","End":"00:53.285","Text":"But it\u0027s possible that this is the only value of x for which the series converges."},{"Start":"00:53.285 ","End":"00:58.440","Text":"Possibility B, this possibility is pretty much the other extreme,"},{"Start":"00:58.440 ","End":"01:02.710","Text":"is that it converges for all x,"},{"Start":"01:06.110 ","End":"01:13.640","Text":"and sometimes we say that it converges for x between minus infinity and infinity,"},{"Start":"01:13.640 ","End":"01:16.540","Text":"which is just another way of saying for all x."},{"Start":"01:16.540 ","End":"01:20.280","Text":"Both of these are not what happens in our particular series;"},{"Start":"01:20.280 ","End":"01:23.390","Text":"our particular series actually belongs to Case C,"},{"Start":"01:23.390 ","End":"01:25.460","Text":"the third of the possibilities,"},{"Start":"01:25.460 ","End":"01:28.205","Text":"and this is a little bit more involved."},{"Start":"01:28.205 ","End":"01:33.970","Text":"Possibility C says that the series converges"},{"Start":"01:33.970 ","End":"01:42.510","Text":"for x between minus R and R,"},{"Start":"01:42.510 ","End":"01:46.605","Text":"and R is some positive number,"},{"Start":"01:46.605 ","End":"01:50.595","Text":"I will give a meaning to it and a name for this R later."},{"Start":"01:50.595 ","End":"02:05.670","Text":"It diverges for x"},{"Start":"02:05.670 ","End":"02:10.035","Text":"less than minus R or x bigger than"},{"Start":"02:10.035 ","End":"02:18.085","Text":"R. In the case that x is exactly 1 of these, anything goes."},{"Start":"02:18.085 ","End":"02:19.785","Text":"Let me write this down."},{"Start":"02:19.785 ","End":"02:25.525","Text":"For x equals plus or minus R,"},{"Start":"02:25.525 ","End":"02:28.485","Text":"I\u0027ll say briefly, we don\u0027t know,"},{"Start":"02:28.485 ","End":"02:30.605","Text":"that we have to check an individual,"},{"Start":"02:30.605 ","End":"02:32.344","Text":"we can\u0027t say in general,"},{"Start":"02:32.344 ","End":"02:36.650","Text":"which means that any 1 of 3 possibilities could occur: It could be"},{"Start":"02:36.650 ","End":"02:41.900","Text":"that it converges absolutely."},{"Start":"02:41.900 ","End":"02:45.870","Text":"It could be that it converges conditionally."},{"Start":"02:49.550 ","End":"02:54.740","Text":"The third possibility is that it actually diverges."},{"Start":"02:54.740 ","End":"03:00.150","Text":"Anything goes; all 3 possibilities we mentioned here could be."},{"Start":"03:00.680 ","End":"03:03.785","Text":"I\u0027ll point out that in our case,"},{"Start":"03:03.785 ","End":"03:09.380","Text":"we are in Case C with R equals 1."},{"Start":"03:09.380 ","End":"03:14.395","Text":"If you look at it, it fits, it converges."},{"Start":"03:14.395 ","End":"03:16.640","Text":"We don\u0027t know about the absolutely,"},{"Start":"03:16.640 ","End":"03:21.890","Text":"but we certainly saw that it converges between minus 1 and 1."},{"Start":"03:21.890 ","End":"03:25.715","Text":"It turns out that when x was equal to plus 1,"},{"Start":"03:25.715 ","End":"03:29.940","Text":"we had that it diverges,"},{"Start":"03:29.940 ","End":"03:31.605","Text":"where at minus 1,"},{"Start":"03:31.605 ","End":"03:39.060","Text":"it converged conditionally, actually, if you go back and look."},{"Start":"03:39.060 ","End":"03:45.830","Text":"Outside, it diverges, at least according to what I stated,"},{"Start":"03:45.830 ","End":"03:47.930","Text":"which I didn\u0027t explain how I got to this,"},{"Start":"03:47.930 ","End":"03:53.135","Text":"but this is what we have in our particular case."},{"Start":"03:53.135 ","End":"03:56.810","Text":"Now I want to next introduce a concept to give a name for"},{"Start":"03:56.810 ","End":"04:01.530","Text":"this R called radius of convergence."},{"Start":"04:04.280 ","End":"04:08.990","Text":"Radius of convergence in each of the 3 cases is defined as follows."},{"Start":"04:08.990 ","End":"04:13.055","Text":"If this happens, then we say that the radius of convergence,"},{"Start":"04:13.055 ","End":"04:16.410","Text":"we\u0027ll call it R. Here,"},{"Start":"04:16.410 ","End":"04:18.745","Text":"we just say R equals 0;"},{"Start":"04:18.745 ","End":"04:21.440","Text":"radius of convergence is 0."},{"Start":"04:21.440 ","End":"04:23.690","Text":"When it converges for all x,"},{"Start":"04:23.690 ","End":"04:28.510","Text":"we say that the radius of convergence is infinity."},{"Start":"04:28.510 ","End":"04:34.325","Text":"In this case, the R that\u0027s here is the radius of convergence."},{"Start":"04:34.325 ","End":"04:42.065","Text":"So that\u0027s the concept of radius of convergence,"},{"Start":"04:42.065 ","End":"04:44.060","Text":"and in the 3 cases."},{"Start":"04:44.060 ","End":"04:47.570","Text":"In our case, the radius of convergence was 1,"},{"Start":"04:47.570 ","End":"04:51.195","Text":"although I didn\u0027t show you how I got to this result."},{"Start":"04:51.195 ","End":"04:53.405","Text":"This is a theorem we can accept,"},{"Start":"04:53.405 ","End":"05:00.175","Text":"and we\u0027re going to use this without proof in what follows."},{"Start":"05:00.175 ","End":"05:02.255","Text":"I\u0027ll tell you what\u0027s coming next."},{"Start":"05:02.255 ","End":"05:05.015","Text":"What\u0027s coming next is I\u0027m going to give you a formula,"},{"Start":"05:05.015 ","End":"05:08.455","Text":"how to figure out the radius of convergence,"},{"Start":"05:08.455 ","End":"05:11.650","Text":"and in our particular example,"},{"Start":"05:11.650 ","End":"05:14.540","Text":"it will come out to be 1."},{"Start":"05:14.540 ","End":"05:17.120","Text":"When we have the radius of convergence,"},{"Start":"05:17.120 ","End":"05:22.280","Text":"we can then use it to find the region of convergence as follows."},{"Start":"05:22.280 ","End":"05:26.000","Text":"If the radius of convergence comes out 0,"},{"Start":"05:26.000 ","End":"05:34.640","Text":"then we\u0027ll know that the interval of convergence is just x equals 0;"},{"Start":"05:34.640 ","End":"05:39.410","Text":"if we get the answer that the radius is infinity,"},{"Start":"05:39.410 ","End":"05:41.600","Text":"then we\u0027ll know it converges for all x;"},{"Start":"05:41.600 ","End":"05:44.450","Text":"and if we get some other value of R,"},{"Start":"05:44.450 ","End":"05:53.445","Text":"some finite value of R that\u0027s bigger than 0, but not infinity,"},{"Start":"05:53.445 ","End":"05:59.930","Text":"then we\u0027ll know that it converges R. I guess I"},{"Start":"05:59.930 ","End":"06:06.870","Text":"forgot to mention that in this case, it converges absolutely."},{"Start":"06:06.970 ","End":"06:10.810","Text":"So if we get 0 and infinity, we know,"},{"Start":"06:10.810 ","End":"06:12.800","Text":"and if we get something else,"},{"Start":"06:12.800 ","End":"06:14.240","Text":"just a finite one,"},{"Start":"06:14.240 ","End":"06:17.090","Text":"then we know it converges absolutely here,"},{"Start":"06:17.090 ","End":"06:18.845","Text":"we know it diverges here,"},{"Start":"06:18.845 ","End":"06:24.380","Text":"but we have to check manually what happens"},{"Start":"06:24.380 ","End":"06:30.020","Text":"just by whatever methods we can for the particular values R and minus R,"},{"Start":"06:30.020 ","End":"06:31.040","Text":"and in our case,"},{"Start":"06:31.040 ","End":"06:35.315","Text":"we did check and we found that in 1 case it diverged,"},{"Start":"06:35.315 ","End":"06:40.750","Text":"and in the other case we\u0027ve got conditional convergence at minus 1."},{"Start":"06:40.750 ","End":"06:48.750","Text":"What remains is to give the theorem formula for how to find the radius,"},{"Start":"06:48.750 ","End":"06:51.280","Text":"and then some more examples."}],"ID":7959},{"Watched":false,"Name":"Radius of Convergence","Duration":"13m 40s","ChapterTopicVideoID":7929,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"I erased what I don\u0027t need and as I said,"},{"Start":"00:03.180 ","End":"00:09.960","Text":"we\u0027re going to be talking now about a formula for computing the radius of convergence."},{"Start":"00:09.960 ","End":"00:17.970","Text":"I kept our example because we\u0027ll use it and the theorem is the formula for the radius,"},{"Start":"00:17.970 ","End":"00:19.680","Text":"and it\u0027s actually 2 of them."},{"Start":"00:19.680 ","End":"00:20.895","Text":"There\u0027s 2 formulas."},{"Start":"00:20.895 ","End":"00:24.010","Text":"I\u0027ll call them number 1 and number 2."},{"Start":"00:24.010 ","End":"00:31.985","Text":"Number 1 says that the radius of convergence R is given by the limit"},{"Start":"00:31.985 ","End":"00:39.815","Text":"as n goes to infinity of the absolute value of an over an plus 1."},{"Start":"00:39.815 ","End":"00:46.370","Text":"The other formula is given by the limit as n goes to infinity"},{"Start":"00:46.370 ","End":"00:54.260","Text":"of the nth root of 1 over the absolute value of an."},{"Start":"00:54.260 ","End":"00:58.220","Text":"We\u0027re not going to prove it, I\u0027ll just tell you that this is based on"},{"Start":"00:58.220 ","End":"01:03.095","Text":"the ratio test for series of numbers."},{"Start":"01:03.095 ","End":"01:05.985","Text":"This 1 is based on the root test."},{"Start":"01:05.985 ","End":"01:14.705","Text":"I suggest you refer back to the section on theories and convergence."},{"Start":"01:14.705 ","End":"01:16.775","Text":"But it\u0027s not important."},{"Start":"01:16.775 ","End":"01:20.330","Text":"We don\u0027t need the proof at this level we\u0027re"},{"Start":"01:20.330 ","End":"01:25.105","Text":"just going to take it on trust that these are the formulas."},{"Start":"01:25.105 ","End":"01:31.050","Text":"Just for completeness I\u0027ll add that this is formula for a power series."},{"Start":"01:31.810 ","End":"01:34.565","Text":"I\u0027m going to give an example of each,"},{"Start":"01:34.565 ","End":"01:36.830","Text":"and the example for number 1 I\u0027m going to use"},{"Start":"01:36.830 ","End":"01:40.265","Text":"this series I\u0027ve been carrying around with me all this time."},{"Start":"01:40.265 ","End":"01:43.705","Text":"This theory is, of course,"},{"Start":"01:43.705 ","End":"01:48.990","Text":"we have that in this that an,"},{"Start":"01:48.990 ","End":"01:58.815","Text":"if I wanted to write it in the form an is 1 over n. Lets take that as example number 1."},{"Start":"01:58.815 ","End":"02:04.295","Text":"We\u0027ll just say that\u0027s the example above, this one here."},{"Start":"02:04.295 ","End":"02:06.875","Text":"We\u0027re going to use the formula now."},{"Start":"02:06.875 ","End":"02:11.940","Text":"Just note that we have that"},{"Start":"02:11.940 ","End":"02:17.690","Text":"an plus 1 I\u0027ll also need is equal to just 1 over n plus 1."},{"Start":"02:17.690 ","End":"02:19.955","Text":"I\u0027m replacing n by n plus 1."},{"Start":"02:19.955 ","End":"02:29.665","Text":"I have that R is the limit as n goes to infinity of the absolute value of an."},{"Start":"02:29.665 ","End":"02:38.740","Text":"We\u0027ve said is 1 over n and an plus 1 is 1 over n plus 1."},{"Start":"02:38.740 ","End":"02:40.610","Text":"It\u0027s an absolute value."},{"Start":"02:40.610 ","End":"02:44.314","Text":"But in this case we don\u0027t need the absolute value because everything\u0027s positive,"},{"Start":"02:44.314 ","End":"02:47.210","Text":"and this just comes out to be,"},{"Start":"02:47.210 ","End":"02:50.570","Text":"if you remember how to divide fractions,"},{"Start":"02:50.570 ","End":"02:53.960","Text":"we multiply by the inverse fraction."},{"Start":"02:53.960 ","End":"03:02.180","Text":"This division comes out to be n plus 1 over n. This limit can be done in various ways."},{"Start":"03:02.180 ","End":"03:04.710","Text":"The answer is 1."},{"Start":"03:05.200 ","End":"03:08.030","Text":"I\u0027ll just briefly mention how I got this."},{"Start":"03:08.030 ","End":"03:10.925","Text":"One way is because it\u0027s a polynomial over a polynomial."},{"Start":"03:10.925 ","End":"03:17.315","Text":"We can just take the leading terms and say it\u0027s n over n and the limit of that is 1."},{"Start":"03:17.315 ","End":"03:20.750","Text":"Or I could simplify this and say that this expression is"},{"Start":"03:20.750 ","End":"03:24.755","Text":"1 plus 1 over n and then when n goes to infinity,"},{"Start":"03:24.755 ","End":"03:26.360","Text":"1 over infinity is 0,"},{"Start":"03:26.360 ","End":"03:27.965","Text":"so it comes out 1."},{"Start":"03:27.965 ","End":"03:30.710","Text":"That\u0027s the radius of convergence."},{"Start":"03:30.710 ","End":"03:34.025","Text":"I would like to take this example further and actually"},{"Start":"03:34.025 ","End":"03:37.310","Text":"find the range of convergence also."},{"Start":"03:37.310 ","End":"03:41.450","Text":"Remember that we\u0027ve said using the radius we can find the range,"},{"Start":"03:41.450 ","End":"03:44.135","Text":"but I want to do this with a picture that we\u0027re going to use often."},{"Start":"03:44.135 ","End":"03:46.800","Text":"I\u0027ll start with the number line."},{"Start":"03:47.510 ","End":"03:54.994","Text":"X goes in this direction and 0 is always on here because it always converges at 0."},{"Start":"03:54.994 ","End":"03:57.890","Text":"You may also wonder about radius."},{"Start":"03:57.890 ","End":"04:03.690","Text":"It\u0027s like I could take a circle of radius 1 around 0."},{"Start":"04:03.690 ","End":"04:11.120","Text":"Here\u0027s the circle of radius 1 around the 0 and it cuts this point and at this point,"},{"Start":"04:11.120 ","End":"04:13.355","Text":"I don\u0027t really need the circle."},{"Start":"04:13.355 ","End":"04:17.030","Text":"There are also power series in 2-dimensions and it really is a circle,"},{"Start":"04:17.030 ","End":"04:18.920","Text":"but it still use the word radius."},{"Start":"04:18.920 ","End":"04:23.140","Text":"This should be 1 and this would be minus 1,"},{"Start":"04:23.140 ","End":"04:28.085","Text":"and from here to here it\u0027s the radius of convergence 1 and from here to here,"},{"Start":"04:28.085 ","End":"04:30.190","Text":"the distance is the radius of convergence 1."},{"Start":"04:30.190 ","End":"04:32.450","Text":"We don\u0027t really need the circle."},{"Start":"04:32.480 ","End":"04:38.405","Text":"In general, we said that between r and minus r, in other words,"},{"Start":"04:38.405 ","End":"04:45.070","Text":"here not including necessarily the endpoints, the series converges."},{"Start":"04:45.070 ","End":"04:48.500","Text":"In fact it even converges absolutely,"},{"Start":"04:48.500 ","End":"04:55.200","Text":"but often we don\u0027t really care whether it\u0027s absolutely or not but I\u0027m just mentioning it."},{"Start":"04:55.200 ","End":"05:02.250","Text":"Here it converges, and outside when it\u0027s bigger than 1, it diverges."},{"Start":"05:02.440 ","End":"05:05.410","Text":"Perhaps I\u0027ll indicate that in a different color,"},{"Start":"05:05.410 ","End":"05:06.730","Text":"so green for good,"},{"Start":"05:06.730 ","End":"05:11.080","Text":"convergence, red for bad diverges."},{"Start":"05:11.080 ","End":"05:17.389","Text":"Here also diverges. In general,"},{"Start":"05:17.389 ","End":"05:24.265","Text":"we often put question marks here and here like so and like so,"},{"Start":"05:24.265 ","End":"05:25.990","Text":"which in the previous theorem says,"},{"Start":"05:25.990 ","End":"05:28.210","Text":"we have to check where it could be convergent,"},{"Start":"05:28.210 ","End":"05:31.510","Text":"divergent, and convergent could be absolutely or conditionally."},{"Start":"05:31.510 ","End":"05:33.685","Text":"Anything could happen here and here."},{"Start":"05:33.685 ","End":"05:39.520","Text":"In our particular example we actually checked and we found that at 1, it diverges."},{"Start":"05:39.520 ","End":"05:46.330","Text":"Also put d for diverge and at minus 1 it converge conditionally in fact."},{"Start":"05:46.520 ","End":"05:50.025","Text":"We use this to get the region."},{"Start":"05:50.025 ","End":"05:52.140","Text":"If I didn\u0027t know this I would just say, okay,"},{"Start":"05:52.140 ","End":"05:57.135","Text":"we got the radius is 1 and so it\u0027s from 1 to minus 1 or vice versa."},{"Start":"05:57.135 ","End":"06:06.825","Text":"I would write minus 1 less than x, less than 1."},{"Start":"06:06.825 ","End":"06:10.320","Text":"The only thing then to check because of these question marks or whether"},{"Start":"06:10.320 ","End":"06:15.620","Text":"this inequality could be less than or equal to and the same here."},{"Start":"06:15.620 ","End":"06:17.885","Text":"We already did that and it turned out that"},{"Start":"06:17.885 ","End":"06:21.689","Text":"the minus 1 is included and the 1 isn\u0027t included."},{"Start":"06:21.700 ","End":"06:25.445","Text":"That\u0027s how we would do the problem in general."},{"Start":"06:25.445 ","End":"06:27.880","Text":"Find the radius, write the range,"},{"Start":"06:27.880 ","End":"06:30.200","Text":"and then just check the endpoints."},{"Start":"06:30.200 ","End":"06:34.160","Text":"Unless the radius comes out 0 or infinity in which case it\u0027s just a"},{"Start":"06:34.160 ","End":"06:38.120","Text":"0 or all x as we stated previously."},{"Start":"06:38.120 ","End":"06:42.540","Text":"I want to move on now to example 2 which is going to use the second formula."},{"Start":"06:43.580 ","End":"06:49.695","Text":"Example 2 is going to be,"},{"Start":"06:49.695 ","End":"06:55.570","Text":"I\u0027m going to take an to equal 1 over 2^n."},{"Start":"06:55.870 ","End":"06:58.010","Text":"You might also ask,"},{"Start":"06:58.010 ","End":"07:00.080","Text":"how did we know which formula to use?"},{"Start":"07:00.080 ","End":"07:05.450","Text":"Well, by experience, but often both formulas work."},{"Start":"07:05.450 ","End":"07:10.745","Text":"I could have just as easily or almost as easily done example 1 using this formula."},{"Start":"07:10.745 ","End":"07:16.175","Text":"One indication is exponents powers which we have here,"},{"Start":"07:16.175 ","End":"07:21.815","Text":"and that\u0027s often a good indication to use the second formula."},{"Start":"07:21.815 ","End":"07:24.560","Text":"Although in both of these examples,"},{"Start":"07:24.560 ","End":"07:29.840","Text":"we could have used both of these and you would have got the answer."},{"Start":"07:29.840 ","End":"07:39.425","Text":"In this case, we know that R is the limit as n goes to infinity"},{"Start":"07:39.425 ","End":"07:49.350","Text":"of the nth root of 1 over the absolute value of an."},{"Start":"07:49.350 ","End":"07:53.325","Text":"Let me just write"},{"Start":"07:53.325 ","End":"07:59.580","Text":"the absolute value, 1 over an."},{"Start":"07:59.580 ","End":"08:04.240","Text":"I\u0027m just going to write it as, where is it?"},{"Start":"08:04.240 ","End":"08:06.800","Text":"Yeah, 1 over 2^n."},{"Start":"08:07.920 ","End":"08:12.400","Text":"Strictly speaking, the absolute value was just down here."},{"Start":"08:12.400 ","End":"08:17.265","Text":"I just want to mention this is a slight variant of this."},{"Start":"08:17.265 ","End":"08:21.010","Text":"I could also just write 1 over"},{"Start":"08:21.010 ","End":"08:26.605","Text":"the nth root of absolute value of an actually it might be simpler."},{"Start":"08:26.605 ","End":"08:32.230","Text":"Again, the limit as n goes to infinity,"},{"Start":"08:32.230 ","End":"08:33.760","Text":"almost practically the same."},{"Start":"08:33.760 ","End":"08:36.100","Text":"I sometimes use this one sometimes the other."},{"Start":"08:36.100 ","End":"08:39.430","Text":"Anyway, back to our example."},{"Start":"08:39.430 ","End":"08:46.115","Text":"This is equal to the limit and the nth root."},{"Start":"08:46.115 ","End":"08:51.640","Text":"1 over 1 over is just bringing the denominator up to the numerator,"},{"Start":"08:51.640 ","End":"08:54.655","Text":"so it\u0027s 2^n, we don\u0027t need the absolute value."},{"Start":"08:54.655 ","End":"08:56.200","Text":"Could throw that out everything\u0027s positive,"},{"Start":"08:56.200 ","End":"08:59.020","Text":"so it\u0027s 1 over 1 over is just this,"},{"Start":"08:59.020 ","End":"09:00.490","Text":"and the nth root of 2 to the n,"},{"Start":"09:00.490 ","End":"09:01.720","Text":"well, it canceled each other out."},{"Start":"09:01.720 ","End":"09:03.280","Text":"The power of n in the nth root,"},{"Start":"09:03.280 ","End":"09:07.175","Text":"this is just the limit of 2,"},{"Start":"09:07.175 ","End":"09:11.290","Text":"and the limit of a constant is just that constant."},{"Start":"09:11.290 ","End":"09:16.830","Text":"In this example, we got the radius of convergence to be 2."},{"Start":"09:16.830 ","End":"09:24.710","Text":"Let\u0027s do it to the end and show also what is the region range, interval of convergence."},{"Start":"09:24.710 ","End":"09:34.770","Text":"I\u0027ll just make some space here and I just copy pasted this picture."},{"Start":"09:34.770 ","End":"09:36.845","Text":"I just need minor modifications."},{"Start":"09:36.845 ","End":"09:41.060","Text":"In this case, this radius is 2,"},{"Start":"09:41.060 ","End":"09:46.825","Text":"so we\u0027re going 2 units from 0 in either direction."},{"Start":"09:46.825 ","End":"09:50.475","Text":"This is 2, this is minus 2,"},{"Start":"09:50.475 ","End":"09:56.850","Text":"and we just have to again check what happens exactly at 2 and at minus 2."},{"Start":"09:56.850 ","End":"10:03.575","Text":"Perhaps I should have said not just what an is but the series is the sum of an x^n"},{"Start":"10:03.575 ","End":"10:10.260","Text":"is the sum 1 over 2^n, x^n."},{"Start":"10:10.260 ","End":"10:14.730","Text":"That\u0027s the series corresponding to this. We want to do 2 things."},{"Start":"10:14.730 ","End":"10:18.435","Text":"We want to substitute x equals"},{"Start":"10:18.435 ","End":"10:23.660","Text":"2 and see what we get and x equals minus 2 and see what we get."},{"Start":"10:23.660 ","End":"10:29.060","Text":"I know already the interval of convergence"},{"Start":"10:29.060 ","End":"10:35.135","Text":"will be something like x between minus 2 and 2."},{"Start":"10:35.135 ","End":"10:39.200","Text":"The only question mark is like these question marks."},{"Start":"10:39.200 ","End":"10:42.260","Text":"It could possibly be less than or equal to,"},{"Start":"10:42.260 ","End":"10:45.710","Text":"and it could possibly be less than or equal to,"},{"Start":"10:45.710 ","End":"10:47.015","Text":"and that\u0027s what we\u0027ll check."},{"Start":"10:47.015 ","End":"10:52.085","Text":"We substitute x equals 2 in here and we get the sum"},{"Start":"10:52.085 ","End":"10:58.150","Text":"from 0 to infinity of 1 over 2^n times x is 2,"},{"Start":"10:58.150 ","End":"11:03.740","Text":"so it\u0027s 2^n which is just the sum from 0 to infinity."},{"Start":"11:03.740 ","End":"11:05.965","Text":"This cancels to1."},{"Start":"11:05.965 ","End":"11:11.245","Text":"Now, this definitely diverges."},{"Start":"11:11.245 ","End":"11:15.930","Text":"I mean, 1 plus 1 plus 1 plus 1 plus 1, it doesn\u0027t converge,"},{"Start":"11:15.930 ","End":"11:20.345","Text":"it goes to infinity but also because the general term doesn\u0027t go to 0,"},{"Start":"11:20.345 ","End":"11:28.055","Text":"the general term is just 1 always and it does not have a limit of 0, so it diverges."},{"Start":"11:28.055 ","End":"11:34.550","Text":"Minus 2, we get the sum from 0 to"},{"Start":"11:34.550 ","End":"11:43.115","Text":"infinity of 1 over 2^n times minus 2^n."},{"Start":"11:43.115 ","End":"11:45.650","Text":"If we cancel 2^n,"},{"Start":"11:45.650 ","End":"11:48.200","Text":"we just get the sum."},{"Start":"11:48.200 ","End":"11:50.135","Text":"This is minus 1 times 2."},{"Start":"11:50.135 ","End":"11:52.280","Text":"We just get minus 1^n."},{"Start":"11:52.280 ","End":"12:00.465","Text":"It\u0027s times 2^n over 2^n and this also diverges."},{"Start":"12:00.465 ","End":"12:06.920","Text":"The reason this diverges is that the general term is either minus 1 or plus 1,"},{"Start":"12:06.920 ","End":"12:09.005","Text":"but it does not go to 0."},{"Start":"12:09.005 ","End":"12:13.010","Text":"This series, each term is either plus or minus 1,"},{"Start":"12:13.010 ","End":"12:15.290","Text":"so it doesn\u0027t tend to 0 either."},{"Start":"12:15.290 ","End":"12:19.925","Text":"In both cases you could say the general term does not tend to 0, so it divergence,"},{"Start":"12:19.925 ","End":"12:25.900","Text":"so we can erase the question marks."},{"Start":"12:25.900 ","End":"12:30.195","Text":"This is not and this is not,"},{"Start":"12:30.195 ","End":"12:34.265","Text":"and we already checked that this was a divergence."},{"Start":"12:34.265 ","End":"12:40.275","Text":"Here we have divergence and so radius is"},{"Start":"12:40.275 ","End":"12:47.105","Text":"2 and the range or interval region of convergence is this."},{"Start":"12:47.105 ","End":"12:48.680","Text":"That will be a typical question."},{"Start":"12:48.680 ","End":"12:55.325","Text":"A typical question that you\u0027ll have in the exercises will be, given this series,"},{"Start":"12:55.325 ","End":"13:04.815","Text":"find its radius of convergence and the range or interval of convergence."},{"Start":"13:04.815 ","End":"13:07.430","Text":"Then you would say, yeah, this is the radius and this is"},{"Start":"13:07.430 ","End":"13:12.730","Text":"the range or interval or region of convergence."},{"Start":"13:12.730 ","End":"13:15.230","Text":"That\u0027s about it for the tutorial,"},{"Start":"13:15.230 ","End":"13:18.125","Text":"there\u0027s plenty of exercises."},{"Start":"13:18.125 ","End":"13:22.235","Text":"I just took a look and I would like to say that I prefer"},{"Start":"13:22.235 ","End":"13:29.030","Text":"this form of the second formula."},{"Start":"13:29.030 ","End":"13:31.345","Text":"In the first formula,"},{"Start":"13:31.345 ","End":"13:35.870","Text":"we have this and in the second formula I\u0027ll use this form not"},{"Start":"13:35.870 ","End":"13:41.100","Text":"this to be consistent with the exercises. That\u0027s it."}],"ID":7960},{"Watched":false,"Name":"Exercise 1 part a","Duration":"25m 44s","ChapterTopicVideoID":7899,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.900","Text":"In this exercise, we want to find the region of convergence of this infinite series."},{"Start":"00:06.900 ","End":"00:10.680","Text":"What this means, these are functions is to find for which"},{"Start":"00:10.680 ","End":"00:14.070","Text":"x the series converges. Let\u0027s give it a name."},{"Start":"00:14.070 ","End":"00:15.720","Text":"First of all, a general term a,"},{"Start":"00:15.720 ","End":"00:19.860","Text":"n will, and I just copied this."},{"Start":"00:19.860 ","End":"00:27.044","Text":"The test I want to use is going to be the root test."},{"Start":"00:27.044 ","End":"00:31.515","Text":"The root test applies to positive series."},{"Start":"00:31.515 ","End":"00:33.420","Text":"This is not a positive series,"},{"Start":"00:33.420 ","End":"00:38.330","Text":"so let\u0027s apply it to the absolute value of a n and that might help us."},{"Start":"00:38.330 ","End":"00:44.105","Text":"The absolute value of a n is 1 over 4n plus 1."},{"Start":"00:44.105 ","End":"00:47.345","Text":"I can put the absolute value in here."},{"Start":"00:47.345 ","End":"00:49.130","Text":"This of course is positive,"},{"Start":"00:49.130 ","End":"00:51.080","Text":"so I don\u0027t need the absolute value."},{"Start":"00:51.080 ","End":"00:57.140","Text":"I\u0027ll put 1 minus x over 1 plus x and absolute value to the power of n. Now,"},{"Start":"00:57.140 ","End":"01:01.640","Text":"the root test for this series says that we have to figure out"},{"Start":"01:01.640 ","End":"01:07.010","Text":"first the limit as n goes to infinity of the nth root of a,"},{"Start":"01:07.010 ","End":"01:12.965","Text":"n of absolute value of a n in this case,"},{"Start":"01:12.965 ","End":"01:19.080","Text":"and this is the limit as n goes to infinity."},{"Start":"01:19.250 ","End":"01:22.020","Text":"Now, I\u0027ll break it up into 2."},{"Start":"01:22.020 ","End":"01:28.350","Text":"The first bit is the nth root of 1 over"},{"Start":"01:28.350 ","End":"01:38.334","Text":"4n plus 1 and the second bit is 1 minus x over 1 plus x."},{"Start":"01:38.334 ","End":"01:43.730","Text":"In absolute value, the nth root of the nth power just cancel."},{"Start":"01:47.630 ","End":"01:50.490","Text":"This doesn\u0027t involve n,"},{"Start":"01:50.490 ","End":"01:53.025","Text":"so this is 1 minus x"},{"Start":"01:53.025 ","End":"02:02.160","Text":"over 1 plus x times the limit of n as n goes to infinity of this thing,"},{"Start":"02:02.160 ","End":"02:06.910","Text":"the nth root of 1 over 4n plus 1."},{"Start":"02:07.580 ","End":"02:11.610","Text":"I\u0027ll do this bit as a side exercise."},{"Start":"02:11.610 ","End":"02:16.650","Text":"Suppose that I label them,"},{"Start":"02:16.650 ","End":"02:26.765","Text":"I want to use a n. Let\u0027s say bn is the nth root of 1 over 4n plus 1."},{"Start":"02:26.765 ","End":"02:30.755","Text":"Now, if I could just take the logarithm of this,"},{"Start":"02:30.755 ","End":"02:32.360","Text":"it would be a lot easier to work with."},{"Start":"02:32.360 ","End":"02:39.330","Text":"I\u0027m going to use the fact that bn is in general,"},{"Start":"02:39.330 ","End":"02:44.845","Text":"any number is e^power of natural log of that number."},{"Start":"02:44.845 ","End":"02:47.900","Text":"If this thing has a limit,"},{"Start":"02:47.900 ","End":"02:55.775","Text":"then we could say that the limit as n goes to infinity of bn"},{"Start":"02:55.775 ","End":"03:01.850","Text":"is e^ power of the limit as n"},{"Start":"03:01.850 ","End":"03:08.370","Text":"goes to infinity of the natural log of bn."},{"Start":"03:19.001 ","End":"03:25.440","Text":"Natural log of bn is 1 over n,"},{"Start":"03:25.440 ","End":"03:28.680","Text":"because this is to the power of 1 over n,"},{"Start":"03:28.680 ","End":"03:39.210","Text":"so I can put it like that times natural log of what\u0027s inside."},{"Start":"03:39.210 ","End":"03:43.730","Text":"I could actually say 1 over is also an exponent minus 1 and"},{"Start":"03:43.730 ","End":"03:47.900","Text":"show it I can get minus 1 times the natural log of this,"},{"Start":"03:47.900 ","End":"03:52.490","Text":"or simply minus natural log of 4n plus 1."},{"Start":"03:52.490 ","End":"03:53.870","Text":"I did this a bit hastily,"},{"Start":"03:53.870 ","End":"03:56.970","Text":"but you could easily check this."},{"Start":"03:57.170 ","End":"04:02.090","Text":"Now what I can say is that the limit of this,"},{"Start":"04:02.090 ","End":"04:05.130","Text":"as n goes to infinity"},{"Start":"04:07.010 ","End":"04:15.750","Text":"of minus natural log of 4n plus 1 over n,"},{"Start":"04:15.750 ","End":"04:19.800","Text":"I claim that this equals 0."},{"Start":"04:19.800 ","End":"04:26.950","Text":"I\u0027ll have to explain this exercise a bit more involved than I had anticipated."},{"Start":"04:26.950 ","End":"04:31.480","Text":"We use another standard trick here of replacing the integer"},{"Start":"04:31.480 ","End":"04:36.220","Text":"and the natural number n by a variable x and that x go to infinity."},{"Start":"04:36.220 ","End":"04:42.295","Text":"What we want to do is compute the limit as x goes to infinity"},{"Start":"04:42.295 ","End":"04:50.615","Text":"of minus natural log of 4 x plus 1 over x."},{"Start":"04:50.615 ","End":"04:52.610","Text":"This we\u0027re verifying here,"},{"Start":"04:52.610 ","End":"04:56.620","Text":"but I put a question mark here because this is what we\u0027re showing now."},{"Start":"04:56.620 ","End":"04:59.585","Text":"Now, when x goes to infinity,"},{"Start":"04:59.585 ","End":"05:03.385","Text":"we have a case of infinity over infinity."},{"Start":"05:03.385 ","End":"05:08.850","Text":"L\u0027Hopital, or that with a minus if you want."},{"Start":"05:08.850 ","End":"05:13.450","Text":"We\u0027re going to use L\u0027Hopital\u0027s rule."},{"Start":"05:14.330 ","End":"05:21.035","Text":"That means we can differentiate top and bottom separately and the limit will be the same."},{"Start":"05:21.035 ","End":"05:25.700","Text":"We\u0027ll get the limit as x goes to infinity."},{"Start":"05:25.700 ","End":"05:28.360","Text":"I\u0027ll leave the minus here. It doesn\u0027t matter."},{"Start":"05:28.360 ","End":"05:34.670","Text":"The derivative of the numerator is 1 over 4x plus 1."},{"Start":"05:34.670 ","End":"05:38.755","Text":"The derivative of the denominator is just 1."},{"Start":"05:38.755 ","End":"05:45.210","Text":"Clearly we just get minus 1 over 4x plus 1 infinity here."},{"Start":"05:45.210 ","End":"05:47.715","Text":"This limit is 0,"},{"Start":"05:47.715 ","End":"05:49.920","Text":"so this thing has been verified."},{"Start":"05:49.920 ","End":"05:59.615","Text":"If this limit is 0, then this limit now becomes e^power of 0,"},{"Start":"05:59.615 ","End":"06:03.415","Text":"because we figured this limit and this is equal to 1."},{"Start":"06:03.415 ","End":"06:14.190","Text":"Back here, we get that this thing is just an absolute value of 1 minus x over 1 plus x."},{"Start":"06:14.190 ","End":"06:18.065","Text":"Now, I\u0027m going to subdivide into 3 separate cases."},{"Start":"06:18.065 ","End":"06:22.810","Text":"We\u0027re going to take the case where this is less than 1,"},{"Start":"06:22.810 ","End":"06:24.870","Text":"when it\u0027s equal to 1,"},{"Start":"06:24.870 ","End":"06:27.240","Text":"and when it\u0027s greater than 1."},{"Start":"06:27.240 ","End":"06:34.320","Text":"Let me call those case A, case B,"},{"Start":"06:34.320 ","End":"06:42.360","Text":"and case C. I\u0027ll deal first of all with case A where this is less than 1."},{"Start":"06:42.360 ","End":"06:45.815","Text":"Now, in case A by the root test,"},{"Start":"06:45.815 ","End":"06:47.600","Text":"if this is less than 1,"},{"Start":"06:47.600 ","End":"06:54.620","Text":"then that means that the series absolute value"},{"Start":"06:54.620 ","End":"07:01.035","Text":"a n converges."},{"Start":"07:01.035 ","End":"07:06.100","Text":"If this converges, then our original series,"},{"Start":"07:06.100 ","End":"07:16.030","Text":"sum of an write it 1 to infinity,"},{"Start":"07:16.030 ","End":"07:20.095","Text":"1 to infinity also converges."},{"Start":"07:20.095 ","End":"07:22.420","Text":"Actually, it more than converges."},{"Start":"07:22.420 ","End":"07:25.840","Text":"It converges absolutely, because that\u0027s what"},{"Start":"07:25.840 ","End":"07:30.580","Text":"absolute convergence means that the absolute value series converges,"},{"Start":"07:30.580 ","End":"07:32.920","Text":"but that\u0027s more than we\u0027re asked for."},{"Start":"07:32.920 ","End":"07:35.815","Text":"Certainly in case a, we have convergence."},{"Start":"07:35.815 ","End":"07:41.110","Text":"The thing is, I want to interpret what it means in terms of x,"},{"Start":"07:41.110 ","End":"07:42.835","Text":"what range is x in."},{"Start":"07:42.835 ","End":"07:44.290","Text":"I need to solve now,"},{"Start":"07:44.290 ","End":"07:48.730","Text":"the inequality absolute value of 1 minus x"},{"Start":"07:48.730 ","End":"07:55.015","Text":"over 1 plus x is less than 1 and see what it means about x."},{"Start":"07:55.015 ","End":"08:01.990","Text":"An absolute value less than something means that this is between this and minus this."},{"Start":"08:01.990 ","End":"08:05.935","Text":"In other words, this is equivalent to minus 1,"},{"Start":"08:05.935 ","End":"08:10.570","Text":"less than 1 minus x over 1 plus x,"},{"Start":"08:10.570 ","End":"08:13.855","Text":"and that is less than 1."},{"Start":"08:13.855 ","End":"08:19.240","Text":"I\u0027ve got 2 inequalities to solve because this is a double inequality."},{"Start":"08:19.240 ","End":"08:24.880","Text":"There\u0027s something I should have mentioned right at the start but I could mention it now,"},{"Start":"08:24.880 ","End":"08:27.040","Text":"a denominator can\u0027t be 0,"},{"Start":"08:27.040 ","End":"08:28.540","Text":"so x can\u0027t be minus 1."},{"Start":"08:28.540 ","End":"08:30.295","Text":"I mean, even in the original series,"},{"Start":"08:30.295 ","End":"08:32.830","Text":"it doesn\u0027t make sense if x is minus 1."},{"Start":"08:32.830 ","End":"08:37.375","Text":"Right off the bat, I can say x is not equal to minus 1."},{"Start":"08:37.375 ","End":"08:42.160","Text":"Whatever I get in the solution of the inequality I have to rule out,"},{"Start":"08:42.160 ","End":"08:44.770","Text":"the minus 1 case for x."},{"Start":"08:44.770 ","End":"08:49.240","Text":"It won\u0027t be in the region of convergence."},{"Start":"08:49.240 ","End":"08:53.539","Text":"Now let\u0027s get to it, just scroll down."},{"Start":"08:55.050 ","End":"08:58.840","Text":"I\u0027ll say that do the first bit,"},{"Start":"08:58.840 ","End":"09:01.990","Text":"which is minus 1, less than 1,"},{"Start":"09:01.990 ","End":"09:04.780","Text":"minus x over 1 plus x."},{"Start":"09:04.780 ","End":"09:12.490","Text":"We\u0027ll solve this, and then we\u0027ll put an AND for the other inequality and solve it."},{"Start":"09:12.490 ","End":"09:21.260","Text":"The other inequality is 1 minus x over 1 plus x is less than 1."},{"Start":"09:22.670 ","End":"09:27.660","Text":"What I suggest here is move the minus 1 over to"},{"Start":"09:27.660 ","End":"09:33.640","Text":"the other side and compare it to 0 and I can actually reverse the inequality also."},{"Start":"09:33.640 ","End":"09:39.370","Text":"I get 1 minus x over 1 plus x plus 1 that\u0027s on the right-hand side,"},{"Start":"09:39.370 ","End":"09:44.950","Text":"so it\u0027s bigger than 0 and then do a common denominator for this."},{"Start":"09:44.950 ","End":"09:46.765","Text":"I think we can do this mentally."},{"Start":"09:46.765 ","End":"09:49.975","Text":"I put it all over 1 plus x,"},{"Start":"09:49.975 ","End":"09:54.460","Text":"I get 1 minus x plus 1 plus x,"},{"Start":"09:54.460 ","End":"09:59.090","Text":"so it just comes out to be 2."},{"Start":"09:59.700 ","End":"10:02.125","Text":"That\u0027s going to be bigger than 0."},{"Start":"10:02.125 ","End":"10:06.850","Text":"For fraction is bigger than 0 and the numerator is bigger than 0,"},{"Start":"10:06.850 ","End":"10:09.520","Text":"then so the denominator has to be bigger than 0"},{"Start":"10:09.520 ","End":"10:13.840","Text":"also because positive over negative is negative,"},{"Start":"10:13.840 ","End":"10:18.265","Text":"so we want 1 plus x also bigger than 0,"},{"Start":"10:18.265 ","End":"10:25.420","Text":"and that gives us that x is bigger than minus 1."},{"Start":"10:25.420 ","End":"10:27.130","Text":"That\u0027s this part."},{"Start":"10:27.130 ","End":"10:30.445","Text":"Now let\u0027s do the other part of the AND."},{"Start":"10:30.445 ","End":"10:32.350","Text":"Using the same trick,"},{"Start":"10:32.350 ","End":"10:40.735","Text":"I just bring to the other side 1 minus x over 1 plus x minus 1 less than 0,"},{"Start":"10:40.735 ","End":"10:45.070","Text":"give it a common denominator of 1 plus x."},{"Start":"10:45.070 ","End":"10:47.245","Text":"We can do this mentally."},{"Start":"10:47.245 ","End":"10:48.910","Text":"It\u0027s 1 minus x,"},{"Start":"10:48.910 ","End":"10:52.135","Text":"subtract 1 plus x."},{"Start":"10:52.135 ","End":"10:58.315","Text":"The 1 \u0027s will cancel, it will give us minus 2x less than 0."},{"Start":"10:58.315 ","End":"11:01.195","Text":"When we have an expression like this,"},{"Start":"11:01.195 ","End":"11:05.440","Text":"one way to do it is to take the number line,"},{"Start":"11:05.440 ","End":"11:10.480","Text":"let\u0027s say here, and on"},{"Start":"11:10.480 ","End":"11:13.345","Text":"this number line we\u0027ll mark the points"},{"Start":"11:13.345 ","End":"11:16.990","Text":"which make either the numerator or the denominator 0."},{"Start":"11:16.990 ","End":"11:21.430","Text":"The numerator is 0 when x is 0,"},{"Start":"11:21.430 ","End":"11:25.240","Text":"and the denominator is 0 when x is minus 1."},{"Start":"11:25.240 ","End":"11:32.590","Text":"I\u0027m marking 0 and minus 1 here and what I have to"},{"Start":"11:32.590 ","End":"11:39.700","Text":"do is check within each region whether this is positive or negative."},{"Start":"11:39.700 ","End":"11:44.380","Text":"It doesn\u0027t change in each of these 3 separate intervals,"},{"Start":"11:44.380 ","End":"11:47.090","Text":"you just have to take a sample point."},{"Start":"11:47.790 ","End":"11:52.885","Text":"Let me choose for this part for x bigger than 0,"},{"Start":"11:52.885 ","End":"11:55.480","Text":"I\u0027ll just label the regions."},{"Start":"11:55.480 ","End":"11:57.790","Text":"This is x bigger than 0."},{"Start":"11:57.790 ","End":"12:00.820","Text":"This is minus one less than x,"},{"Start":"12:00.820 ","End":"12:05.965","Text":"less than 0, and this is x less than minus 1."},{"Start":"12:05.965 ","End":"12:08.365","Text":"Now I\u0027ll choose a sample point in each."},{"Start":"12:08.365 ","End":"12:16.960","Text":"Suppose that here I\u0027ll choose 1 and here I\u0027ll"},{"Start":"12:16.960 ","End":"12:27.310","Text":"choose minus a half and here I\u0027ll choose minus 2."},{"Start":"12:27.310 ","End":"12:29.380","Text":"First, do this mentally."},{"Start":"12:29.380 ","End":"12:32.770","Text":"If we plug in minus 2 into here,"},{"Start":"12:32.770 ","End":"12:37.540","Text":"the numerator becomes 4,"},{"Start":"12:37.540 ","End":"12:42.580","Text":"the denominator is minus 1 and"},{"Start":"12:42.580 ","End":"12:47.935","Text":"so the quotient is negative and we\u0027re collecting negatives,"},{"Start":"12:47.935 ","End":"12:50.605","Text":"so good for us."},{"Start":"12:50.605 ","End":"12:53.425","Text":"If we put in minus a half,"},{"Start":"12:53.425 ","End":"12:57.085","Text":"the numerator becomes plus 1,"},{"Start":"12:57.085 ","End":"13:05.090","Text":"the denominator is plus a half."},{"Start":"13:05.340 ","End":"13:09.670","Text":"We get positive over positive,"},{"Start":"13:09.670 ","End":"13:13.945","Text":"which is positive, and that is bad for us."},{"Start":"13:13.945 ","End":"13:15.745","Text":"Not what we\u0027re collecting."},{"Start":"13:15.745 ","End":"13:18.260","Text":"We want negative."},{"Start":"13:24.810 ","End":"13:27.745","Text":"When we have x equals 1,"},{"Start":"13:27.745 ","End":"13:30.820","Text":"then the numerator is minus 2,"},{"Start":"13:30.820 ","End":"13:32.740","Text":"the denominator is 2,"},{"Start":"13:32.740 ","End":"13:36.790","Text":"negative over positive is negative and that is good."},{"Start":"13:36.790 ","End":"13:38.425","Text":"We want negative."},{"Start":"13:38.425 ","End":"13:42.115","Text":"What we get for this part,"},{"Start":"13:42.115 ","End":"13:45.940","Text":"I write it over here close to this one, deliberately."},{"Start":"13:45.940 ","End":"13:52.525","Text":"We have x is less than minus 2,"},{"Start":"13:52.525 ","End":"13:57.295","Text":"OR, OR meaning it\u0027s both a possible,"},{"Start":"13:57.295 ","End":"14:01.150","Text":"OR x is bigger than 0."},{"Start":"14:01.150 ","End":"14:06.040","Text":"I combine these, I\u0027ll put brackets around here with an AND."},{"Start":"14:06.040 ","End":"14:08.135","Text":"That\u0027s the and from here."},{"Start":"14:08.135 ","End":"14:11.880","Text":"Now I\u0027ve learned how to do these combinations of AND, OR."},{"Start":"14:11.880 ","End":"14:16.955","Text":"The way we do this is to do a little sketch of the number line."},{"Start":"14:16.955 ","End":"14:26.150","Text":"Let\u0027s say this is our number line and I need to put on it all the interesting numbers,"},{"Start":"14:26.150 ","End":"14:27.230","Text":"all the numbers that appear."},{"Start":"14:27.230 ","End":"14:29.000","Text":"We have a minus 1,"},{"Start":"14:29.000 ","End":"14:30.995","Text":"a 0, minus 2,"},{"Start":"14:30.995 ","End":"14:38.290","Text":"so minus 2, minus 1, and 0."},{"Start":"14:38.290 ","End":"14:45.480","Text":"Now to say x is bigger than minus 1,"},{"Start":"14:45.480 ","End":"14:50.345","Text":"I can indicate this by putting a hollow circle"},{"Start":"14:50.345 ","End":"14:57.260","Text":"and an arrow going all the way to the right of minus 1 and endlessly,"},{"Start":"14:57.260 ","End":"14:59.625","Text":"and then for this,"},{"Start":"14:59.625 ","End":"15:03.655","Text":"OR, I\u0027ll use a different color,"},{"Start":"15:03.655 ","End":"15:06.740","Text":"less than minus 2."},{"Start":"15:08.520 ","End":"15:15.680","Text":"The hollow means not including the minus 2 or x bigger than 0."},{"Start":"15:16.320 ","End":"15:22.300","Text":"This is the second bit from here and now"},{"Start":"15:22.300 ","End":"15:28.095","Text":"the AND means that we have to find the area where they overlap."},{"Start":"15:28.095 ","End":"15:32.250","Text":"The overlap is from 0 onward."},{"Start":"15:32.250 ","End":"15:36.550","Text":"This is where they both overlap."},{"Start":"15:36.550 ","End":"15:43.900","Text":"Basically that gives us from 0 and to the right,"},{"Start":"15:43.900 ","End":"15:49.285","Text":"which just means x bigger than 0."},{"Start":"15:49.285 ","End":"15:55.280","Text":"That\u0027s the answer to the inequality."},{"Start":"15:55.290 ","End":"15:58.540","Text":"I can say that in part a,"},{"Start":"15:58.540 ","End":"16:02.305","Text":"when x is bigger than 0"},{"Start":"16:02.305 ","End":"16:07.550","Text":"then the sum of"},{"Start":"16:08.130 ","End":"16:15.130","Text":"the original series converges it\u0027s also absolutely."},{"Start":"16:15.130 ","End":"16:18.340","Text":"Let\u0027s see if it converges anywhere else."},{"Start":"16:18.340 ","End":"16:23.485","Text":"We have x bigger than 0 all ready and let\u0027s continue with part b."},{"Start":"16:23.485 ","End":"16:29.570","Text":"Part b was when this thing was equal to 1."},{"Start":"16:29.790 ","End":"16:34.190","Text":"Let\u0027s write that down."},{"Start":"16:34.290 ","End":"16:44.425","Text":"Case b, absolute value of 1 minus x over 1 plus x is equal to 1."},{"Start":"16:44.425 ","End":"16:52.300","Text":"If the absolute value is equal to 1 then it itself must equal plus or minus 1."},{"Start":"16:52.300 ","End":"16:58.450","Text":"We have either 1 minus x over 1 plus x equals"},{"Start":"16:58.450 ","End":"17:06.220","Text":"1 or 1 minus x over 1 plus x is equal to minus 1."},{"Start":"17:06.220 ","End":"17:08.784","Text":"Now, in the first case,"},{"Start":"17:08.784 ","End":"17:16.160","Text":"if we multiply, we get 1 minus x equals 1 plus x."},{"Start":"17:16.710 ","End":"17:21.910","Text":"If you solve this you get that x equals"},{"Start":"17:21.910 ","End":"17:26.980","Text":"0 because you could bring the 1 to this side and get 0 equals to x,"},{"Start":"17:26.980 ","End":"17:29.965","Text":"so x equals 0."},{"Start":"17:29.965 ","End":"17:33.220","Text":"If you solve this or try to,"},{"Start":"17:33.220 ","End":"17:42.895","Text":"you get 1 minus x equals minus 1 and minus x."},{"Start":"17:42.895 ","End":"17:47.950","Text":"If I add x to both sides I\u0027ve got 1 equals minus"},{"Start":"17:47.950 ","End":"17:53.780","Text":"1 which is impossible so it cannot be."},{"Start":"17:53.780 ","End":"17:57.865","Text":"There\u0027s no way that this is going to equal minus 1."},{"Start":"17:57.865 ","End":"18:00.880","Text":"All we\u0027re left with is x equals 0."},{"Start":"18:00.880 ","End":"18:05.110","Text":"Now, x equals 0 we have to now"},{"Start":"18:05.110 ","End":"18:11.755","Text":"check specifically for this value what happens with the original series."},{"Start":"18:11.755 ","End":"18:18.070","Text":"Now, the original series or an if you recall,"},{"Start":"18:18.070 ","End":"18:27.220","Text":"was 1 over 4n plus 1 times 1 minus x"},{"Start":"18:27.220 ","End":"18:31.015","Text":"over 1 plus x^n."},{"Start":"18:31.015 ","End":"18:33.115","Text":"But if x equals 0,"},{"Start":"18:33.115 ","End":"18:38.920","Text":"an is just 1 over 4n plus 1,"},{"Start":"18:38.920 ","End":"18:42.565","Text":"because this is 0, this is 1, 1^n is 1."},{"Start":"18:42.565 ","End":"18:47.605","Text":"We want to know what happens to the sum of this."},{"Start":"18:47.605 ","End":"18:55.375","Text":"The sum n equals 1 to infinity 1 over 4n plus 1."},{"Start":"18:55.375 ","End":"19:02.290","Text":"Now, I claim this diverges and I suggest just use"},{"Start":"19:02.290 ","End":"19:12.080","Text":"the limit comparison test"},{"Start":"19:13.110 ","End":"19:23.605","Text":"with the sum n goes from 1 to infinity of 1 over n, the harmonic series."},{"Start":"19:23.605 ","End":"19:28.510","Text":"Now, the harmonic series is well-known to be divergent."},{"Start":"19:28.510 ","End":"19:31.210","Text":"This series diverges and by"},{"Start":"19:31.210 ","End":"19:35.440","Text":"the limit comparison test these 2 either both converge or both diverge."},{"Start":"19:35.440 ","End":"19:39.040","Text":"This one is also divergent."},{"Start":"19:39.040 ","End":"19:46.480","Text":"But this is the sum of an because 1 over 4n plus 1 is an."},{"Start":"19:46.480 ","End":"19:50.005","Text":"In this case also we get that the sum,"},{"Start":"19:50.005 ","End":"19:54.685","Text":"the original series is divergent,"},{"Start":"19:54.685 ","End":"19:58.465","Text":"and so case b adds nothing"},{"Start":"19:58.465 ","End":"20:03.625","Text":"new to the convergent case which so far stands at x bigger than 0."},{"Start":"20:03.625 ","End":"20:06.400","Text":"Now, I\u0027m going to erase case b and we\u0027ll move"},{"Start":"20:06.400 ","End":"20:09.815","Text":"on to case c where this thing is bigger than 1."},{"Start":"20:09.815 ","End":"20:16.845","Text":"I erase some stuff and now I\u0027m going to Change this to a bigger than 1."},{"Start":"20:16.845 ","End":"20:22.570","Text":"This is k c. Let me just put this at the side. I still need it."},{"Start":"20:22.570 ","End":"20:28.749","Text":"In this case again with the root test we know that the sum"},{"Start":"20:28.749 ","End":"20:37.800","Text":"of the absolute value of an from 1 to infinity, this diverges."},{"Start":"20:37.800 ","End":"20:45.955","Text":"But we can\u0027t conclude from this that the sum of the original series an diverges."},{"Start":"20:45.955 ","End":"20:48.850","Text":"There\u0027s something called conditional convergence,"},{"Start":"20:48.850 ","End":"20:53.860","Text":"like the original series could converge but the absolute value could diverge."},{"Start":"20:53.860 ","End":"20:56.650","Text":"We seem to be stuck here."},{"Start":"20:56.650 ","End":"21:02.440","Text":"I claim that in this case the sum of an also"},{"Start":"21:02.440 ","End":"21:11.155","Text":"diverges but I\u0027ll show you why it diverges and I\u0027ll do it by contradiction."},{"Start":"21:11.155 ","End":"21:17.680","Text":"Suppose that this thing converges."},{"Start":"21:17.680 ","End":"21:22.675","Text":"I should have written the word suppose and I\u0027ll write the word if."},{"Start":"21:22.675 ","End":"21:29.815","Text":"If this thing does converge that implies that the general term goes to 0,"},{"Start":"21:29.815 ","End":"21:34.600","Text":"that\u0027s a necessary condition that when n goes to"},{"Start":"21:34.600 ","End":"21:41.035","Text":"infinity that an goes to 0, this limit."},{"Start":"21:41.035 ","End":"21:46.180","Text":"Now, if an goes to 0 then"},{"Start":"21:46.180 ","End":"21:53.485","Text":"also the limit of absolute value of an also goes to 0."},{"Start":"21:53.485 ","End":"21:57.595","Text":"It\u0027s pretty clear if an keeps getting closer and close to 0,"},{"Start":"21:57.595 ","End":"22:02.620","Text":"and regardless of the sign even if I throw all the minuses to the other side,"},{"Start":"22:02.620 ","End":"22:06.039","Text":"they\u0027re still going to get closer and closer to 0."},{"Start":"22:06.039 ","End":"22:11.740","Text":"But actually, now I\u0027m going to make a computation that this is not true."},{"Start":"22:11.740 ","End":"22:15.800","Text":"This is actually equals to infinity."},{"Start":"22:18.330 ","End":"22:24.085","Text":"We reach a contradiction that it\u0027s not 0 it\u0027s infinity,"},{"Start":"22:24.085 ","End":"22:26.515","Text":"and so this thing can\u0027t converge."},{"Start":"22:26.515 ","End":"22:30.265","Text":"Perhaps I should write the words I will show."},{"Start":"22:30.265 ","End":"22:36.220","Text":"I\u0027ll move this. Here\u0027s our"},{"Start":"22:36.220 ","End":"22:43.090","Text":"an and our absolute value of an as we had before was 1 over 4n plus 1,"},{"Start":"22:43.090 ","End":"22:49.060","Text":"absolute value of this thing to the power of n. Now,"},{"Start":"22:49.060 ","End":"22:52.465","Text":"just to make it a bit shorter,"},{"Start":"22:52.465 ","End":"22:58.960","Text":"let me call this whole absolute value thing q."},{"Start":"22:58.960 ","End":"23:03.370","Text":"This is q^n."},{"Start":"23:03.370 ","End":"23:09.985","Text":"I can write it as q^n over 4n plus 1."},{"Start":"23:09.985 ","End":"23:15.430","Text":"What do I know? I know that q from here is bigger than 1."},{"Start":"23:15.430 ","End":"23:20.080","Text":"Q is bigger than 1 here."},{"Start":"23:20.080 ","End":"23:28.060","Text":"All I have to do is show this limit as n goes to infinity is infinity."},{"Start":"23:28.060 ","End":"23:30.550","Text":"It\u0027s not immediately obvious because"},{"Start":"23:30.550 ","End":"23:34.225","Text":"the numerator goes to infinity and so does the denominator."},{"Start":"23:34.225 ","End":"23:37.270","Text":"I\u0027m going to use the trick of moving from n to x,"},{"Start":"23:37.270 ","End":"23:42.820","Text":"and then we\u0027ll use L\u0027Hopital\u0027s theorem. Here it goes."},{"Start":"23:42.820 ","End":"23:48.550","Text":"What I want, instead of the limit as"},{"Start":"23:48.550 ","End":"23:54.745","Text":"n goes to infinity of q to the n over 4n plus 1,"},{"Start":"23:54.745 ","End":"24:00.025","Text":"I\u0027ll take it as the limit of a variable as x goes to infinity"},{"Start":"24:00.025 ","End":"24:07.165","Text":"of q to the x over 4x plus 1."},{"Start":"24:07.165 ","End":"24:14.170","Text":"Now, I\u0027m going to use L\u0027Hopital because we have an infinity over infinity case here."},{"Start":"24:14.170 ","End":"24:17.065","Text":"I wrote the name L,Hospital."},{"Start":"24:17.065 ","End":"24:19.420","Text":"We\u0027re going to replace this limit by"},{"Start":"24:19.420 ","End":"24:22.599","Text":"a different limit where we differentiate top and bottom."},{"Start":"24:22.599 ","End":"24:28.340","Text":"We get the limit as x goes to infinity."},{"Start":"24:28.520 ","End":"24:31.245","Text":"We differentiate the bottom,"},{"Start":"24:31.245 ","End":"24:33.675","Text":"the derivative is 4."},{"Start":"24:33.675 ","End":"24:38.575","Text":"The derivative of q^x is the general formula for a^x."},{"Start":"24:38.575 ","End":"24:43.390","Text":"It\u0027s just this thing to the x times natural log of q,"},{"Start":"24:43.390 ","End":"24:45.175","Text":"which is just some constant,"},{"Start":"24:45.175 ","End":"24:49.120","Text":"actually a positive constant since q is bigger than 1."},{"Start":"24:49.120 ","End":"24:52.880","Text":"Now, this thing is a constant."},{"Start":"24:53.100 ","End":"24:55.915","Text":"Because q is bigger than 1,"},{"Start":"24:55.915 ","End":"25:00.190","Text":"something bigger than 1^x goes to infinity."},{"Start":"25:00.190 ","End":"25:03.220","Text":"It\u0027s an exponential function with base bigger than 1,"},{"Start":"25:03.220 ","End":"25:09.760","Text":"so this thing is equal to infinity and therefore the limit is equal to infinity."},{"Start":"25:09.760 ","End":"25:11.995","Text":"We\u0027ve just shown that."},{"Start":"25:11.995 ","End":"25:15.250","Text":"All the rest of it we\u0027ve said before proves that"},{"Start":"25:15.250 ","End":"25:20.065","Text":"the original series diverges and so all we\u0027re left with,"},{"Start":"25:20.065 ","End":"25:25.879","Text":"the only convergent part is when x is bigger than 0."},{"Start":"25:25.879 ","End":"25:32.310","Text":"The fine lances of the interval of convergence is x bigger than 0."},{"Start":"25:32.310 ","End":"25:33.690","Text":"As we noted before,"},{"Start":"25:33.690 ","End":"25:36.330","Text":"we get a bonus is that not only do they converge,"},{"Start":"25:36.330 ","End":"25:38.310","Text":"but it converges absolutely,"},{"Start":"25:38.310 ","End":"25:44.170","Text":"but that\u0027s not required. We are done."}],"ID":7961},{"Watched":false,"Name":"Exercise 1 part b","Duration":"6m 11s","ChapterTopicVideoID":7900,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.495","Text":"In this exercise, we want to find the region of convergence of this series,"},{"Start":"00:06.495 ","End":"00:09.090","Text":"which is a series of functions in x."},{"Start":"00:09.090 ","End":"00:14.655","Text":"What we want to do is find for which values of x does this series converge."},{"Start":"00:14.655 ","End":"00:19.755","Text":"Right off the bat, I can tell you that x cannot equal 5."},{"Start":"00:19.755 ","End":"00:22.740","Text":"Because when x equals 5, this thing doesn\u0027t even make sense."},{"Start":"00:22.740 ","End":"00:27.115","Text":"So we have to exclude that from any region of convergence."},{"Start":"00:27.115 ","End":"00:31.910","Text":"Now what I\u0027d like to do here because of exponents and factorials is use what"},{"Start":"00:31.910 ","End":"00:36.815","Text":"is called the ratio test for convergence."},{"Start":"00:36.815 ","End":"00:40.250","Text":"But there\u0027s a snag because the ratio test works"},{"Start":"00:40.250 ","End":"00:43.970","Text":"for positive series or non-negative series."},{"Start":"00:43.970 ","End":"00:47.210","Text":"Here this denominator, this x minus 5,"},{"Start":"00:47.210 ","End":"00:49.054","Text":"could be positive or negative."},{"Start":"00:49.054 ","End":"00:55.820","Text":"So what we do is the standard trick is to start off by taking the absolute value of"},{"Start":"00:55.820 ","End":"00:59.345","Text":"the series and then"},{"Start":"00:59.345 ","End":"01:03.860","Text":"concluding stuff about this and inferring then about the original series."},{"Start":"01:03.860 ","End":"01:11.240","Text":"The absolute value of the series would be the sum and goes from 1 to infinity."},{"Start":"01:11.240 ","End":"01:13.340","Text":"Now everything here is positive,"},{"Start":"01:13.340 ","End":"01:15.860","Text":"except possibly the x minus 5."},{"Start":"01:15.860 ","End":"01:19.390","Text":"So we get 2^n as is,"},{"Start":"01:19.390 ","End":"01:21.140","Text":"n factorial as is,"},{"Start":"01:21.140 ","End":"01:26.300","Text":"but here I need an absolute value of x minus 5 to the power"},{"Start":"01:26.300 ","End":"01:31.695","Text":"of n. Let\u0027s just call the general term a_n."},{"Start":"01:31.695 ","End":"01:36.230","Text":"What we have here is the sum of a_n,"},{"Start":"01:36.230 ","End":"01:40.210","Text":"where a_n is this."},{"Start":"01:40.210 ","End":"01:44.280","Text":"Now the ratio test talks about"},{"Start":"01:44.280 ","End":"01:49.530","Text":"the limit as n"},{"Start":"01:49.530 ","End":"01:54.560","Text":"goes to infinity of a_n plus 1 over a_n."},{"Start":"01:54.560 ","End":"01:57.290","Text":"According to what this comes out,"},{"Start":"01:57.290 ","End":"02:00.960","Text":"we can tell about convergence."},{"Start":"02:00.960 ","End":"02:04.175","Text":"I think I\u0027ll remind you what the ratio test says."},{"Start":"02:04.175 ","End":"02:09.200","Text":"If this limit is less than 1,"},{"Start":"02:09.200 ","End":"02:17.675","Text":"then the original series converges some a_n."},{"Start":"02:17.675 ","End":"02:20.705","Text":"If it\u0027s equal to 1,"},{"Start":"02:20.705 ","End":"02:21.965","Text":"we don\u0027t know."},{"Start":"02:21.965 ","End":"02:23.480","Text":"Could be anything."},{"Start":"02:23.480 ","End":"02:32.075","Text":"If it\u0027s greater than 1, then it diverges."},{"Start":"02:32.075 ","End":"02:35.605","Text":"Okay. Let\u0027s see what we get in our case."},{"Start":"02:35.605 ","End":"02:37.500","Text":"Before doing the limit,"},{"Start":"02:37.500 ","End":"02:42.880","Text":"let\u0027s just see what is a_n plus 1 over a_n."},{"Start":"02:42.880 ","End":"02:45.130","Text":"What does this equal?"},{"Start":"02:45.130 ","End":"02:52.220","Text":"So a_n plus 1 would be what we get if we replace n by n plus 1."},{"Start":"02:52.220 ","End":"03:02.775","Text":"So it\u0027s 2 to the power of n plus 1 over n plus 1 factorial."},{"Start":"03:02.775 ","End":"03:09.500","Text":"Here, absolute value of x minus 5 to the power of n plus 1."},{"Start":"03:09.500 ","End":"03:15.109","Text":"Now dividing by a_n is like multiplying by the inverse."},{"Start":"03:15.109 ","End":"03:25.100","Text":"I can multiply this on the denominator 2^n and on the numerator n factorial x"},{"Start":"03:25.100 ","End":"03:30.260","Text":"minus 5 to the power of n. Now I\u0027m going"},{"Start":"03:30.260 ","End":"03:35.750","Text":"to rewrite the first factor so that things will cancel."},{"Start":"03:35.750 ","End":"03:38.990","Text":"What I will do is rewrite 2^n plus 1 as"},{"Start":"03:38.990 ","End":"03:42.530","Text":"2 times 2^n because it wanted to cancel with this."},{"Start":"03:42.530 ","End":"03:47.360","Text":"On the denominator, n plus 1 factorial."},{"Start":"03:47.360 ","End":"03:52.400","Text":"It\u0027s well-known that this is n plus 1 times n factorial."},{"Start":"03:52.400 ","End":"03:56.300","Text":"The product of the numbers from 1 to n plus 1 is just n plus"},{"Start":"03:56.300 ","End":"03:59.780","Text":"1 times the product from 1 to n. Again,"},{"Start":"03:59.780 ","End":"04:01.325","Text":"here there\u0027s an exponent,"},{"Start":"04:01.325 ","End":"04:08.735","Text":"so I\u0027ll write it as x minus 5 in absolute value and then x minus 5 to the power of"},{"Start":"04:08.735 ","End":"04:13.595","Text":"n. This I just copy as is and"},{"Start":"04:13.595 ","End":"04:20.700","Text":"factorial x minus 5^n over 2^n."},{"Start":"04:20.700 ","End":"04:22.785","Text":"Now I can do a lot of canceling."},{"Start":"04:22.785 ","End":"04:29.145","Text":"2^n with 2^n, n factorial with n factorial,"},{"Start":"04:29.145 ","End":"04:33.640","Text":"x minus 5^n with x minus 5^n."},{"Start":"04:33.640 ","End":"04:39.440","Text":"Now I can write the limit as n goes to infinity of a_n plus"},{"Start":"04:39.440 ","End":"04:45.080","Text":"1 over a_n as the limit of whatever remains here,"},{"Start":"04:45.080 ","End":"04:55.040","Text":"which is just 2 over n plus 1 times absolute value of x minus 5,"},{"Start":"04:55.040 ","End":"04:57.545","Text":"n goes to infinity."},{"Start":"04:57.545 ","End":"04:59.735","Text":"Now as far as n goes,"},{"Start":"04:59.735 ","End":"05:06.275","Text":"this and this are constants and n plus 1 goes to infinity also."},{"Start":"05:06.275 ","End":"05:10.885","Text":"So basically we get 2 over infinity,"},{"Start":"05:10.885 ","End":"05:13.545","Text":"we get a constant over infinity,"},{"Start":"05:13.545 ","End":"05:17.280","Text":"and this is going to equal 0."},{"Start":"05:17.280 ","End":"05:20.775","Text":"Because this is 0."},{"Start":"05:20.775 ","End":"05:25.130","Text":"We are in this case because 0 is less than 1,"},{"Start":"05:25.130 ","End":"05:29.400","Text":"and so this converges."},{"Start":"05:29.400 ","End":"05:35.515","Text":"But don\u0027t forget that we were dealing with the absolute value series."},{"Start":"05:35.515 ","End":"05:41.305","Text":"So if the absolute value series converges,"},{"Start":"05:41.305 ","End":"05:43.180","Text":"then the original series,"},{"Start":"05:43.180 ","End":"05:48.500","Text":"I\u0027ll just write it, converges absolutely,"},{"Start":"05:48.810 ","End":"05:52.600","Text":"which is better, but it\u0027s not what we were asked for."},{"Start":"05:52.600 ","End":"05:55.390","Text":"In short, whatever x is,"},{"Start":"05:55.390 ","End":"05:59.710","Text":"this thing converges except for the x not equal to 5."},{"Start":"05:59.710 ","End":"06:01.825","Text":"So the region of convergence,"},{"Start":"06:01.825 ","End":"06:05.575","Text":"the answer would be x not equal to 5."},{"Start":"06:05.575 ","End":"06:11.750","Text":"All x except 5. We\u0027re done."}],"ID":7962},{"Watched":false,"Name":"Exercise 1 part c","Duration":"16m 17s","ChapterTopicVideoID":7901,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"In this exercise we want to find the region of convergence of this series."},{"Start":"00:04.755 ","End":"00:12.449","Text":"I note right away that x cannot equal 4 so this is outside the region of convergence."},{"Start":"00:12.449 ","End":"00:16.290","Text":"It isn\u0027t even defined when x is 4."},{"Start":"00:16.290 ","End":"00:21.450","Text":"I\u0027d like to use either the ratio test or the root test,"},{"Start":"00:21.450 ","End":"00:25.000","Text":"but both of them need a non-negative series."},{"Start":"00:25.000 ","End":"00:30.440","Text":"What I\u0027ll do is if the original series is the sum of an,"},{"Start":"00:30.440 ","End":"00:32.180","Text":"we\u0027ll let this be an."},{"Start":"00:32.180 ","End":"00:39.920","Text":"What I\u0027m going to do is take the absolute value of the series,"},{"Start":"00:39.920 ","End":"00:44.540","Text":"other words, the sum from 1 to infinity of"},{"Start":"00:44.540 ","End":"00:50.595","Text":"absolute value of an and see if this converges or not."},{"Start":"00:50.595 ","End":"00:54.770","Text":"From this, I\u0027ll be able to deduce some things about the original series."},{"Start":"00:54.770 ","End":"00:58.115","Text":"Like I said, you could use the root test or the ratio test."},{"Start":"00:58.115 ","End":"01:02.065","Text":"I\u0027m going to use the ratio test."},{"Start":"01:02.065 ","End":"01:07.924","Text":"The ratio test tells us to look at the limit."},{"Start":"01:07.924 ","End":"01:12.215","Text":"In fact, for the ratio test we need to be strictly positive,"},{"Start":"01:12.215 ","End":"01:15.515","Text":"which we are, not just non-negative."},{"Start":"01:15.515 ","End":"01:19.130","Text":"Nothing here is 0, the numerator is not 0."},{"Start":"01:19.130 ","End":"01:22.835","Text":"We take the limit as n goes to infinity,"},{"Start":"01:22.835 ","End":"01:30.180","Text":"normally would be an plus 1 over an but our general term is absolute value"},{"Start":"01:30.180 ","End":"01:37.310","Text":"so we take the limit of the absolute value and see what this is."},{"Start":"01:37.310 ","End":"01:45.215","Text":"Then there are possibilities that it could be greater than 1,"},{"Start":"01:45.215 ","End":"01:49.695","Text":"it could be equal to 1,"},{"Start":"01:49.695 ","End":"01:53.710","Text":"and it could be less than 1."},{"Start":"01:58.010 ","End":"02:03.325","Text":"In this case, the absolute value series converges."},{"Start":"02:03.325 ","End":"02:10.625","Text":"This is the good case we want because then the original series also converges absolutely."},{"Start":"02:10.625 ","End":"02:17.030","Text":"When this is equal to 1 then we just don\u0027t know."},{"Start":"02:17.030 ","End":"02:19.500","Text":"I\u0027ll put a question mark."},{"Start":"02:20.020 ","End":"02:26.480","Text":"Let\u0027s first of all see what is an plus 1/an in absolute value."},{"Start":"02:26.480 ","End":"02:31.320","Text":"Before we take the limit what this equals is,"},{"Start":"02:31.320 ","End":"02:39.990","Text":"we\u0027re taking the absolute value so we don\u0027t need the minus 1 here."},{"Start":"02:39.990 ","End":"02:47.610","Text":"We just replace n by n plus 1 for the numerator."},{"Start":"02:47.610 ","End":"02:50.780","Text":"Then I guess at this we don\u0027t need at all in the absolute value."},{"Start":"02:50.780 ","End":"03:00.335","Text":"So we get 1 plus 2 because I\u0027m replacing n by n plus 1,"},{"Start":"03:00.335 ","End":"03:06.410","Text":"times 10^n plus 1 times x"},{"Start":"03:06.410 ","End":"03:12.870","Text":"minus 4^n plus 1."},{"Start":"03:12.870 ","End":"03:17.175","Text":"Then I\u0027m dividing by an and if I\u0027m dividing by"},{"Start":"03:17.175 ","End":"03:23.465","Text":"an then I\u0027m multiplying by the reciprocal."},{"Start":"03:23.465 ","End":"03:26.670","Text":"I\u0027m multiplying by just this thing here,"},{"Start":"03:26.670 ","End":"03:34.650","Text":"n plus 1, 10^n, x minus 4^n."},{"Start":"03:34.650 ","End":"03:36.180","Text":"Because of the absolute value,"},{"Start":"03:36.180 ","End":"03:40.440","Text":"I\u0027m ignoring the minus 1 to the power of."},{"Start":"03:40.440 ","End":"03:44.290","Text":"Then a lot of stuff will cancel here."},{"Start":"03:44.290 ","End":"03:52.600","Text":"But if we simplify this we\u0027ll get n plus 1 plus"},{"Start":"03:52.600 ","End":"04:00.055","Text":"2 and then we\u0027ll get"},{"Start":"04:00.055 ","End":"04:08.445","Text":"times 1/10 from the 10^n/10^n plus 1."},{"Start":"04:08.445 ","End":"04:16.200","Text":"We\u0027ll also get 1 over absolute value of x minus 4."},{"Start":"04:17.030 ","End":"04:24.380","Text":"Now, if I want to take the limit as n goes to"},{"Start":"04:24.380 ","End":"04:31.755","Text":"infinity of absolute value of an plus 1/an is just equal to,"},{"Start":"04:31.755 ","End":"04:35.390","Text":"this bit has nothing to do with n. It\u0027s a constant."},{"Start":"04:35.390 ","End":"04:37.040","Text":"I can pull it outside."},{"Start":"04:37.040 ","End":"04:43.105","Text":"It\u0027s 1/10 times absolute value of x minus 4."},{"Start":"04:43.105 ","End":"04:45.565","Text":"Then I need the limit."},{"Start":"04:45.565 ","End":"04:51.130","Text":"As n goes to infinity of n"},{"Start":"04:51.130 ","End":"04:58.430","Text":"plus 1 plus 2."},{"Start":"04:58.430 ","End":"05:00.740","Text":"I claim that this is exactly equal to 1,"},{"Start":"05:00.740 ","End":"05:05.120","Text":"this limit, and I can show you why."},{"Start":"05:05.120 ","End":"05:06.500","Text":"I\u0027ll do it here at the side."},{"Start":"05:06.500 ","End":"05:12.335","Text":"n plus 1 plus 2 is equal to n,"},{"Start":"05:12.335 ","End":"05:18.520","Text":"instead of the plus 1 I can write it as plus 2 minus 1 plus 2."},{"Start":"05:18.520 ","End":"05:19.830","Text":"Why is this good?"},{"Start":"05:19.830 ","End":"05:22.890","Text":"Because I can write n plus 2 plus 2 is 1"},{"Start":"05:22.890 ","End":"05:30.930","Text":"minus 1 plus 2."},{"Start":"05:30.930 ","End":"05:38.090","Text":"In the limit what I\u0027ll get is 1 minus 1/infinity plus 2,"},{"Start":"05:38.090 ","End":"05:46.760","Text":"so to speak, and this is equal to 1 because this goes to infinity."},{"Start":"05:46.760 ","End":"05:48.110","Text":"1 over infinity is 0,"},{"Start":"05:48.110 ","End":"05:52.055","Text":"1 minus 0 is 1 so this is 1."},{"Start":"05:52.055 ","End":"05:59.435","Text":"Our limit is just equal to 1/10,"},{"Start":"05:59.435 ","End":"06:03.360","Text":"absolute value of x minus 4."},{"Start":"06:05.830 ","End":"06:10.490","Text":"Let\u0027s look for where this is less than 1."},{"Start":"06:10.490 ","End":"06:18.095","Text":"If I take 1/10 times absolute value of x minus 4,"},{"Start":"06:18.095 ","End":"06:27.680","Text":"less than 1, what I can do is invert the both sides."},{"Start":"06:27.680 ","End":"06:32.660","Text":"If I take the reciprocal of 2 positive things it reverses the inequality."},{"Start":"06:32.660 ","End":"06:39.340","Text":"So I get 10 times absolute value of x minus 4 is bigger than 1."},{"Start":"06:39.340 ","End":"06:43.455","Text":"1 over something that\u0027s less than 1 it\u0027s got to be bigger than 1 basically."},{"Start":"06:43.455 ","End":"06:49.800","Text":"That will give us that absolute value of x minus"},{"Start":"06:49.800 ","End":"06:56.700","Text":"4 is bigger than 1/10."},{"Start":"06:56.700 ","End":"07:02.900","Text":"When we have the absolute value bigger than something, in general,"},{"Start":"07:02.900 ","End":"07:09.920","Text":"this will give us that either x is bigger than 4 and"},{"Start":"07:09.920 ","End":"07:18.135","Text":"1/10 or x is less than 4 minus 1/10,"},{"Start":"07:18.135 ","End":"07:21.480","Text":"which is 3 and 9/10."},{"Start":"07:21.480 ","End":"07:26.580","Text":"I suppose I should have worked in decimals, it doesn\u0027t matter."},{"Start":"07:26.580 ","End":"07:29.890","Text":"Let me highlight this."},{"Start":"07:30.080 ","End":"07:34.870","Text":"In these 2 intervals"},{"Start":"07:34.870 ","End":"07:40.430","Text":"we have that the series of absolute values converges and so the original series,"},{"Start":"07:40.430 ","End":"07:46.530","Text":"write it for short Sigma an, converges absolutely."},{"Start":"07:46.620 ","End":"07:49.270","Text":"We weren\u0027t really asked about absolutely,"},{"Start":"07:49.270 ","End":"07:50.710","Text":"so I\u0027ll put that in brackets."},{"Start":"07:50.710 ","End":"07:55.300","Text":"But certainly we found where it converges absolutely."},{"Start":"07:55.300 ","End":"07:58.630","Text":"Now, the question is what happens at the borders and what happens"},{"Start":"07:58.630 ","End":"08:03.340","Text":"between 3 9/10 and 4 1/10."},{"Start":"08:03.340 ","End":"08:08.470","Text":"Let\u0027s check at each of these points what happens to the original series."},{"Start":"08:08.470 ","End":"08:13.210","Text":"If x equals 4 1/10,"},{"Start":"08:13.210 ","End":"08:17.830","Text":"then what we get is that the sum of an,"},{"Start":"08:17.830 ","End":"08:21.445","Text":"I don\u0027t always write n goes from 1 to infinity, it gets tedious,"},{"Start":"08:21.445 ","End":"08:26.785","Text":"is equal to the series."},{"Start":"08:26.785 ","End":"08:29.230","Text":"Where is the original series?"},{"Start":"08:29.230 ","End":"08:33.280","Text":"I\u0027ve lost it. Here it is."},{"Start":"08:33.280 ","End":"08:39.040","Text":"What we get is minus 1 to the power of n plus"},{"Start":"08:39.040 ","End":"08:45.940","Text":"1 over n plus 1."},{"Start":"08:45.940 ","End":"08:49.270","Text":"Now, if x is 4 1/10,"},{"Start":"08:49.270 ","End":"08:52.630","Text":"4 1/10 minus 4 is 1/10,"},{"Start":"08:52.630 ","End":"08:55.990","Text":"and 1/10 to the n times 10 to the n is 1."},{"Start":"08:55.990 ","End":"08:59.005","Text":"This bit just becomes 1,"},{"Start":"08:59.005 ","End":"09:03.260","Text":"so, actually this is all the series."},{"Start":"09:03.270 ","End":"09:09.490","Text":"I\u0027ll just write the n goes from 1 to infinity here."},{"Start":"09:09.490 ","End":"09:17.635","Text":"Let me just note that if n is 3 and 9/10,"},{"Start":"09:17.635 ","End":"09:25.735","Text":"then here I have minus 1/10."},{"Start":"09:25.735 ","End":"09:29.290","Text":"Here I\u0027ll get minus 1 to the n, so,"},{"Start":"09:29.290 ","End":"09:32.785","Text":"altogether after cancellation I\u0027ll just get minus 1."},{"Start":"09:32.785 ","End":"09:38.590","Text":"I\u0027m going to write that now that if x is equal to 3 and 9/10,"},{"Start":"09:38.590 ","End":"09:43.930","Text":"then an is the sum of"},{"Start":"09:43.930 ","End":"09:50.515","Text":"just minus 1 over n plus 1,"},{"Start":"09:50.515 ","End":"09:54.010","Text":"because we had, remember a minus 1 to the n in the denominator, so,"},{"Start":"09:54.010 ","End":"09:56.995","Text":"just minus 1 unrelated to n,"},{"Start":"09:56.995 ","End":"09:59.110","Text":"also from 1 to infinity."},{"Start":"09:59.110 ","End":"10:01.720","Text":"Maybe write it here also."},{"Start":"10:01.720 ","End":"10:03.985","Text":"1 to infinity."},{"Start":"10:03.985 ","End":"10:08.380","Text":"Now, this is like the harmonic series."},{"Start":"10:08.380 ","End":"10:10.270","Text":"If you pull out the minus 1 in front,"},{"Start":"10:10.270 ","End":"10:12.190","Text":"1 over n plus 1 or 1 over n,"},{"Start":"10:12.190 ","End":"10:13.750","Text":"it\u0027s just shifted by 1."},{"Start":"10:13.750 ","End":"10:16.880","Text":"This is divergent."},{"Start":"10:17.730 ","End":"10:23.470","Text":"Basically the harmonic series or slight variant of it."},{"Start":"10:23.470 ","End":"10:32.860","Text":"This is convergent because it\u0027s an alternating Leibniz series. Look that up."},{"Start":"10:32.860 ","End":"10:34.060","Text":"If we have something,"},{"Start":"10:34.060 ","End":"10:42.550","Text":"a general term goes to 0 and it alternates plus and minus."},{"Start":"10:42.550 ","End":"10:46.150","Text":"Sometimes it goes to 0 monotonically,"},{"Start":"10:46.150 ","End":"10:49.435","Text":"decreases to 0 and then we take a plus and minus."},{"Start":"10:49.435 ","End":"10:51.280","Text":"We get a convergent series."},{"Start":"10:51.280 ","End":"10:54.175","Text":"It\u0027s actually conditionally convergent."},{"Start":"10:54.175 ","End":"11:02.410","Text":"Meanwhile, what we have is also x equals 4 1/10 is in."},{"Start":"11:02.410 ","End":"11:04.795","Text":"I\u0027ll highlight this."},{"Start":"11:04.795 ","End":"11:09.025","Text":"This is also where the original series converges."},{"Start":"11:09.025 ","End":"11:18.500","Text":"All we have left to check is between 3 and 3 9/10 and 4 1/10."},{"Start":"11:18.510 ","End":"11:24.610","Text":"Actually, these 2 correspond to the case where this thing is equal to 1,"},{"Start":"11:24.610 ","End":"11:25.885","Text":"and the third case,"},{"Start":"11:25.885 ","End":"11:27.100","Text":"it doesn\u0027t matter where x is."},{"Start":"11:27.100 ","End":"11:28.150","Text":"I want to take the third case,"},{"Start":"11:28.150 ","End":"11:30.595","Text":"is this being bigger than 1."},{"Start":"11:30.595 ","End":"11:35.260","Text":"We\u0027re going to check if 1 over 10,"},{"Start":"11:35.260 ","End":"11:39.070","Text":"x minus 4 an absolute value is bigger than 1,"},{"Start":"11:39.070 ","End":"11:47.380","Text":"which happens to be between 3 and 9 1/10 and 4 1/10."},{"Start":"11:47.380 ","End":"11:48.940","Text":"If this is bigger than 1,"},{"Start":"11:48.940 ","End":"11:54.235","Text":"I\u0027m going to show that our original series is divergent."},{"Start":"11:54.235 ","End":"11:58.420","Text":"First, let\u0027s look at the absolute value series,"},{"Start":"11:58.420 ","End":"12:00.040","Text":"the general term an,"},{"Start":"12:00.040 ","End":"12:02.470","Text":"somehow I\u0027ve lost it."},{"Start":"12:02.470 ","End":"12:04.465","Text":"I\u0027ll just rewrite it."},{"Start":"12:04.465 ","End":"12:12.115","Text":"It\u0027s 1 over, what was it now?"},{"Start":"12:12.115 ","End":"12:21.980","Text":"It was n plus 1 times 10 to"},{"Start":"12:21.980 ","End":"12:27.490","Text":"the n. Absolute value of x minus 4 to"},{"Start":"12:27.490 ","End":"12:33.925","Text":"the power of n. Now if I label this as q,"},{"Start":"12:33.925 ","End":"12:38.035","Text":"I get that q is bigger than 1 and I can rewrite this"},{"Start":"12:38.035 ","End":"12:46.300","Text":"as just q to the n over n plus 1."},{"Start":"12:46.300 ","End":"12:53.245","Text":"Now, we\u0027ve done a similar thing to this before."},{"Start":"12:53.245 ","End":"12:58.190","Text":"This in fact tends to infinity."},{"Start":"13:01.620 ","End":"13:04.120","Text":"As n goes to infinity."},{"Start":"13:04.120 ","End":"13:06.320","Text":"I\u0027ll quickly show you again at the side."},{"Start":"13:06.320 ","End":"13:10.380","Text":"To prove things like this, we can change n to an x,"},{"Start":"13:10.380 ","End":"13:19.780","Text":"so I could look at the limit as x goes to infinity of q to the x over x plus 1,"},{"Start":"13:19.830 ","End":"13:23.260","Text":"and now what I can do, is say by L\u0027Hopital\u0027s."},{"Start":"13:23.260 ","End":"13:26.200","Text":"Well, you have to notice first,"},{"Start":"13:26.200 ","End":"13:27.730","Text":"when x goes to infinity,"},{"Start":"13:27.730 ","End":"13:30.370","Text":"if a number is larger than 1 and to the power of x,"},{"Start":"13:30.370 ","End":"13:31.990","Text":"it also goes to infinity."},{"Start":"13:31.990 ","End":"13:34.150","Text":"We have an infinity over infinity,"},{"Start":"13:34.150 ","End":"13:37.180","Text":"and using L\u0027Hopital\u0027s rule we can differentiate"},{"Start":"13:37.180 ","End":"13:40.810","Text":"numerator and denominator and get a different limit."},{"Start":"13:40.810 ","End":"13:42.415","Text":"X goes to infinity."},{"Start":"13:42.415 ","End":"13:44.515","Text":"The derivative of this, if you look it up,"},{"Start":"13:44.515 ","End":"13:48.355","Text":"is just q to the x natural log of q,"},{"Start":"13:48.355 ","End":"13:51.190","Text":"and the derivative of this is 1,"},{"Start":"13:51.190 ","End":"13:52.900","Text":"and when x goes to infinity,"},{"Start":"13:52.900 ","End":"13:54.265","Text":"this is just a constant."},{"Start":"13:54.265 ","End":"13:57.175","Text":"In fact, it\u0027s a positive constant when q is bigger than 1,"},{"Start":"13:57.175 ","End":"14:01.525","Text":"and q to the x goes to infinity."},{"Start":"14:01.525 ","End":"14:05.545","Text":"This is infinity that explains this."},{"Start":"14:05.545 ","End":"14:07.870","Text":"I don\u0027t care that it\u0027s infinity,"},{"Start":"14:07.870 ","End":"14:11.890","Text":"but what I do care about,"},{"Start":"14:11.890 ","End":"14:15.910","Text":"is that an does not turn to 0."},{"Start":"14:15.910 ","End":"14:21.020","Text":"Yeah. Does not turn to 0 as n goes to infinity."},{"Start":"14:23.730 ","End":"14:28.330","Text":"I\u0027ve used this to remark before, trick if you like."},{"Start":"14:28.330 ","End":"14:32.184","Text":"That if the absolute value of an doesn\u0027t go to 0,"},{"Start":"14:32.184 ","End":"14:38.335","Text":"then the an itself doesn\u0027t turn to 0."},{"Start":"14:38.335 ","End":"14:40.750","Text":"Because if a series goes to 0,"},{"Start":"14:40.750 ","End":"14:45.130","Text":"then absolute value also goes to 0 because it\u0027s getting closer and closer to 0."},{"Start":"14:45.130 ","End":"14:48.775","Text":"It doesn\u0027t matter if you shift it from left to right or right to left,"},{"Start":"14:48.775 ","End":"14:51.355","Text":"it\u0027s still going to get closer and closer to 0,"},{"Start":"14:51.355 ","End":"14:54.820","Text":"and if a general term doesn\u0027t go to 0,"},{"Start":"14:54.820 ","End":"14:59.590","Text":"then that means that the series is divergent because one of"},{"Start":"14:59.590 ","End":"15:07.300","Text":"the basic properties of convergence series that the general term turns to 0."},{"Start":"15:07.300 ","End":"15:10.180","Text":"This is usually used as a divergence test."},{"Start":"15:10.180 ","End":"15:12.550","Text":"If you want to show a series is divergent and you show"},{"Start":"15:12.550 ","End":"15:16.555","Text":"the general term doesn\u0027t turn to 0, then you\u0027ve got it."},{"Start":"15:16.555 ","End":"15:26.140","Text":"We\u0027ve got nothing new to add to our collection of xs for which the series is convergence,"},{"Start":"15:26.140 ","End":"15:28.400","Text":"so, I\u0027ll just summarize."},{"Start":"15:29.190 ","End":"15:32.800","Text":"It can say that the series,"},{"Start":"15:32.800 ","End":"15:34.720","Text":"the sum of an,"},{"Start":"15:34.720 ","End":"15:40.420","Text":"our original one, converges for,"},{"Start":"15:40.420 ","End":"15:42.530","Text":"if I don\u0027t care about absolutely,"},{"Start":"15:42.530 ","End":"15:45.350","Text":"I can combine these and say,"},{"Start":"15:45.350 ","End":"15:49.820","Text":"x bigger or equal to 4 1/10,"},{"Start":"15:49.820 ","End":"15:51.990","Text":"because I\u0027ve included the 4 tenths,"},{"Start":"15:51.990 ","End":"15:59.620","Text":"or x less than 3 and 9 tenths."},{"Start":"15:59.620 ","End":"16:04.360","Text":"In fact, it converges absolutely except that exactly 4 1/10,"},{"Start":"16:04.360 ","End":"16:06.695","Text":"where it converges only conditionally."},{"Start":"16:06.695 ","End":"16:14.300","Text":"But I would leave my answer like this."},{"Start":"16:14.610 ","End":"16:17.900","Text":"We are done."}],"ID":7963},{"Watched":false,"Name":"Exercise 1 part d","Duration":"10m 51s","ChapterTopicVideoID":7902,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this exercise, we want to find the region of convergence of the following series."},{"Start":"00:05.175 ","End":"00:09.585","Text":"Notice it contains a natural logarithm here."},{"Start":"00:09.585 ","End":"00:13.829","Text":"The first thing I want to do is find out if there\u0027s any restrictions"},{"Start":"00:13.829 ","End":"00:18.165","Text":"on x as far as the domain where this series isn\u0027t even defined."},{"Start":"00:18.165 ","End":"00:21.210","Text":"For let the general term be a_n,"},{"Start":"00:21.210 ","End":"00:27.510","Text":"which is 1 over n. I can slightly rewrite this as"},{"Start":"00:27.510 ","End":"00:35.170","Text":"the natural log of nx all to the power of 4."},{"Start":"00:35.780 ","End":"00:40.945","Text":"There\u0027s a couple of things I have to make sure because of the natural log."},{"Start":"00:40.945 ","End":"00:45.485","Text":"I have to have a positive nx."},{"Start":"00:45.485 ","End":"00:48.575","Text":"If nx is positive,"},{"Start":"00:48.575 ","End":"00:52.670","Text":"then that means that x has to be positive."},{"Start":"00:52.670 ","End":"00:57.350","Text":"First of all, we\u0027re restricting ourselves to positive x on the region."},{"Start":"00:57.350 ","End":"01:01.475","Text":"The other thing that could go wrong is that this thing could be 0."},{"Start":"01:01.475 ","End":"01:03.545","Text":"When can this be 0?"},{"Start":"01:03.545 ","End":"01:05.485","Text":"If nx is 1,"},{"Start":"01:05.485 ","End":"01:13.010","Text":"so nx cannot equal 1 which means that x cannot equal 1 over n,"},{"Start":"01:13.010 ","End":"01:18.170","Text":"but it can equal that for any n because all the terms have to be defined."},{"Start":"01:18.170 ","End":"01:22.320","Text":"I mean, x is not equal to 1,"},{"Start":"01:22.320 ","End":"01:26.010","Text":"1/2, 1/3, 1/4 and so on."},{"Start":"01:26.010 ","End":"01:28.280","Text":"We\u0027re positive, I mean,"},{"Start":"01:28.280 ","End":"01:32.005","Text":"x is positive and not equal to 1 of these things."},{"Start":"01:32.005 ","End":"01:37.800","Text":"Then we still have to look within this where it converges."},{"Start":"01:37.800 ","End":"01:47.195","Text":"What I\u0027m going to do is use what is called the integral test for convergence of a series."},{"Start":"01:47.195 ","End":"01:50.345","Text":"To use it, we require, first of all,"},{"Start":"01:50.345 ","End":"01:57.485","Text":"that a_n be a positive series or at least non-negative, I\u0027m not sure."},{"Start":"01:57.485 ","End":"02:01.130","Text":"In our case we are positive because n is positive,"},{"Start":"02:01.130 ","End":"02:05.690","Text":"and we\u0027ve made sure that the natural log is not 0,"},{"Start":"02:05.690 ","End":"02:08.945","Text":"anything not 0 to the power of 4 is positive."},{"Start":"02:08.945 ","End":"02:16.060","Text":"We also need a_n to be a decreasing series."},{"Start":"02:16.060 ","End":"02:19.410","Text":"That\u0027s also true."},{"Start":"02:19.410 ","End":"02:20.700","Text":"I\u0027m not going to do it rigidly,"},{"Start":"02:20.700 ","End":"02:27.450","Text":"but because nx increases,"},{"Start":"02:27.450 ","End":"02:29.840","Text":"for a given x, x is going to be constant here."},{"Start":"02:29.840 ","End":"02:36.635","Text":"But as n increases and x also increases because x is positive."},{"Start":"02:36.635 ","End":"02:41.210","Text":"The natural log is an increasing function also increases into the power of"},{"Start":"02:41.210 ","End":"02:46.290","Text":"4 still increases, and n increases."},{"Start":"02:46.290 ","End":"02:50.170","Text":"Altogether successive terms get smaller and smaller."},{"Start":"02:52.910 ","End":"03:00.760","Text":"The last thing we need is that a_n doesn\u0027t just decrease but it actually goes down to 0."},{"Start":"03:01.430 ","End":"03:05.150","Text":"The reason that this goes to 0, well,"},{"Start":"03:05.150 ","End":"03:07.145","Text":"if we look at the denominator here,"},{"Start":"03:07.145 ","End":"03:10.200","Text":"n goes to infinity."},{"Start":"03:10.600 ","End":"03:20.389","Text":"Also, natural log of nx for a given x also goes to infinity."},{"Start":"03:20.389 ","End":"03:22.235","Text":"The reason for this,"},{"Start":"03:22.235 ","End":"03:24.575","Text":"I can go through in some intermediate steps,"},{"Start":"03:24.575 ","End":"03:28.935","Text":"n goes to infinity. I made some room."},{"Start":"03:28.935 ","End":"03:32.015","Text":"Then nx also goes to infinity,"},{"Start":"03:32.015 ","End":"03:38.560","Text":"and the natural logarithm of nx goes to infinity."},{"Start":"03:38.560 ","End":"03:40.760","Text":"If I take it to the power of 4,"},{"Start":"03:40.760 ","End":"03:42.740","Text":"it still goes to infinity."},{"Start":"03:42.740 ","End":"03:49.295","Text":"What I have is 1 over infinity which is 0,"},{"Start":"03:49.295 ","End":"03:51.950","Text":"infinity times infinity if you like,"},{"Start":"03:51.950 ","End":"03:54.500","Text":"or 1 over infinity times infinity to the fourth,"},{"Start":"03:54.500 ","End":"03:56.810","Text":"anyway, it just comes out to be 0."},{"Start":"03:56.810 ","End":"04:00.455","Text":"All the conditions of the integral test are met."},{"Start":"04:00.455 ","End":"04:08.255","Text":"By the convergence of the series."},{"Start":"04:08.255 ","End":"04:11.720","Text":"I\u0027ll write it again,1 to infinity."},{"Start":"04:11.720 ","End":"04:13.580","Text":"Well, I\u0027ll just write a_n."},{"Start":"04:13.580 ","End":"04:20.629","Text":"This converges, if and only if a certain integral converges."},{"Start":"04:20.629 ","End":"04:21.830","Text":"What is that integral?"},{"Start":"04:21.830 ","End":"04:32.030","Text":"It\u0027s the integral from 1 to infinity of what we get here if we replace n by a variable."},{"Start":"04:32.030 ","End":"04:39.390","Text":"Now, x is considered to be like a constant when we take the limit."},{"Start":"04:39.390 ","End":"04:41.170","Text":"I mean, x varies,"},{"Start":"04:41.170 ","End":"04:44.419","Text":"but when we take the limit, it\u0027s a constant."},{"Start":"04:44.419 ","End":"04:47.285","Text":"We\u0027re letting n go to infinity."},{"Start":"04:47.285 ","End":"04:49.925","Text":"We replace n by a variable,"},{"Start":"04:49.925 ","End":"04:55.240","Text":"not the x. I\u0027ll replace it by t. It\u0027s 1"},{"Start":"04:55.240 ","End":"05:04.500","Text":"over t natural log^4 times tx,"},{"Start":"05:04.500 ","End":"05:06.900","Text":"and this is going to be dt."},{"Start":"05:06.900 ","End":"05:12.920","Text":"As I said, x is a constant at least as far as t goes."},{"Start":"05:12.920 ","End":"05:16.800","Text":"I forgot to write the word converges."},{"Start":"05:17.270 ","End":"05:20.220","Text":"This converges if this integral converges."},{"Start":"05:20.220 ","End":"05:21.920","Text":"What do we mean by an integral converging?"},{"Start":"05:21.920 ","End":"05:25.775","Text":"Well, it\u0027s an improper integral and notice the infinity here."},{"Start":"05:25.775 ","End":"05:29.825","Text":"What we do is we take some other letter and that it turn to infinity."},{"Start":"05:29.825 ","End":"05:35.400","Text":"This integral is equal to the limit,"},{"Start":"05:35.400 ","End":"05:39.910","Text":"let\u0027s use letter b."},{"Start":"05:39.920 ","End":"05:43.140","Text":"It\u0027s a lower limit, I replace it by a,"},{"Start":"05:43.140 ","End":"05:48.585","Text":"and upper limit I replace it by b of same thing,"},{"Start":"05:48.585 ","End":"05:57.800","Text":"1 over t natural log^4 of tx, dt."},{"Start":"05:57.800 ","End":"06:00.720","Text":"I think we\u0027ll do this with a substitution."},{"Start":"06:00.800 ","End":"06:05.400","Text":"We let u equal natural log of tx,"},{"Start":"06:05.400 ","End":"06:08.620","Text":"I\u0027d actually rather call it xt,"},{"Start":"06:08.620 ","End":"06:10.750","Text":"t is my variable,"},{"Start":"06:10.750 ","End":"06:12.920","Text":"some constant times t,"},{"Start":"06:12.920 ","End":"06:18.630","Text":"Then du is equal to,"},{"Start":"06:18.630 ","End":"06:22.280","Text":"the derivative of this is 1"},{"Start":"06:22.280 ","End":"06:29.419","Text":"over xt because of the natural log and the inner derivative,"},{"Start":"06:29.419 ","End":"06:32.010","Text":"which is x dt."},{"Start":"06:32.010 ","End":"06:36.930","Text":"The x cancels with the x,"},{"Start":"06:36.930 ","End":"06:39.400","Text":"it\u0027s just 1 over t dt."},{"Start":"06:39.500 ","End":"06:44.090","Text":"Also, I\u0027m going to substitute the limits when t equals 1,"},{"Start":"06:44.090 ","End":"06:45.860","Text":"let\u0027s see what u equals,"},{"Start":"06:45.860 ","End":"06:47.705","Text":"and when t equals b,"},{"Start":"06:47.705 ","End":"06:50.105","Text":"let\u0027s see what u equals."},{"Start":"06:50.105 ","End":"06:57.885","Text":"t equals 1, then u is natural log of x."},{"Start":"06:57.885 ","End":"07:00.515","Text":"When t equals b,"},{"Start":"07:00.515 ","End":"07:08.910","Text":"then u equals natural log of xb or bx."},{"Start":"07:09.580 ","End":"07:15.595","Text":"We get, and I need some space,"},{"Start":"07:15.595 ","End":"07:21.295","Text":"the limit as b goes to infinity."},{"Start":"07:21.295 ","End":"07:30.270","Text":"The integral, this time it\u0027s integral du from natural log of x,"},{"Start":"07:30.270 ","End":"07:32.430","Text":"x is the constant here,"},{"Start":"07:32.430 ","End":"07:37.245","Text":"to natural log of bx."},{"Start":"07:37.245 ","End":"07:40.420","Text":"Put brackets, it doesn\u0027t matter."},{"Start":"07:40.790 ","End":"07:47.655","Text":"Now dt over t is du,"},{"Start":"07:47.655 ","End":"07:50.730","Text":"I just get du over."},{"Start":"07:50.730 ","End":"07:53.790","Text":"This thing is u^4."},{"Start":"07:53.790 ","End":"07:56.060","Text":"Now we can start doing this."},{"Start":"07:56.060 ","End":"08:00.860","Text":"The integral of 1 over u^4 du is equal,"},{"Start":"08:00.860 ","End":"08:05.675","Text":"to maybe do a quick exercise at the side,"},{"Start":"08:05.675 ","End":"08:09.425","Text":"u^ minus 4, I won\u0027t even bother writing the du."},{"Start":"08:09.425 ","End":"08:11.690","Text":"Raise the power by 1."},{"Start":"08:11.690 ","End":"08:17.640","Text":"It\u0027s u^ minus 3 and divide by minus 3, another constant."},{"Start":"08:18.320 ","End":"08:24.905","Text":"We can take the 1 over minus 3 in front of the integral and the limit,"},{"Start":"08:24.905 ","End":"08:32.280","Text":"so we got minus 1/3 the integral of,"},{"Start":"08:32.280 ","End":"08:34.725","Text":"sorry, not the integral,"},{"Start":"08:34.725 ","End":"08:38.640","Text":"of 1 over u cubed,"},{"Start":"08:38.640 ","End":"08:40.765","Text":"that\u0027s from the u^ minus 3,"},{"Start":"08:40.765 ","End":"08:45.335","Text":"taken between natural log of"},{"Start":"08:45.335 ","End":"08:53.240","Text":"bx and natural log of x. I forgot the limit."},{"Start":"08:53.240 ","End":"08:56.400","Text":"Fix that, put the limit in."},{"Start":"09:00.740 ","End":"09:03.820","Text":"The minus 1/3 is not affecting converges,"},{"Start":"09:03.820 ","End":"09:05.430","Text":"I\u0027m just dragging it along."},{"Start":"09:05.430 ","End":"09:11.540","Text":"It\u0027s minus 1/3 the limit as b goes to infinity."},{"Start":"09:11.540 ","End":"09:15.545","Text":"If I substitute natural log of bx,"},{"Start":"09:15.545 ","End":"09:21.710","Text":"I get 1 over natural log cubed of"},{"Start":"09:21.710 ","End":"09:29.700","Text":"bx minus 1 over natural log cubed of x."},{"Start":"09:30.040 ","End":"09:34.865","Text":"Just as before or similarly to before,"},{"Start":"09:34.865 ","End":"09:41.315","Text":"I\u0027m claiming that this denominator goes to infinity."},{"Start":"09:41.315 ","End":"09:43.940","Text":"Look, b goes to infinity."},{"Start":"09:43.940 ","End":"09:45.650","Text":"Now x is positive,"},{"Start":"09:45.650 ","End":"09:49.175","Text":"so bx also goes to infinity."},{"Start":"09:49.175 ","End":"09:51.440","Text":"The natural log of infinity is infinity."},{"Start":"09:51.440 ","End":"09:56.575","Text":"The natural log of bx also goes to infinity."},{"Start":"09:56.575 ","End":"10:05.720","Text":"Therefore, natural log cubed of bx goes to infinity."},{"Start":"10:05.720 ","End":"10:08.810","Text":"If the denominator goes to infinity,"},{"Start":"10:08.810 ","End":"10:13.995","Text":"then this goes to infinity,"},{"Start":"10:13.995 ","End":"10:17.650","Text":"so 1 over it goes to 0."},{"Start":"10:19.370 ","End":"10:22.010","Text":"We see that this thing converges."},{"Start":"10:22.010 ","End":"10:26.960","Text":"We can actually compute the limit because the limit will just equal,"},{"Start":"10:26.960 ","End":"10:29.600","Text":"the minus with the minus will cancel,"},{"Start":"10:29.600 ","End":"10:38.375","Text":"and I\u0027ll just get 1 over 3 times natural log cubed of x."},{"Start":"10:38.375 ","End":"10:41.810","Text":"It doesn\u0027t matter, it\u0027s a finite number and this thing converges."},{"Start":"10:41.810 ","End":"10:47.835","Text":"Because this converges, then our original series converges,"},{"Start":"10:47.835 ","End":"10:51.570","Text":"and, yeah, we\u0027re done."}],"ID":7964},{"Watched":false,"Name":"Exercise 1 part e","Duration":"6m 58s","ChapterTopicVideoID":7903,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we want to find the region of convergence of this series."},{"Start":"00:04.230 ","End":"00:06.300","Text":"In other words, you see x here."},{"Start":"00:06.300 ","End":"00:10.275","Text":"We have to find out for which x this series converges."},{"Start":"00:10.275 ","End":"00:13.470","Text":"Sometimes we also like to know what it converges absolutely,"},{"Start":"00:13.470 ","End":"00:16.470","Text":"but that was not a requirement here."},{"Start":"00:16.470 ","End":"00:22.755","Text":"Now, I\u0027d like you to go and look up p-series."},{"Start":"00:22.755 ","End":"00:31.270","Text":"I want to strike the term p-series in the chapter on series of numbers."},{"Start":"00:31.490 ","End":"00:36.405","Text":"I\u0027ll just remind you of the main result that if we have a series,"},{"Start":"00:36.405 ","End":"00:40.800","Text":"the sum n goes from 1 to infinity or it could"},{"Start":"00:40.800 ","End":"00:45.604","Text":"be from some other number higher than 1 of 1"},{"Start":"00:45.604 ","End":"00:49.925","Text":"over n to the power of p. It\u0027s usually given in the form of p."},{"Start":"00:49.925 ","End":"00:55.955","Text":"Then this has 2 possibilities."},{"Start":"00:55.955 ","End":"01:01.430","Text":"If p is bigger than 1, then it converges,"},{"Start":"01:01.430 ","End":"01:09.350","Text":"and if p is less than or equal to 1, it diverges."},{"Start":"01:09.350 ","End":"01:14.565","Text":"Now the bigger than 1 part\u0027s going to help us because I want to take my series."},{"Start":"01:14.565 ","End":"01:16.970","Text":"Our series is not exactly a p-series."},{"Start":"01:16.970 ","End":"01:19.235","Text":"I mean, I could replace x by p,"},{"Start":"01:19.235 ","End":"01:21.080","Text":"and that\u0027s a cosmetic difference,"},{"Start":"01:21.080 ","End":"01:22.610","Text":"but there\u0027s here an alternation,"},{"Start":"01:22.610 ","End":"01:28.840","Text":"a minus 1 to the n. What I\u0027d like to do is divide it into more cases."},{"Start":"01:28.840 ","End":"01:35.125","Text":"I want to take the sum n equals 1 to infinity of minus 1 to the n"},{"Start":"01:35.125 ","End":"01:41.265","Text":"over n to the x. Yeah,"},{"Start":"01:41.265 ","End":"01:42.970","Text":"I\u0027m going to take 3 cases."},{"Start":"01:42.970 ","End":"01:44.695","Text":"I\u0027m going to take first of all,"},{"Start":"01:44.695 ","End":"01:48.685","Text":"the case where x is bigger than 1."},{"Start":"01:48.685 ","End":"01:50.860","Text":"If x is bigger than 1,"},{"Start":"01:50.860 ","End":"01:57.255","Text":"then the absolute values of this series is exactly this series,"},{"Start":"01:57.255 ","End":"02:01.870","Text":"just replace x with p,"},{"Start":"02:01.870 ","End":"02:04.870","Text":"and so the absolute value converges."},{"Start":"02:04.870 ","End":"02:14.920","Text":"We have absolute convergence of the original series."},{"Start":"02:15.300 ","End":"02:20.020","Text":"The next case I want to consider is when x is positive,"},{"Start":"02:20.020 ","End":"02:22.590","Text":"but they want to be in this case,"},{"Start":"02:22.590 ","End":"02:26.535","Text":"so I\u0027ll go just up to an including 1."},{"Start":"02:26.535 ","End":"02:29.615","Text":"Now I claim that in this case,"},{"Start":"02:29.615 ","End":"02:32.800","Text":"I have conditional convergence and that this is"},{"Start":"02:32.800 ","End":"02:39.435","Text":"a Leibniz alternating series so it\u0027s conditional convergence."},{"Start":"02:39.435 ","End":"02:44.140","Text":"As for the alternating Leibniz series, first of all,"},{"Start":"02:44.140 ","End":"02:49.335","Text":"it\u0027s alternating, that\u0027s clear because of the minus 1 to the end."},{"Start":"02:49.335 ","End":"02:54.345","Text":"Now, it also has to decrease."},{"Start":"02:54.345 ","End":"03:01.590","Text":"In general, the function n to the x is"},{"Start":"03:01.590 ","End":"03:05.730","Text":"an increasing function when n"},{"Start":"03:05.730 ","End":"03:12.320","Text":"grows because whenever you have something positive,"},{"Start":"03:12.320 ","End":"03:14.700","Text":"it doesn\u0027t matter how small."},{"Start":"03:14.830 ","End":"03:21.850","Text":"This is strange. This is the variable as n goes to infinity,"},{"Start":"03:21.850 ","End":"03:30.920","Text":"as n increases, the bigger n is,"},{"Start":"03:30.920 ","End":"03:32.480","Text":"the bigger n to the power of x."},{"Start":"03:32.480 ","End":"03:38.430","Text":"Any positive power means that it\u0027s increasing."},{"Start":"03:38.810 ","End":"03:42.770","Text":"It also increases to infinity because the limit of"},{"Start":"03:42.770 ","End":"03:46.895","Text":"this as infinity even n to the power of 0.0001,"},{"Start":"03:46.895 ","End":"03:48.530","Text":"even if x is very small,"},{"Start":"03:48.530 ","End":"03:50.090","Text":"as n goes to infinity,"},{"Start":"03:50.090 ","End":"03:52.174","Text":"this goes to infinity."},{"Start":"03:52.174 ","End":"03:56.370","Text":"It increases to infinity,"},{"Start":"03:56.370 ","End":"04:01.515","Text":"which means that 1 over it goes to 0,"},{"Start":"04:01.515 ","End":"04:07.305","Text":"so 1 over n to the x decreases to"},{"Start":"04:07.305 ","End":"04:14.640","Text":"0 as n increases the same."},{"Start":"04:14.640 ","End":"04:17.465","Text":"I will just slightly rearranged s here,"},{"Start":"04:17.465 ","End":"04:20.030","Text":"n to the power of x increases to infinity,"},{"Start":"04:20.030 ","End":"04:24.125","Text":"so the reciprocal decreases to 0, as n increases,"},{"Start":"04:24.125 ","End":"04:29.730","Text":"x is fixed or arbitrary but fixed."},{"Start":"04:29.730 ","End":"04:33.500","Text":"The remaining case is when x is negative."},{"Start":"04:33.500 ","End":"04:35.615","Text":"If x is negative,"},{"Start":"04:35.615 ","End":"04:38.030","Text":"then minus x is positive."},{"Start":"04:38.030 ","End":"04:44.360","Text":"What we get is from here is a series of minus"},{"Start":"04:44.360 ","End":"04:52.235","Text":"1 to the n times"},{"Start":"04:52.235 ","End":"04:55.190","Text":"n to the power of minus x,"},{"Start":"04:55.190 ","End":"04:59.400","Text":"and this minus x is positive."},{"Start":"05:02.020 ","End":"05:08.915","Text":"Now, the reason that this is divergent,"},{"Start":"05:08.915 ","End":"05:14.660","Text":"and I\u0027m claiming that is definitely divergent is the simplest divergence test."},{"Start":"05:14.660 ","End":"05:18.095","Text":"The general term does not go to 0."},{"Start":"05:18.095 ","End":"05:24.445","Text":"I mean, n to the power of anything positive is always bigger than 1."},{"Start":"05:24.445 ","End":"05:29.620","Text":"Not only, I mean, even if x is,"},{"Start":"05:30.250 ","End":"05:34.190","Text":"sorry, less than or equal to 0, I wanted to write here,"},{"Start":"05:34.190 ","End":"05:38.835","Text":"even if x is 0,"},{"Start":"05:38.835 ","End":"05:42.165","Text":"I still get this bit is just 1,"},{"Start":"05:42.165 ","End":"05:44.040","Text":"and I\u0027ve just got minus 1 to the n,"},{"Start":"05:44.040 ","End":"05:45.930","Text":"which just goes plus or minus 1."},{"Start":"05:45.930 ","End":"05:55.380","Text":"It doesn\u0027t go to 0. It only makes it worse if I put x minus a 1/2 or minus 1 or minus 2,"},{"Start":"05:55.380 ","End":"05:59.750","Text":"and minus x becomes a plus power and"},{"Start":"05:59.750 ","End":"06:03.710","Text":"n to the power of anything positive goes to infinity."},{"Start":"06:03.710 ","End":"06:08.195","Text":"Either this bit here either sticks at 1 or goes to infinity depending."},{"Start":"06:08.195 ","End":"06:13.200","Text":"In any event, it does not go to 0, the general term."},{"Start":"06:14.870 ","End":"06:18.610","Text":"I\u0027ll write this is divergent,"},{"Start":"06:21.260 ","End":"06:25.430","Text":"and I\u0027ll just write here that"},{"Start":"06:25.430 ","End":"06:34.115","Text":"the general term does not tend to 0 as n goes to infinity,"},{"Start":"06:34.115 ","End":"06:37.475","Text":"which is a requirement for a convergent series."},{"Start":"06:37.475 ","End":"06:41.915","Text":"If I had to just say what is the region of convergence of the series,"},{"Start":"06:41.915 ","End":"06:46.610","Text":"I would say that it is convergent for both of these cases."},{"Start":"06:46.610 ","End":"06:50.975","Text":"I would just say for x bigger than 0,"},{"Start":"06:50.975 ","End":"06:53.315","Text":"but if you wanted to split it up further,"},{"Start":"06:53.315 ","End":"06:57.064","Text":"bigger than 1 is absolute convergence and here just conditional."},{"Start":"06:57.064 ","End":"06:59.850","Text":"I\u0027m done with this exercise."}],"ID":7965},{"Watched":false,"Name":"Exercise 1 part f","Duration":"7m 2s","ChapterTopicVideoID":7904,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we want to find the region of convergence of the following series."},{"Start":"00:05.250 ","End":"00:09.000","Text":"Let\u0027s first of all make sure that it\u0027s"},{"Start":"00:09.000 ","End":"00:15.475","Text":"defined and that we find the domain of definition for this series."},{"Start":"00:15.475 ","End":"00:17.910","Text":"Well, we can\u0027t divide by 0,"},{"Start":"00:17.910 ","End":"00:19.975","Text":"that\u0027s the only problem that\u0027s likely to occur."},{"Start":"00:19.975 ","End":"00:23.930","Text":"We must have that x is not equal"},{"Start":"00:23.930 ","End":"00:31.400","Text":"to minus n. Also,"},{"Start":"00:31.400 ","End":"00:32.630","Text":"x is not equal to,"},{"Start":"00:32.630 ","End":"00:33.995","Text":"let\u0027s see, I bringing this to the other side,"},{"Start":"00:33.995 ","End":"00:38.235","Text":"not equal to 1 minus n. That would make this 1 0."},{"Start":"00:38.235 ","End":"00:40.340","Text":"But n is variable,"},{"Start":"00:40.340 ","End":"00:45.855","Text":"so minus n would give us minus 1,"},{"Start":"00:45.855 ","End":"00:49.440","Text":"minus 2, minus 3, etc."},{"Start":"00:49.440 ","End":"00:54.450","Text":"1 minus n, which start off when n is 1 with 0,"},{"Start":"00:54.450 ","End":"00:56.085","Text":"and then minus 1."},{"Start":"00:56.085 ","End":"01:00.960","Text":"Basically, these are the forbidden values."},{"Start":"01:00.960 ","End":"01:03.555","Text":"Let me just move them over a bit."},{"Start":"01:03.555 ","End":"01:08.675","Text":"Got to say that x is not equal to any of these."},{"Start":"01:08.675 ","End":"01:12.530","Text":"So x can\u0027t be a non positive integer,"},{"Start":"01:12.530 ","End":"01:14.125","Text":"if you wanted it in words."},{"Start":"01:14.125 ","End":"01:17.645","Text":"Other than that, the way we\u0027re going to do this"},{"Start":"01:17.645 ","End":"01:22.790","Text":"is by decomposing this to partial fractions."},{"Start":"01:22.790 ","End":"01:24.485","Text":"Now, it sounds awful,"},{"Start":"01:24.485 ","End":"01:26.855","Text":"but let me just give you an example of what I mean."},{"Start":"01:26.855 ","End":"01:32.390","Text":"Notice that these 2 factors are almost the same, just this is 1 less."},{"Start":"01:32.390 ","End":"01:40.430","Text":"Suppose I did something like 1/4 times 3."},{"Start":"01:40.430 ","End":"01:48.860","Text":"I\u0027m claiming that this is equal to 1/3 minus 1/4."},{"Start":"01:48.860 ","End":"01:51.270","Text":"Because if I cross-multiply,"},{"Start":"01:51.270 ","End":"01:55.020","Text":"I get 4 minus 3,"},{"Start":"01:55.020 ","End":"01:56.980","Text":"which is 1, over 3 times 4,"},{"Start":"01:56.980 ","End":"01:59.065","Text":"which is the same as 4 times 3."},{"Start":"01:59.065 ","End":"02:01.405","Text":"Now, with this numerical example,"},{"Start":"02:01.405 ","End":"02:04.360","Text":"I\u0027m claiming that the same thing happens here,"},{"Start":"02:04.360 ","End":"02:07.150","Text":"that 1/x plus n,"},{"Start":"02:07.150 ","End":"02:10.315","Text":"x plus n minus 1,"},{"Start":"02:10.315 ","End":"02:11.950","Text":"where I see x plus n,"},{"Start":"02:11.950 ","End":"02:13.615","Text":"it\u0027s like the 4."},{"Start":"02:13.615 ","End":"02:18.279","Text":"So it\u0027s equal to 1 over the second 1,"},{"Start":"02:18.279 ","End":"02:20.650","Text":"x plus n minus 1,"},{"Start":"02:20.650 ","End":"02:26.330","Text":"minus 1/x plus n. If you multiply this out,"},{"Start":"02:26.330 ","End":"02:28.880","Text":"you get this minus this,"},{"Start":"02:28.880 ","End":"02:31.325","Text":"so we get minus minus 1 is just the 1,"},{"Start":"02:31.325 ","End":"02:34.215","Text":"and this times this can reverse the order."},{"Start":"02:34.215 ","End":"02:36.470","Text":"If I use this equation,"},{"Start":"02:36.470 ","End":"02:37.880","Text":"now I can expand."},{"Start":"02:37.880 ","End":"02:40.490","Text":"This is what we call a telescoping series."},{"Start":"02:40.490 ","End":"02:43.540","Text":"This series converges if and only if"},{"Start":"02:43.540 ","End":"02:53.115","Text":"the sequence of partial sums, S_n converges."},{"Start":"02:53.115 ","End":"02:59.895","Text":"What is S_n?"},{"Start":"02:59.895 ","End":"03:05.820","Text":"I don\u0027t want to use this n. I\u0027ll use N,"},{"Start":"03:05.820 ","End":"03:10.254","Text":"where S_N is the sum,"},{"Start":"03:10.254 ","End":"03:14.360","Text":"not to infinity, but just to some finite, number n,"},{"Start":"03:14.360 ","End":"03:17.070","Text":"of the same thing,"},{"Start":"03:18.190 ","End":"03:24.335","Text":"of 1/x plus n,"},{"Start":"03:24.335 ","End":"03:27.820","Text":"x plus n minus 1."},{"Start":"03:27.820 ","End":"03:33.080","Text":"Now this, I can compute because if I use this breakup,"},{"Start":"03:33.080 ","End":"03:41.175","Text":"S_N, what this equals,"},{"Start":"03:41.175 ","End":"03:43.765","Text":"and I\u0027m going to actually write it out,"},{"Start":"03:43.765 ","End":"03:51.670","Text":"for n is 1, we get 1/x,"},{"Start":"03:51.670 ","End":"03:56.090","Text":"minus 1/x plus 1."},{"Start":"03:56.640 ","End":"03:59.785","Text":"Then if x is 2,"},{"Start":"03:59.785 ","End":"04:13.720","Text":"we have 1/x plus 1 minus 1/x"},{"Start":"04:13.720 ","End":"04:19.850","Text":"plus 2 plus, and so on."},{"Start":"04:19.850 ","End":"04:23.250","Text":"I\u0027ll write 1 more to see if you can see."},{"Start":"04:23.250 ","End":"04:26.580","Text":"When n is 3, 3 minus 1 is 2,"},{"Start":"04:26.580 ","End":"04:28.830","Text":"so 1/x plus 2"},{"Start":"04:28.830 ","End":"04:33.309","Text":"minus 1/x plus"},{"Start":"04:41.390 ","End":"04:51.350","Text":"3, n is 3, and so on, up to when n is N,"},{"Start":"04:51.350 ","End":"04:58.590","Text":"1/x plus N minus 1"},{"Start":"05:00.260 ","End":"05:09.365","Text":"minus 1/x plus N. Now,"},{"Start":"05:09.365 ","End":"05:11.005","Text":"this cancels with this,"},{"Start":"05:11.005 ","End":"05:13.720","Text":"and this cancels with this, and so on."},{"Start":"05:13.720 ","End":"05:16.390","Text":"Everything cancels in pairs."},{"Start":"05:16.390 ","End":"05:18.460","Text":"This cancels with the 1 on the next 1,"},{"Start":"05:18.460 ","End":"05:21.490","Text":"and the 1 with here cancels with this 1,"},{"Start":"05:21.490 ","End":"05:29.395","Text":"so all we\u0027re left with is that S_N is 1/x minus"},{"Start":"05:29.395 ","End":"05:32.664","Text":"1/x plus N."},{"Start":"05:32.664 ","End":"05:37.450","Text":"Now I can say that the limit of the sequence S_N,"},{"Start":"05:37.450 ","End":"05:44.470","Text":"the limit is N goes to infinity of S and is equal to,"},{"Start":"05:44.470 ","End":"05:47.380","Text":"when n goes to infinity, N plus something,"},{"Start":"05:47.380 ","End":"05:48.760","Text":"x is like a constant,"},{"Start":"05:48.760 ","End":"05:50.710","Text":"is also going to infinity,"},{"Start":"05:50.710 ","End":"05:55.105","Text":"so 1 over infinity is 0 is just 1 over x."},{"Start":"05:55.105 ","End":"05:59.800","Text":"Like I said, if in general,"},{"Start":"05:59.800 ","End":"06:04.240","Text":"the limit of the sequence of partial sums converges,"},{"Start":"06:04.240 ","End":"06:07.135","Text":"then the original series converges."},{"Start":"06:07.135 ","End":"06:12.520","Text":"In general, we say that S_N is"},{"Start":"06:12.520 ","End":"06:18.225","Text":"the sum from 1 to N of a_n from the original series,"},{"Start":"06:18.225 ","End":"06:21.090","Text":"and S_N, as a sequence,"},{"Start":"06:21.090 ","End":"06:25.080","Text":"converges if and only if a_n,"},{"Start":"06:25.080 ","End":"06:28.140","Text":"as the series, converges."},{"Start":"06:28.140 ","End":"06:34.300","Text":"The answer is that the convergence,"},{"Start":"06:36.230 ","End":"06:38.389","Text":"and not only convergence,"},{"Start":"06:38.389 ","End":"06:39.500","Text":"we know what the sum is,"},{"Start":"06:39.500 ","End":"06:44.150","Text":"to 1/ for all x except to,"},{"Start":"06:44.150 ","End":"06:47.660","Text":"I can say for x not equal to 0,"},{"Start":"06:47.660 ","End":"06:49.130","Text":"minus 1, minus 2,"},{"Start":"06:49.130 ","End":"06:51.770","Text":"which was the original restriction, etc."},{"Start":"06:51.770 ","End":"07:02.670","Text":"All x except the negative integers and 0. That is it."}],"ID":7966},{"Watched":false,"Name":"Exercise 2 part a","Duration":"2m 15s","ChapterTopicVideoID":7905,"CourseChapterTopicPlaylistID":4250,"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 exercise, we have to check the uniform conversion of the series."},{"Start":"00:05.010 ","End":"00:07.750","Text":"It\u0027s a series of functions."},{"Start":"00:07.760 ","End":"00:13.200","Text":"For all x, which we can sometimes write as minus infinity to infinity."},{"Start":"00:13.200 ","End":"00:18.555","Text":"Our main tool is the Weierstrass M-test."},{"Start":"00:18.555 ","End":"00:24.004","Text":"I take the absolute value of typical term,"},{"Start":"00:24.004 ","End":"00:29.460","Text":"cosine nx over n-squared."},{"Start":"00:29.460 ","End":"00:31.925","Text":"For each n, this is a function of x."},{"Start":"00:31.925 ","End":"00:37.850","Text":"What we want to do is make it less than or equal to some number that doesn\u0027t depend on"},{"Start":"00:37.850 ","End":"00:45.865","Text":"x in such a way that the series we get converges."},{"Start":"00:45.865 ","End":"00:48.980","Text":"Now, in our case, the obvious thing to do is to say that"},{"Start":"00:48.980 ","End":"00:53.100","Text":"the cosine in absolute value is always less than 1."},{"Start":"00:53.100 ","End":"00:56.300","Text":"I mean, the cosine oscillates between 1 and minus 1,"},{"Start":"00:56.300 ","End":"00:58.520","Text":"but never goes out that bound."},{"Start":"00:58.520 ","End":"01:03.765","Text":"This is less than or equal to 1 over n squared."},{"Start":"01:03.765 ","End":"01:05.790","Text":"Then the question is,"},{"Start":"01:05.790 ","End":"01:15.520","Text":"is the sum from 1 to infinity of 1 over n squared convergent?"},{"Start":"01:15.680 ","End":"01:19.110","Text":"I\u0027ll put a question mark."},{"Start":"01:19.110 ","End":"01:23.225","Text":"Right away, I\u0027ll tell you that the answer is yes."},{"Start":"01:23.225 ","End":"01:25.175","Text":"It\u0027s a familiar series."},{"Start":"01:25.175 ","End":"01:28.250","Text":"But the easiest way to see it, if you haven\u0027t,"},{"Start":"01:28.250 ","End":"01:33.010","Text":"is to take it as a p-series."},{"Start":"01:33.010 ","End":"01:39.695","Text":"This 2 is a p in the p-series,"},{"Start":"01:39.695 ","End":"01:42.830","Text":"which is the generalized harmonic series also,"},{"Start":"01:42.830 ","End":"01:47.420","Text":"instead of just 1 over n, 1^p."},{"Start":"01:47.420 ","End":"01:51.115","Text":"Whenever p is strictly greater than 1,"},{"Start":"01:51.115 ","End":"01:54.360","Text":"then this converges and certainly,"},{"Start":"01:54.360 ","End":"01:56.610","Text":"2 is greater than 1."},{"Start":"01:56.610 ","End":"01:58.725","Text":"P is 2 here."},{"Start":"01:58.725 ","End":"02:00.345","Text":"The answer is yes."},{"Start":"02:00.345 ","End":"02:01.680","Text":"This series converges."},{"Start":"02:01.680 ","End":"02:11.595","Text":"So our original series is uniformly convergent according to the Weierstrass M-test."},{"Start":"02:11.595 ","End":"02:14.050","Text":"That\u0027s all there is to it."}],"ID":7967},{"Watched":false,"Name":"Exercise 2 part b","Duration":"2m 44s","ChapterTopicVideoID":7906,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we need to check the uniform convergence of this series"},{"Start":"00:05.250 ","End":"00:10.740","Text":"of functions in this range between minus 1 and 1 inclusive."},{"Start":"00:10.740 ","End":"00:17.100","Text":"Our main tool is the Weierstrass M-test."},{"Start":"00:17.100 ","End":"00:24.220","Text":"What it says, we\u0027ll take the typical term and take its absolute value,"},{"Start":"00:27.190 ","End":"00:32.640","Text":"x^n over n^3/2 and we want to find that it\u0027s going to be less than or equal to"},{"Start":"00:32.640 ","End":"00:37.890","Text":"some convergent series of numbers, not a function."},{"Start":"00:37.890 ","End":"00:39.990","Text":"In other words, x will not appear here."},{"Start":"00:39.990 ","End":"00:46.115","Text":"Now notice that if x is between minus 1 and 1,"},{"Start":"00:46.115 ","End":"00:51.485","Text":"then the absolute value of x is less than or equal to 1."},{"Start":"00:51.485 ","End":"00:55.730","Text":"So the absolute value of x^n,"},{"Start":"00:55.730 ","End":"00:58.685","Text":"doesn\u0027t matter if I put the n inside or outside,"},{"Start":"00:58.685 ","End":"01:01.925","Text":"it\u0027s also going to be less than or equal to 1."},{"Start":"01:01.925 ","End":"01:07.220","Text":"If I take something between 0 and 1 and I square it, cube it,"},{"Start":"01:07.220 ","End":"01:08.630","Text":"take it to the power of 4, whatever,"},{"Start":"01:08.630 ","End":"01:11.720","Text":"it\u0027s still going to be less than or equal to 1."},{"Start":"01:11.720 ","End":"01:18.600","Text":"So the series I\u0027ll put here is just 1 over n^3/2."},{"Start":"01:18.730 ","End":"01:21.650","Text":"That\u0027s a series of numbers."},{"Start":"01:21.650 ","End":"01:22.850","Text":"There\u0027s no x\u0027s here."},{"Start":"01:22.850 ","End":"01:26.330","Text":"What I\u0027ll have to do is show that this converges,"},{"Start":"01:26.330 ","End":"01:30.330","Text":"but the sum of"},{"Start":"01:30.700 ","End":"01:37.190","Text":"1 over n^3/2 indeed converges and I\u0027ll tell you why in a moment."},{"Start":"01:37.190 ","End":"01:39.859","Text":"Just want to put n equals 1 to infinity."},{"Start":"01:39.859 ","End":"01:44.850","Text":"Because this is a p-series"},{"Start":"01:46.160 ","End":"01:53.510","Text":"with p equals 3/2 which is bigger than 1."},{"Start":"01:53.510 ","End":"02:00.570","Text":"I\u0027ll just remind you a p-series is the sum n goes from 1"},{"Start":"02:00.570 ","End":"02:15.780","Text":"to infinity of 1 over n^p."},{"Start":"02:15.780 ","End":"02:22.290","Text":"It converges if p is bigger than 1."},{"Start":"02:22.290 ","End":"02:29.290","Text":"It also diverges if p is less than or equal to 1."},{"Start":"02:29.290 ","End":"02:33.769","Text":"Anyway, we do have 3/2 bigger than 1, so this converges."},{"Start":"02:33.769 ","End":"02:39.545","Text":"So the Weierstrass M-test says that the original series converges uniformly."},{"Start":"02:39.545 ","End":"02:40.910","Text":"So the answer is yes,"},{"Start":"02:40.910 ","End":"02:44.370","Text":"it converges uniformly in this range."}],"ID":7968},{"Watched":false,"Name":"Exercise 2 part c","Duration":"2m 56s","ChapterTopicVideoID":7907,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.925","Text":"In this exercise, we want to check"},{"Start":"00:02.925 ","End":"00:09.030","Text":"uniform convergence of this series of functions for all x."},{"Start":"00:09.030 ","End":"00:11.205","Text":"That\u0027s what this range means."},{"Start":"00:11.205 ","End":"00:14.340","Text":"We\u0027re going to use our main tool,"},{"Start":"00:14.340 ","End":"00:17.340","Text":"and that is the Weierstass M-test."},{"Start":"00:17.340 ","End":"00:23.835","Text":"What we do is we take the absolute value of a general term."},{"Start":"00:23.835 ","End":"00:29.420","Text":"Of course, we don\u0027t really need the absolute value in this case"},{"Start":"00:29.420 ","End":"00:32.240","Text":"because everything is positive."},{"Start":"00:32.240 ","End":"00:38.705","Text":"But we want to bound this by some series."},{"Start":"00:38.705 ","End":"00:42.589","Text":"In other words, this should depend just on n, not on x,"},{"Start":"00:42.589 ","End":"00:47.705","Text":"a number series, which is part of a convergent series."},{"Start":"00:47.705 ","End":"00:52.940","Text":"Now, the easiest thing to do is just to throw out the x squared"},{"Start":"00:52.940 ","End":"00:58.430","Text":"and say this is less than or equal to 1 over n square root of n."},{"Start":"00:58.430 ","End":"01:05.680","Text":"Because if I decrease the denominator fa a positive fraction,"},{"Start":"01:05.680 ","End":"01:07.675","Text":"then I\u0027m just increasing,"},{"Start":"01:07.675 ","End":"01:09.805","Text":"It can\u0027t get smaller."},{"Start":"01:09.805 ","End":"01:14.290","Text":"This is going to be our general term."},{"Start":"01:14.290 ","End":"01:16.435","Text":"Then the question is,"},{"Start":"01:16.435 ","End":"01:23.060","Text":"is the sum of this 1 over n root n,"},{"Start":"01:23.060 ","End":"01:28.620","Text":"does this or is this convergent?"},{"Start":"01:28.620 ","End":"01:31.840","Text":"I\u0027ll put it as a question mark."},{"Start":"01:32.330 ","End":"01:36.210","Text":"I claim that the answer is yes."},{"Start":"01:36.210 ","End":"01:40.240","Text":"The reason that this is convergent is that"},{"Start":"01:40.240 ","End":"01:44.475","Text":"this is actually a p-series or generalized harmonic series."},{"Start":"01:44.475 ","End":"01:54.060","Text":"You see it better if I write it as the sum of 1 over n to the power of 3 over 2."},{"Start":"01:54.060 ","End":"01:59.405","Text":"3 over 2 is like our p,"},{"Start":"01:59.405 ","End":"02:06.649","Text":"because in general and we have the sum of 1 over n to the power of p"},{"Start":"02:06.649 ","End":"02:11.990","Text":"and p is bigger than 1."},{"Start":"02:11.990 ","End":"02:15.335","Text":"Then we know that this is convergent."},{"Start":"02:15.335 ","End":"02:22.505","Text":"It\u0027s called a p-series or generalized harmonic series from 1 to infinity."},{"Start":"02:22.505 ","End":"02:29.940","Text":"This is the case here because 3 over 2 is bigger than 1."},{"Start":"02:29.940 ","End":"02:31.560","Text":"We\u0027re all right there."},{"Start":"02:31.560 ","End":"02:35.515","Text":"This is convergent by the M-test."},{"Start":"02:35.515 ","End":"02:41.935","Text":"This original series is uniformly convergent."},{"Start":"02:41.935 ","End":"02:44.490","Text":"That\u0027s all there is to do."},{"Start":"02:44.490 ","End":"02:47.840","Text":"Maybe write a few words that this implies that"},{"Start":"02:47.840 ","End":"02:53.510","Text":"the original series is uniformly convergent in all of its range and so on."},{"Start":"02:53.510 ","End":"02:56.640","Text":"We\u0027re done."}],"ID":7969},{"Watched":false,"Name":"Exercise 2 part d","Duration":"11m 19s","ChapterTopicVideoID":7908,"CourseChapterTopicPlaylistID":4250,"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":"In this exercise, we want to check whether this series of"},{"Start":"00:04.890 ","End":"00:09.780","Text":"functions converges uniformly and in this particular range,"},{"Start":"00:09.780 ","End":"00:13.890","Text":"where x is between 1/4 and 4 inclusive."},{"Start":"00:13.890 ","End":"00:20.640","Text":"Main tool for uniform convergence is the Weierstrass M-test."},{"Start":"00:20.640 ","End":"00:24.540","Text":"In case you wondering why it\u0027s called the M-test,"},{"Start":"00:24.540 ","End":"00:35.220","Text":"if our original series is the sum of a_n then what"},{"Start":"00:35.220 ","End":"00:38.400","Text":"Weierstrass\u0027 test says that if we can"},{"Start":"00:38.400 ","End":"00:49.010","Text":"find an upper bound"},{"Start":"00:49.010 ","End":"00:50.960","Text":"for this series of functions,"},{"Start":"00:50.960 ","End":"00:56.160","Text":"that each function will be less than or equal to in absolute value."},{"Start":"00:56.300 ","End":"01:02.614","Text":"Well, constant that depends on n only and not x,"},{"Start":"01:02.614 ","End":"01:11.870","Text":"then if this series converges and goes from 1 to infinity M_n,"},{"Start":"01:11.870 ","End":"01:20.170","Text":"converges, then the original series of functions converges uniformly."},{"Start":"01:20.170 ","End":"01:27.780","Text":"What we have to do in our case is take the absolute value of the general term."},{"Start":"01:28.330 ","End":"01:32.599","Text":"Actually, we don\u0027t need the absolute value."},{"Start":"01:32.599 ","End":"01:34.340","Text":"Well, I\u0027ll just write it."},{"Start":"01:34.340 ","End":"01:43.670","Text":"n plus 1/the square root of n factorial times"},{"Start":"01:43.670 ","End":"01:52.850","Text":"x^n plus x^minus n. We want to find what will M_n be."},{"Start":"01:52.850 ","End":"01:54.830","Text":"Something that depends only on n but not on"},{"Start":"01:54.830 ","End":"01:59.760","Text":"x and then get a convergent series, hopefully."},{"Start":"02:02.900 ","End":"02:09.190","Text":"I\u0027m going to work just with this bit because this is constant as far as x goes."},{"Start":"02:09.190 ","End":"02:10.750","Text":"I\u0027m going to bound this."},{"Start":"02:10.750 ","End":"02:13.375","Text":"I\u0027ll do this as a side exercise."},{"Start":"02:13.375 ","End":"02:17.935","Text":"The function x^n, as"},{"Start":"02:17.935 ","End":"02:25.700","Text":"a function of x with a constant n increases with x."},{"Start":"02:27.930 ","End":"02:31.240","Text":"If it increases with x,"},{"Start":"02:31.240 ","End":"02:38.440","Text":"then I know that means that since x is less than or equal to 4,"},{"Start":"02:38.440 ","End":"02:47.890","Text":"that x^n is less than or equal to 4^n."},{"Start":"02:47.890 ","End":"02:53.500","Text":"Similarly, x^minus n, this might"},{"Start":"02:53.500 ","End":"02:59.800","Text":"be if n is say 2,"},{"Start":"02:59.800 ","End":"03:03.310","Text":"that would be 1/x squared that decreases."},{"Start":"03:03.310 ","End":"03:11.250","Text":"1 over something it decreases with x as x gets bigger,"},{"Start":"03:11.250 ","End":"03:13.005","Text":"it only gets smaller."},{"Start":"03:13.005 ","End":"03:24.465","Text":"Now I\u0027m going to use the other side so that because 1/4 is less than or equal to x,"},{"Start":"03:24.465 ","End":"03:30.060","Text":"then I got to reverse the inequality,"},{"Start":"03:30.060 ","End":"03:46.910","Text":"then 1/4^n is bigger or equal to x^n."},{"Start":"03:46.910 ","End":"03:52.770","Text":"Now, 1/4^minus n, this is 4^minus 1^minus n. This thing"},{"Start":"03:52.770 ","End":"04:00.360","Text":"is the same as 4^n."},{"Start":"04:00.360 ","End":"04:02.535","Text":"I forgot the minus here."},{"Start":"04:02.535 ","End":"04:05.565","Text":"If I just write this backwards,"},{"Start":"04:05.565 ","End":"04:14.860","Text":"this just says that x^minus n is less than or equal to 4^n."},{"Start":"04:14.860 ","End":"04:18.720","Text":"Just looking at the inequality from the other side."},{"Start":"04:19.060 ","End":"04:22.445","Text":"This is less than 4^n,"},{"Start":"04:22.445 ","End":"04:24.770","Text":"this is less than 4^n."},{"Start":"04:24.770 ","End":"04:29.795","Text":"Going back here, this is going to be less than or equal to."},{"Start":"04:29.795 ","End":"04:38.495","Text":"I can put n plus 1/square root of"},{"Start":"04:38.495 ","End":"04:41.660","Text":"n factorial times"},{"Start":"04:41.660 ","End":"04:49.130","Text":"4^n plus 4^n."},{"Start":"04:49.130 ","End":"04:53.640","Text":"This is the M_n of the Weierstrass M-test."},{"Start":"04:53.640 ","End":"04:58.630","Text":"What we have to show is that the series,"},{"Start":"04:58.630 ","End":"05:08.710","Text":"the sum of these is convergent and I can write this as twice n plus 1,"},{"Start":"05:08.710 ","End":"05:16.020","Text":"4^n/square root of n factorial."},{"Start":"05:16.020 ","End":"05:23.115","Text":"I want to know is it convergent?"},{"Start":"05:23.115 ","End":"05:24.975","Text":"Hopefully, yes."},{"Start":"05:24.975 ","End":"05:30.530","Text":"If it is, then the original series will be uniformly convergent."},{"Start":"05:30.530 ","End":"05:33.379","Text":"Now, I claim that the answer is yes,"},{"Start":"05:33.379 ","End":"05:40.280","Text":"but I have to show you and what I\u0027m going do here these various tests,"},{"Start":"05:40.280 ","End":"05:42.695","Text":"notice that this is a positive series."},{"Start":"05:42.695 ","End":"05:45.365","Text":"Lot of tests only work for positive series."},{"Start":"05:45.365 ","End":"05:48.635","Text":"I\u0027m going to use the ratio test."},{"Start":"05:48.635 ","End":"05:50.630","Text":"That\u0027s the 1 that will do it for me here."},{"Start":"05:50.630 ","End":"05:53.960","Text":"Although, you probably could use some other test."},{"Start":"05:53.960 ","End":"05:56.240","Text":"Maybe the root test, I don\u0027t know,"},{"Start":"05:56.240 ","End":"05:58.940","Text":"but certainly, the ratio test will work."},{"Start":"05:58.940 ","End":"06:09.995","Text":"What the ratio test talks about is a_n plus 1/a_n and the limit of this,"},{"Start":"06:09.995 ","End":"06:12.380","Text":"what it tends to."},{"Start":"06:12.380 ","End":"06:17.450","Text":"Basically, if the limit of this at n go to infinity is less than 1,"},{"Start":"06:17.450 ","End":"06:20.150","Text":"then the original series is convergent."},{"Start":"06:20.150 ","End":"06:23.100","Text":"Let\u0027s see what this is equal to."},{"Start":"06:24.740 ","End":"06:29.175","Text":"This will be equal to, let see,"},{"Start":"06:29.175 ","End":"06:34.719","Text":"put n plus 1 instead of n in the general term."},{"Start":"06:36.140 ","End":"06:45.090","Text":"We have twice n plus 2,"},{"Start":"06:45.090 ","End":"06:47.175","Text":"plus 2, and plus 1 plus 1."},{"Start":"06:47.175 ","End":"06:54.470","Text":"Then 4^n plus 1/square root"},{"Start":"06:54.470 ","End":"06:57.650","Text":"of n plus 1 factorial."},{"Start":"06:57.650 ","End":"06:59.375","Text":"That\u0027s the numerator."},{"Start":"06:59.375 ","End":"07:02.450","Text":"Now the denominator is just this,"},{"Start":"07:02.450 ","End":"07:05.240","Text":"but instead of dividing by a fraction,"},{"Start":"07:05.240 ","End":"07:07.729","Text":"we can multiply by the inverse fraction."},{"Start":"07:07.729 ","End":"07:09.875","Text":"I\u0027m putting here square root of"},{"Start":"07:09.875 ","End":"07:13.810","Text":"n factorial/twice n"},{"Start":"07:13.810 ","End":"07:20.420","Text":"plus 1, 4^n."},{"Start":"07:20.420 ","End":"07:25.115","Text":"Now, the 2 cancels with the 2"},{"Start":"07:25.115 ","End":"07:34.890","Text":"and 4^n plus 1 is just 4 times 4^n."},{"Start":"07:34.890 ","End":"07:41.150","Text":"This will cancel with this and this square root of n plus 1 factorial."},{"Start":"07:41.150 ","End":"07:49.095","Text":"n plus 1 factorial remember is n plus 1 times n factorial."},{"Start":"07:49.095 ","End":"07:51.830","Text":"Product of all numbers from 1 to n plus 1,"},{"Start":"07:51.830 ","End":"07:56.895","Text":"you can take the product from 1 to n and then add just the last term, n plus 1."},{"Start":"07:56.895 ","End":"07:59.860","Text":"Let\u0027s see what this comes out to."},{"Start":"08:00.560 ","End":"08:03.940","Text":"Lot of canceling."},{"Start":"08:04.420 ","End":"08:08.200","Text":"We have here after tidying up, let\u0027s see."},{"Start":"08:08.200 ","End":"08:13.175","Text":"We have 4 and the numerator we have n plus"},{"Start":"08:13.175 ","End":"08:20.435","Text":"2 and that\u0027s what we\u0027re left with on the numerator."},{"Start":"08:20.435 ","End":"08:27.200","Text":"On the denominator, we have the square roots can apply to a product,"},{"Start":"08:27.200 ","End":"08:31.800","Text":"so we get the square root of n plus 1."},{"Start":"08:35.960 ","End":"08:40.575","Text":"We still have an n plus 1 here."},{"Start":"08:40.575 ","End":"08:44.260","Text":"This looks like it."},{"Start":"08:44.300 ","End":"08:53.470","Text":"We want the limit of this as n goes to infinity of this thing."},{"Start":"08:53.470 ","End":"08:55.595","Text":"We can say that this is,"},{"Start":"08:55.595 ","End":"09:05.390","Text":"I\u0027ll just write it as 4 and then I could write n plus 2"},{"Start":"09:05.390 ","End":"09:19.320","Text":"plus 1 and then 1/square root of n plus 1."},{"Start":"09:19.900 ","End":"09:23.120","Text":"Now, this is a constant."},{"Start":"09:23.120 ","End":"09:26.490","Text":"I claim that this tends to 1."},{"Start":"09:27.360 ","End":"09:31.050","Text":"I\u0027ll show you the side in a moment."},{"Start":"09:31.050 ","End":"09:38.070","Text":"Here the denominator goes to infinity."},{"Start":"09:39.710 ","End":"09:50.540","Text":"What we\u0027re going to get is 4 times 1 times 1/infinity. This thing is 0."},{"Start":"09:50.540 ","End":"09:57.210","Text":"Basically, this comes out to be just 0,"},{"Start":"09:57.210 ","End":"10:00.265","Text":"which is certainly less than 1,"},{"Start":"10:00.265 ","End":"10:08.420","Text":"which is what we need for the ratio test and I just need to explain this to you."},{"Start":"10:08.420 ","End":"10:10.055","Text":"But after I\u0027ve done that,"},{"Start":"10:10.055 ","End":"10:16.460","Text":"we will then be able to conclude from the ratio test that this series is"},{"Start":"10:16.460 ","End":"10:25.160","Text":"convergent and hence that our original series converges uniformly because of this test."},{"Start":"10:25.160 ","End":"10:28.050","Text":"Why does this go to 1?"},{"Start":"10:29.440 ","End":"10:34.655","Text":"Well, n plus 2 plus 1."},{"Start":"10:34.655 ","End":"10:40.370","Text":"These are polynomial/polynomial,"},{"Start":"10:40.370 ","End":"10:45.500","Text":"on each part you take out the highest power."},{"Start":"10:45.500 ","End":"10:47.495","Text":"If I take out n from here,"},{"Start":"10:47.495 ","End":"10:53.150","Text":"I can get that this is n times 1 plus 2."},{"Start":"10:53.150 ","End":"10:55.350","Text":"Here if I take out n,"},{"Start":"10:55.350 ","End":"11:02.390","Text":"it\u0027s 1 plus 1 and then the n\u0027s cancel."},{"Start":"11:02.390 ","End":"11:07.235","Text":"2 goes to 0, 1 goes to 0,"},{"Start":"11:07.235 ","End":"11:14.415","Text":"so this just goes to 1/1, which equals 1."},{"Start":"11:14.415 ","End":"11:16.485","Text":"That explains that bit."},{"Start":"11:16.485 ","End":"11:18.570","Text":"That was all that was left to show,"},{"Start":"11:18.570 ","End":"11:20.800","Text":"so we are done."}],"ID":7970},{"Watched":false,"Name":"Exercise 2 part e","Duration":"10m 56s","ChapterTopicVideoID":7909,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.889","Text":"In this exercise, we want to check if this series of functions converges"},{"Start":"00:04.889 ","End":"00:11.430","Text":"uniformly on the range x between minus a and a inclusive."},{"Start":"00:11.430 ","End":"00:12.900","Text":"We\u0027re not told what a is,"},{"Start":"00:12.900 ","End":"00:17.025","Text":"it could be anything, any positive number."},{"Start":"00:17.025 ","End":"00:21.180","Text":"Could be a million, I don\u0027t know."},{"Start":"00:21.180 ","End":"00:29.235","Text":"Now, our main tool for doing such things is the Weierstrass M-test."},{"Start":"00:29.235 ","End":"00:31.350","Text":"Within the Weierstrass M-test,"},{"Start":"00:31.350 ","End":"00:34.440","Text":"we take the absolute value of the general term,"},{"Start":"00:34.440 ","End":"00:40.035","Text":"which is natural log of 1 plus x squared over n,"},{"Start":"00:40.035 ","End":"00:44.990","Text":"natural log squared of n. We want to make"},{"Start":"00:44.990 ","End":"00:51.440","Text":"this less than or equal to some number series."},{"Start":"00:51.440 ","End":"00:56.390","Text":"I want some number that depends on n but not on"},{"Start":"00:56.390 ","End":"01:02.885","Text":"x and in such a way that I get a convergence series."},{"Start":"01:02.885 ","End":"01:08.090","Text":"Now, how can I make this less than or equal to something?"},{"Start":"01:08.090 ","End":"01:13.905","Text":"I\u0027d like to remind you of an inequality,"},{"Start":"01:13.905 ","End":"01:17.870","Text":"it\u0027s generally a well-known inequality, and I need it."},{"Start":"01:17.870 ","End":"01:20.515","Text":"I\u0027m pulling it out of a hat."},{"Start":"01:20.515 ","End":"01:26.220","Text":"The natural log of 1 plus x is less than or"},{"Start":"01:26.220 ","End":"01:32.565","Text":"equal to x for x bigger or equal to 0,"},{"Start":"01:32.565 ","End":"01:34.675","Text":"and I want to apply this."},{"Start":"01:34.675 ","End":"01:38.365","Text":"In our case, x is going to be this thing."},{"Start":"01:38.365 ","End":"01:41.470","Text":"Now I claim that this is bigger or equal to 0."},{"Start":"01:41.470 ","End":"01:44.749","Text":"In fact, it\u0027s actually bigger than 0."},{"Start":"01:44.760 ","End":"01:51.010","Text":"First of all, the denominator is positive,"},{"Start":"01:51.010 ","End":"01:55.750","Text":"because n is bigger or equal to 2."},{"Start":"01:55.750 ","End":"01:58.275","Text":"If n is bigger or equal to 2, yeah,"},{"Start":"01:58.275 ","End":"02:00.875","Text":"the series starts at 2,"},{"Start":"02:00.875 ","End":"02:03.190","Text":"because if the series started at 1,"},{"Start":"02:03.190 ","End":"02:05.230","Text":"we\u0027d have 0 in the denominator,"},{"Start":"02:05.230 ","End":"02:06.975","Text":"the natural log of 1 is 0."},{"Start":"02:06.975 ","End":"02:09.170","Text":"If n is bigger or equal to 2,"},{"Start":"02:09.170 ","End":"02:17.480","Text":"then the natural log of"},{"Start":"02:17.480 ","End":"02:22.310","Text":"n is going to be bigger"},{"Start":"02:22.310 ","End":"02:28.465","Text":"or equal to natural log of 2,"},{"Start":"02:28.465 ","End":"02:32.105","Text":"and if I put a square here,"},{"Start":"02:32.105 ","End":"02:39.410","Text":"it\u0027s going to be bigger or equal to this positive constant,"},{"Start":"02:39.410 ","End":"02:44.480","Text":"and n is also positive."},{"Start":"02:44.480 ","End":"02:50.090","Text":"Altogether, I\u0027ve got that n is"},{"Start":"02:50.090 ","End":"02:57.330","Text":"bigger than 0 because the natural log of 2 is bigger than 0."},{"Start":"02:57.350 ","End":"03:07.850","Text":"N natural log squared of n will be bigger than 0 because this is bigger than 0,"},{"Start":"03:07.850 ","End":"03:17.519","Text":"n is bigger than 0, and x squared is certainly bigger or equal to 0."},{"Start":"03:17.950 ","End":"03:21.770","Text":"Actually, it could be 0 because 0 is in the range,"},{"Start":"03:21.770 ","End":"03:25.175","Text":"but certainly this is bigger or equal to 0 because bigger or equal to 0"},{"Start":"03:25.175 ","End":"03:29.075","Text":"over bigger than 0 is bigger or equal to 0."},{"Start":"03:29.075 ","End":"03:34.895","Text":"I can use this property,"},{"Start":"03:34.895 ","End":"03:38.090","Text":"this inequality and say that this is less than or equal"},{"Start":"03:38.090 ","End":"03:43.890","Text":"to just dx and I don\u0027t need the absolute value."},{"Start":"03:45.200 ","End":"03:53.940","Text":"So x squared over n natural log squared of n,"},{"Start":"03:53.940 ","End":"03:57.770","Text":"but I want something that doesn\u0027t depend on x."},{"Start":"03:57.770 ","End":"04:02.970","Text":"Now, if x is between minus a and a,"},{"Start":"04:02.970 ","End":"04:07.575","Text":"then x squared is going to be less than or equal to a squared."},{"Start":"04:07.575 ","End":"04:13.335","Text":"So I can write a squared over n natural log squared of n,"},{"Start":"04:13.335 ","End":"04:16.170","Text":"and now I\u0027m independent of x."},{"Start":"04:16.170 ","End":"04:19.080","Text":"Now the question is,"},{"Start":"04:19.080 ","End":"04:22.135","Text":"how about the series?"},{"Start":"04:22.135 ","End":"04:24.530","Text":"Well, to be consistent,"},{"Start":"04:24.530 ","End":"04:27.115","Text":"I\u0027ll take it also from 2 to infinity."},{"Start":"04:27.115 ","End":"04:30.695","Text":"In fact, I have to take it from 2 to infinity because if n is 1,"},{"Start":"04:30.695 ","End":"04:32.935","Text":"I get a 0 in the denominator,"},{"Start":"04:32.935 ","End":"04:38.930","Text":"a squared over n natural log squared of"},{"Start":"04:38.930 ","End":"04:46.020","Text":"n. I asked, is this convergent?"},{"Start":"04:46.810 ","End":"04:50.180","Text":"If so, then by the M-test,"},{"Start":"04:50.180 ","End":"04:52.865","Text":"this will be uniformly convergent."},{"Start":"04:52.865 ","End":"04:56.180","Text":"How do I know that this is convergent?"},{"Start":"04:56.180 ","End":"04:58.490","Text":"The a squared is a constant,"},{"Start":"04:58.490 ","End":"05:03.525","Text":"so I could bring it out in front of the summation sign."},{"Start":"05:03.525 ","End":"05:08.820","Text":"We\u0027re really concerned with this 1 over n natural log squared of"},{"Start":"05:08.820 ","End":"05:14.540","Text":"n. This has been done before using the integral test,"},{"Start":"05:14.540 ","End":"05:17.670","Text":"perhaps I\u0027ll show you again quickly."},{"Start":"05:19.720 ","End":"05:27.590","Text":"I\u0027ll just write integral test and I\u0027ll just do it briefly because it"},{"Start":"05:27.590 ","End":"05:35.500","Text":"has been done before in the chapter on series with the integral test."},{"Start":"05:35.500 ","End":"05:42.000","Text":"We replace n by x and we ask,"},{"Start":"05:42.000 ","End":"05:50.790","Text":"is the integral from 2 to infinity of 1 over,"},{"Start":"05:50.790 ","End":"05:52.335","Text":"instead of n, I put x,"},{"Start":"05:52.335 ","End":"05:58.700","Text":"x natural log squared of x dx and ask, is this convergent?"},{"Start":"05:58.700 ","End":"06:00.505","Text":"What does it mean convergence?"},{"Start":"06:00.505 ","End":"06:03.310","Text":"Well, it\u0027s an improper integral that has"},{"Start":"06:03.310 ","End":"06:07.060","Text":"an infinity here so that\u0027s why we can talk about convergence"},{"Start":"06:07.060 ","End":"06:10.810","Text":"and I should have mentioned that in the integral test we want"},{"Start":"06:10.810 ","End":"06:15.925","Text":"a positive series which decreases to 0."},{"Start":"06:15.925 ","End":"06:23.375","Text":"I\u0027m claiming that the 1 over n natural log squared of n,"},{"Start":"06:23.375 ","End":"06:25.530","Text":"first of all, it\u0027s bigger than 0,"},{"Start":"06:25.530 ","End":"06:34.010","Text":"it\u0027s positive and it\u0027s decreasing and it tends to 0,"},{"Start":"06:34.010 ","End":"06:40.655","Text":"which I\u0027ll just write this as decreases to 0."},{"Start":"06:40.655 ","End":"06:42.560","Text":"I won\u0027t go into all the details,"},{"Start":"06:42.560 ","End":"06:45.860","Text":"but basically the denominator is increasing"},{"Start":"06:45.860 ","End":"06:49.580","Text":"because n is increasing and natural log of n is increasing and it"},{"Start":"06:49.580 ","End":"06:52.670","Text":"goes to 0 because the denominator goes to"},{"Start":"06:52.670 ","End":"06:57.695","Text":"infinity and so we use the integral test and this integral test,"},{"Start":"06:57.695 ","End":"07:02.960","Text":"by substitution, no, first of all,"},{"Start":"07:02.960 ","End":"07:05.495","Text":"I say that it\u0027s equal to the limit,"},{"Start":"07:05.495 ","End":"07:10.040","Text":"instead of infinity, I put some other number,"},{"Start":"07:10.040 ","End":"07:15.515","Text":"say b, and then take b goes to infinity instead of the infinity."},{"Start":"07:15.515 ","End":"07:25.215","Text":"I\u0027ll just write it as dx over x natural log squared of x. I\u0027m doing this a bit quickly."},{"Start":"07:25.215 ","End":"07:27.720","Text":"Do it by substitution,"},{"Start":"07:27.720 ","End":"07:36.030","Text":"let\u0027s let t equals natural log of x and then dt"},{"Start":"07:36.030 ","End":"07:47.590","Text":"is 1 over x dx or dx over x and you also have to see what the limits go to."},{"Start":"07:52.850 ","End":"08:01.800","Text":"X equals 2 gives me the t as natural log 2, x equals b,"},{"Start":"08:01.800 ","End":"08:06.150","Text":"t equals natural log b and after all this,"},{"Start":"08:06.150 ","End":"08:08.295","Text":"what we get is"},{"Start":"08:08.295 ","End":"08:14.895","Text":"the limit as b"},{"Start":"08:14.895 ","End":"08:20.475","Text":"goes to infinity of the integral and now we\u0027re working in t,"},{"Start":"08:20.475 ","End":"08:24.045","Text":"dx over x is dt,"},{"Start":"08:24.045 ","End":"08:29.280","Text":"natural log squared of x is t squared,"},{"Start":"08:29.280 ","End":"08:36.245","Text":"and here we have from natural log of 2 to natural log of b."},{"Start":"08:36.245 ","End":"08:45.120","Text":"This is equal to the limit as b goes to infinity of,"},{"Start":"08:45.120 ","End":"08:51.820","Text":"the integral of 1 over t squared is minus 1 over t,"},{"Start":"08:53.540 ","End":"09:00.250","Text":"from natural log 2 to natural log of b."},{"Start":"09:00.440 ","End":"09:04.390","Text":"Let\u0027s continue a bit more."},{"Start":"09:10.190 ","End":"09:19.270","Text":"What I like to do is forget about the minus and reverse the order here so I get 1"},{"Start":"09:19.270 ","End":"09:28.570","Text":"over natural log of 2 minus 1 over natural log of b."},{"Start":"09:34.700 ","End":"09:38.550","Text":"Limit, I forgot to write that."},{"Start":"09:38.550 ","End":"09:40.565","Text":"This is a constant,"},{"Start":"09:40.565 ","End":"09:44.915","Text":"this goes to infinity, so altogether,"},{"Start":"09:44.915 ","End":"09:51.215","Text":"this converges to 1 over natural log of 2."},{"Start":"09:51.215 ","End":"09:52.640","Text":"The answer is not important."},{"Start":"09:52.640 ","End":"09:58.480","Text":"The fact that it is convergent is important and so that settles that."},{"Start":"09:58.480 ","End":"10:01.145","Text":"By the integral test,"},{"Start":"10:01.145 ","End":"10:05.375","Text":"this series is convergent."},{"Start":"10:05.375 ","End":"10:06.730","Text":"We did it without the a squared,"},{"Start":"10:06.730 ","End":"10:09.740","Text":"the a squared makes no difference to convergence."},{"Start":"10:09.740 ","End":"10:17.855","Text":"This series would actually converge to a squared over natural log of 2."},{"Start":"10:17.855 ","End":"10:20.840","Text":"No, it wouldn\u0027t. Sorry, that\u0027s the integral."},{"Start":"10:20.840 ","End":"10:27.740","Text":"Sorry. The integral says that either both of these are convergent or both are divergent."},{"Start":"10:27.740 ","End":"10:29.390","Text":"If the integral converges,"},{"Start":"10:29.390 ","End":"10:32.890","Text":"the series converges and vice versa,"},{"Start":"10:32.890 ","End":"10:35.835","Text":"but it doesn\u0027t equal the same thing."},{"Start":"10:35.835 ","End":"10:43.360","Text":"I can conclude that the sum of the series is this. Sorry about that."},{"Start":"10:44.540 ","End":"10:47.670","Text":"We are done."},{"Start":"10:47.670 ","End":"10:49.410","Text":"By the Weierstrass M-test,"},{"Start":"10:49.410 ","End":"10:56.590","Text":"the original series converges uniformly and we\u0027re done."}],"ID":7971},{"Watched":false,"Name":"Exercise 2 part f","Duration":"12m 37s","ChapterTopicVideoID":7910,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.850","Text":"In this exercise, you want to check this series of functions and see if it\u0027s uniformly"},{"Start":"00:05.850 ","End":"00:10.770","Text":"convergent in the range from"},{"Start":"00:10.770 ","End":"00:16.950","Text":"minus infinity to infinity means the whole number line for all x."},{"Start":"00:16.950 ","End":"00:25.090","Text":"The only real tool that we have is the Weierstrass M-test."},{"Start":"00:30.770 ","End":"00:32.910","Text":"This is a function of x,"},{"Start":"00:32.910 ","End":"00:34.170","Text":"but it also depends on n,"},{"Start":"00:34.170 ","End":"00:43.350","Text":"so let\u0027s just call it the nth function, f_n of x."},{"Start":"00:43.350 ","End":"00:51.030","Text":"Now, if we have such a situation and if we can find a set of numbers,"},{"Start":"00:51.030 ","End":"00:56.590","Text":"and here\u0027s where the name for the M-test comes from."},{"Start":"00:58.400 ","End":"01:02.090","Text":"Well, not constants as far as n goes,"},{"Start":"01:02.090 ","End":"01:03.740","Text":"but constants as far as x goes."},{"Start":"01:03.740 ","End":"01:07.805","Text":"In other words, each function f_n is going to be less than or equal to some number,"},{"Start":"01:07.805 ","End":"01:16.605","Text":"which depends only on n. If we can find such a series of numbers,"},{"Start":"01:16.605 ","End":"01:27.880","Text":"and if we can show that the sum of M_n as a series of numbers converges,"},{"Start":"01:29.600 ","End":"01:36.120","Text":"then we get that the sum of f_n of x,"},{"Start":"01:36.120 ","End":"01:45.850","Text":"which is what we want, converges uniformly."},{"Start":"01:50.870 ","End":"01:56.990","Text":"This has to hold true for x in the particular range."},{"Start":"01:56.990 ","End":"02:01.085","Text":"In our case, we have to show that this is true for all x."},{"Start":"02:01.085 ","End":"02:05.510","Text":"In our case, it\u0027s not quite clear because what we have"},{"Start":"02:05.510 ","End":"02:11.375","Text":"to bound this function, we sometimes say."},{"Start":"02:11.375 ","End":"02:19.730","Text":"How do we bound n squared x over 1 plus n^7 x squared?"},{"Start":"02:19.730 ","End":"02:24.945","Text":"I\u0027m going to say this is less than or equal to something that doesn\u0027t depend on x,"},{"Start":"02:24.945 ","End":"02:30.005","Text":"M, which only depends on n. How do I do this?"},{"Start":"02:30.005 ","End":"02:31.860","Text":"Well, in this exercise,"},{"Start":"02:31.860 ","End":"02:34.495","Text":"I\u0027ll show you a new trick."},{"Start":"02:34.495 ","End":"02:45.140","Text":"What we can do is let this M be the maximum value of this function for all x."},{"Start":"02:45.140 ","End":"02:47.720","Text":"What we do is we want to find"},{"Start":"02:47.720 ","End":"02:55.905","Text":"the maximum of f_n"},{"Start":"02:55.905 ","End":"03:00.100","Text":"of x as a function of x."},{"Start":"03:01.280 ","End":"03:10.480","Text":"This is equal to n squared x over 1 plus n^7 x squared."},{"Start":"03:10.480 ","End":"03:16.055","Text":"Remember this is a function of x. N is a constant, we treat it as."},{"Start":"03:16.055 ","End":"03:20.375","Text":"We\u0027re going to use techniques of calculus to find the maximum,"},{"Start":"03:20.375 ","End":"03:25.245","Text":"choose the derivative, and find critical points."},{"Start":"03:25.245 ","End":"03:33.475","Text":"F_n prime of x using the quotient rule will be the denominator squared."},{"Start":"03:33.475 ","End":"03:38.000","Text":"Then we have the derivative of the numerator,"},{"Start":"03:38.000 ","End":"03:44.725","Text":"which is n squared times the denominator,"},{"Start":"03:44.725 ","End":"03:52.145","Text":"which is 1 plus n^7 x squared minus"},{"Start":"03:52.145 ","End":"04:00.755","Text":"the numerator as is n squared x times the derivative of the denominator,"},{"Start":"04:00.755 ","End":"04:11.850","Text":"which is 2n^7 x."},{"Start":"04:11.850 ","End":"04:13.610","Text":"Let\u0027s just simplify a bit."},{"Start":"04:13.610 ","End":"04:16.069","Text":"Take n squared out of the brackets."},{"Start":"04:16.069 ","End":"04:18.695","Text":"I\u0027ve got n squared,"},{"Start":"04:18.695 ","End":"04:24.875","Text":"and then I have 1 plus,"},{"Start":"04:24.875 ","End":"04:28.775","Text":"here n^7 x squared, and here,"},{"Start":"04:28.775 ","End":"04:30.815","Text":"after I\u0027ve gotten rid of the n squared,"},{"Start":"04:30.815 ","End":"04:35.199","Text":"I have 2n^7 x squared."},{"Start":"04:35.199 ","End":"04:42.120","Text":"Altogether, I will get minus n^7 x squared."},{"Start":"04:42.120 ","End":"04:44.415","Text":"I have plus 1 minus 2 of them,"},{"Start":"04:44.415 ","End":"04:49.740","Text":"so it\u0027s minus 1 of them over the denominator."},{"Start":"04:49.740 ","End":"04:56.030","Text":"I don\u0027t really need this because I\u0027m looking for critical points where it\u0027s 0,"},{"Start":"04:56.030 ","End":"04:58.715","Text":"I just care about the numerator."},{"Start":"04:58.715 ","End":"05:00.695","Text":"This is also a constant."},{"Start":"05:00.695 ","End":"05:11.015","Text":"So what I get is that 1 minus n^7 x squared equals 0."},{"Start":"05:11.015 ","End":"05:14.540","Text":"Then if I bring this to the other side,"},{"Start":"05:14.540 ","End":"05:24.050","Text":"divide by n^7, I get that x squared is 1 over n^7."},{"Start":"05:24.050 ","End":"05:27.070","Text":"Then I take the square root,"},{"Start":"05:27.070 ","End":"05:32.060","Text":"so I have that x is equal to plus or"},{"Start":"05:32.060 ","End":"05:38.740","Text":"minus the square root of 1 over n^7."},{"Start":"05:38.740 ","End":"05:41.150","Text":"How do I know which is the maximum,"},{"Start":"05:41.150 ","End":"05:42.830","Text":"which is the minimum?"},{"Start":"05:42.830 ","End":"05:46.960","Text":"Well, we could use the second derivative test,"},{"Start":"05:46.960 ","End":"05:49.990","Text":"but it has a shortcut."},{"Start":"05:53.390 ","End":"06:00.100","Text":"Normally, I would say f_n double-prime and then I would substitute"},{"Start":"06:00.100 ","End":"06:08.200","Text":"the critical points here and see if I get plus or minus the critical points."},{"Start":"06:08.200 ","End":"06:11.530","Text":"The thing is that this gets very messy,"},{"Start":"06:11.530 ","End":"06:16.990","Text":"and there\u0027s a shortcut that if our original function came from a quotient and"},{"Start":"06:16.990 ","End":"06:22.239","Text":"we got f_n prime using derivative of quotient,"},{"Start":"06:22.239 ","End":"06:27.970","Text":"instead of differentiating all of this,"},{"Start":"06:27.970 ","End":"06:33.640","Text":"we just have to differentiate the numerator as far as checking for critical points."},{"Start":"06:33.640 ","End":"06:38.710","Text":"I\u0027m going to use the shortcut and say I want"},{"Start":"06:38.710 ","End":"06:46.310","Text":"to take the derivative of n squared,"},{"Start":"06:46.310 ","End":"06:50.505","Text":"1 minus n squared x squared prime."},{"Start":"06:50.505 ","End":"06:53.540","Text":"Instead of f_n double-prime, the shortcut,"},{"Start":"06:53.540 ","End":"07:00.130","Text":"maybe I\u0027ll write the word shortcut."},{"Start":"07:02.540 ","End":"07:05.460","Text":"Instead of f_n double-prime,"},{"Start":"07:05.460 ","End":"07:06.959","Text":"we take this instead,"},{"Start":"07:06.959 ","End":"07:10.650","Text":"just the derivative of the numerator part."},{"Start":"07:10.790 ","End":"07:16.005","Text":"This is equal to n squared is a constant,"},{"Start":"07:16.005 ","End":"07:27.465","Text":"so I just get n squared times minus,"},{"Start":"07:27.465 ","End":"07:33.855","Text":"this is a 7, so minus 2x,"},{"Start":"07:33.855 ","End":"07:39.510","Text":"so it\u0027s minus 2n^7 x,"},{"Start":"07:39.510 ","End":"07:42.090","Text":"and this is equal"},{"Start":"07:42.090 ","End":"07:59.070","Text":"to minus 2n^9 x."},{"Start":"07:59.070 ","End":"08:01.985","Text":"Now, this thing is positive,"},{"Start":"08:01.985 ","End":"08:07.170","Text":"so basically this comes out to be"},{"Start":"08:09.370 ","End":"08:19.125","Text":"positive if x is negative,"},{"Start":"08:19.125 ","End":"08:24.480","Text":"and negative if x is positive."},{"Start":"08:24.480 ","End":"08:33.370","Text":"The plus is the minimum"},{"Start":"08:34.480 ","End":"08:40.715","Text":"and the minus is the maximum."},{"Start":"08:40.715 ","End":"08:46.710","Text":"The maximum is when we take the positive 1."},{"Start":"08:49.360 ","End":"09:00.650","Text":"Our x maximum is"},{"Start":"09:00.650 ","End":"09:01.880","Text":"when x is positive,"},{"Start":"09:01.880 ","End":"09:07.820","Text":"so it\u0027s the square root of 1 over n^7,"},{"Start":"09:07.820 ","End":"09:12.635","Text":"and we can also write this as n^3 1/2,"},{"Start":"09:12.635 ","End":"09:20.880","Text":"minus 3.5 in decimal."},{"Start":"09:21.040 ","End":"09:25.655","Text":"Now, I don\u0027t need this."},{"Start":"09:25.655 ","End":"09:28.525","Text":"Just erase this."},{"Start":"09:28.525 ","End":"09:33.030","Text":"What I do need is f_n"},{"Start":"09:33.030 ","End":"09:38.990","Text":"of x_max because I don\u0027t want the value where the maximum is obtained,"},{"Start":"09:38.990 ","End":"09:41.915","Text":"I want the actual maximum value."},{"Start":"09:41.915 ","End":"09:45.290","Text":"I highlighted it earlier when you weren\u0027t looking."},{"Start":"09:45.290 ","End":"09:55.065","Text":"What we want is to plug this into here and what we get is n"},{"Start":"09:55.065 ","End":"10:00.690","Text":"squared n^minus"},{"Start":"10:00.690 ","End":"10:09.120","Text":"3.5 over 1 plus n^7,"},{"Start":"10:09.120 ","End":"10:14.680","Text":"and x squared is,"},{"Start":"10:15.770 ","End":"10:18.765","Text":"well, it\u0027s 1 over n^7,"},{"Start":"10:18.765 ","End":"10:23.050","Text":"is n to the minus 7."},{"Start":"10:24.170 ","End":"10:28.920","Text":"This comes out to be just,"},{"Start":"10:28.920 ","End":"10:30.584","Text":"this with this cancels,"},{"Start":"10:30.584 ","End":"10:39.290","Text":"it\u0027s 1/2 of n^ minus 1 1/2,"},{"Start":"10:39.290 ","End":"10:46.940","Text":"which is 1 over n^1.5,"},{"Start":"10:46.940 ","End":"10:51.240","Text":"of the minus 1.5,"},{"Start":"10:51.240 ","End":"10:53.890","Text":"I\u0027ll put it in the denominator."},{"Start":"10:54.050 ","End":"10:58.230","Text":"This is the M_n from here."},{"Start":"10:58.230 ","End":"11:00.180","Text":"Just to emphasize that."},{"Start":"11:00.180 ","End":"11:09.195","Text":"What I\u0027m looking for is the sum of the series M_n and see if it converges."},{"Start":"11:09.195 ","End":"11:12.255","Text":"What we\u0027re reduced to now is asking,"},{"Start":"11:12.255 ","End":"11:21.435","Text":"is the sum of 1/2 times 1"},{"Start":"11:21.435 ","End":"11:27.690","Text":"over n^1.5 from 1"},{"Start":"11:27.690 ","End":"11:32.740","Text":"to infinity; is this convergent?"},{"Start":"11:35.150 ","End":"11:39.390","Text":"I claim that the answer is yes."},{"Start":"11:39.390 ","End":"11:42.325","Text":"The 1/2 is not important."},{"Start":"11:42.325 ","End":"11:47.255","Text":"Basically, we\u0027ve seen this before we use the p-series."},{"Start":"11:47.255 ","End":"11:51.400","Text":"A p-series is a generalized harmonic series,"},{"Start":"11:51.400 ","End":"11:54.085","Text":"is instead of 1 over n,"},{"Start":"11:54.085 ","End":"11:58.610","Text":"we have 1 over n^p, and it"},{"Start":"11:58.610 ","End":"12:06.355","Text":"converges if p is greater than 1."},{"Start":"12:06.355 ","End":"12:07.910","Text":"Now, in our case,"},{"Start":"12:07.910 ","End":"12:13.295","Text":"we have p equals 1.5,"},{"Start":"12:13.295 ","End":"12:17.500","Text":"which is certainly greater than 1,"},{"Start":"12:17.500 ","End":"12:23.350","Text":"and so that explains why this is convergent,"},{"Start":"12:23.350 ","End":"12:27.815","Text":"and everything falls into place now because if this series is convergent,"},{"Start":"12:27.815 ","End":"12:35.105","Text":"this sum converges, and so our series of functions converges uniformly,"},{"Start":"12:35.105 ","End":"12:38.250","Text":"and we are done."}],"ID":7972},{"Watched":false,"Name":"Exercise 3 part a","Duration":"6m 28s","ChapterTopicVideoID":7911,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"In this exercise, we have to find the radius and the range of conversion of this series."},{"Start":"00:05.970 ","End":"00:14.750","Text":"Now, in general, a series is the sum of a_n x to the n. In our case,"},{"Start":"00:14.750 ","End":"00:17.990","Text":"a_n is just the coefficient of x to the n,"},{"Start":"00:17.990 ","End":"00:21.485","Text":"which is 1 over n plus 1."},{"Start":"00:21.485 ","End":"00:30.630","Text":"Now, 1 of the formulas for radius of convergence R is the limit"},{"Start":"00:30.630 ","End":"00:35.655","Text":"as n goes to infinity"},{"Start":"00:35.655 ","End":"00:42.100","Text":"of the absolute value of a_n over a_n plus 1."},{"Start":"00:42.100 ","End":"00:44.595","Text":"Let\u0027s see what we get in our case."},{"Start":"00:44.595 ","End":"00:47.235","Text":"It\u0027s going to be the limit."},{"Start":"00:47.235 ","End":"00:54.070","Text":"Now, a_n is 1 over n plus 1,"},{"Start":"00:54.620 ","End":"01:03.430","Text":"and a_n plus 1 is 1 over n plus 2."},{"Start":"01:04.070 ","End":"01:11.190","Text":"Everything\u0027s positive, and also I can divide the fraction in the denominator."},{"Start":"01:11.190 ","End":"01:14.250","Text":"I can turn it into multiplication."},{"Start":"01:14.250 ","End":"01:22.435","Text":"Basically, we get the limit of n plus 2 over n plus 1."},{"Start":"01:22.435 ","End":"01:26.100","Text":"N goes to infinity."},{"Start":"01:26.950 ","End":"01:31.790","Text":"There are several ways of doing this, basically,"},{"Start":"01:31.790 ","End":"01:34.190","Text":"you could just treat them as polynomial over"},{"Start":"01:34.190 ","End":"01:40.130","Text":"polynomial and just look at the leading coefficients in top and bottom."},{"Start":"01:40.130 ","End":"01:50.530","Text":"That\u0027s n over n. Limit of n over n, which equals 1."},{"Start":"01:52.220 ","End":"01:55.390","Text":"If the radius of convergence is 1,"},{"Start":"01:55.390 ","End":"01:59.589","Text":"let\u0027s sketch this on the number line,"},{"Start":"01:59.589 ","End":"02:02.300","Text":"so that this be the number line."},{"Start":"02:02.300 ","End":"02:06.825","Text":"Let\u0027s mark on 1 and minus 1."},{"Start":"02:06.825 ","End":"02:09.000","Text":"Zero is in the middle,"},{"Start":"02:09.000 ","End":"02:11.264","Text":"and let\u0027s say this is 0,"},{"Start":"02:11.264 ","End":"02:13.140","Text":"so I\u0027ve got 1 either way,"},{"Start":"02:13.140 ","End":"02:18.615","Text":"so this is 1, and this is minus 1."},{"Start":"02:18.615 ","End":"02:23.480","Text":"What I know, if I put some dotted lines here,"},{"Start":"02:24.230 ","End":"02:26.925","Text":"decided to go with coloring."},{"Start":"02:26.925 ","End":"02:29.260","Text":"Within the radius of convergence,"},{"Start":"02:29.260 ","End":"02:31.495","Text":"within plus or minus 1,"},{"Start":"02:31.495 ","End":"02:34.585","Text":"we can say that here it converges,"},{"Start":"02:34.585 ","End":"02:41.475","Text":"here it diverges, and here too, it diverges."},{"Start":"02:41.475 ","End":"02:45.680","Text":"The only question is, what happens exactly at these 2 points here?"},{"Start":"02:45.680 ","End":"02:48.120","Text":"We don\u0027t know as yet,"},{"Start":"02:48.120 ","End":"02:54.415","Text":"so we have to substitute in the original series and see what happens."},{"Start":"02:54.415 ","End":"02:56.909","Text":"Let\u0027s take this 1 first,"},{"Start":"02:56.909 ","End":"02:59.490","Text":"x equals 1, and later,"},{"Start":"02:59.490 ","End":"03:01.350","Text":"we will do the minus 1."},{"Start":"03:01.350 ","End":"03:07.310","Text":"When x equals 1, then the original series we get is the sum."},{"Start":"03:07.310 ","End":"03:17.045","Text":"N goes from 0 to infinity of x to the n, which is 1."},{"Start":"03:17.045 ","End":"03:21.630","Text":"It\u0027s 1 over n plus 1."},{"Start":"03:23.470 ","End":"03:28.110","Text":"What this is, if I write it out,"},{"Start":"03:28.110 ","End":"03:30.900","Text":"is just when n is 0, we get 1."},{"Start":"03:30.900 ","End":"03:32.295","Text":"When is 1, we get 2."},{"Start":"03:32.295 ","End":"03:39.620","Text":"We just get 1 plus a 1/2 plus a 1/3 plus so on."},{"Start":"03:39.620 ","End":"03:44.480","Text":"It\u0027s just the harmonic series shifted by 1."},{"Start":"03:44.480 ","End":"03:46.820","Text":"N starts from 0."},{"Start":"03:46.820 ","End":"03:56.215","Text":"It\u0027s exactly the same as if we start at 1 and then just put 1 over n. It\u0027s shifted."},{"Start":"03:56.215 ","End":"03:58.790","Text":"This would give us exactly the same thing."},{"Start":"03:58.790 ","End":"04:00.800","Text":"1 plus 1/2 plus 1/3,"},{"Start":"04:00.800 ","End":"04:05.510","Text":"and we know that this is the harmonic series,"},{"Start":"04:05.510 ","End":"04:06.920","Text":"and we\u0027ve seen it before,"},{"Start":"04:06.920 ","End":"04:09.810","Text":"and we know that it diverges."},{"Start":"04:13.360 ","End":"04:23.340","Text":"This point, I\u0027ll call it just d for diverges."},{"Start":"04:23.340 ","End":"04:25.275","Text":"Sorry, that\u0027s this 1."},{"Start":"04:25.275 ","End":"04:28.145","Text":"Now the next 1, the next question mark,"},{"Start":"04:28.145 ","End":"04:37.020","Text":"which is the minus 1 gives us the sum and goes from 0 to infinity of when x is minus 1,"},{"Start":"04:37.020 ","End":"04:41.310","Text":"we get minus 1 to the n over n plus 1,"},{"Start":"04:41.310 ","End":"04:43.955","Text":"and if you see what this is,"},{"Start":"04:43.955 ","End":"04:46.790","Text":"it\u0027s like this, except that we have alternating signs."},{"Start":"04:46.790 ","End":"04:50.180","Text":"When n is 0, this is1."},{"Start":"04:50.180 ","End":"04:51.900","Text":"For the evens, it\u0027s 1,"},{"Start":"04:51.900 ","End":"04:53.595","Text":"and for the odds, it\u0027s minus 1."},{"Start":"04:53.595 ","End":"05:00.735","Text":"We get 1 minus 1/2 plus 1/3 minus 1/4 plus, and so on."},{"Start":"05:00.735 ","End":"05:09.500","Text":"This is actually convergent because this is a Leibniz alternating series."},{"Start":"05:09.500 ","End":"05:14.360","Text":"We think about it, the alternating is from the signs,"},{"Start":"05:14.360 ","End":"05:16.400","Text":"but it alternates this series."},{"Start":"05:16.400 ","End":"05:19.100","Text":"Now this series, essentially,"},{"Start":"05:19.100 ","End":"05:21.280","Text":"it\u0027s the 1 over n plus 1."},{"Start":"05:21.280 ","End":"05:26.025","Text":"Now, the alternating we have, decreasing we have,"},{"Start":"05:26.025 ","End":"05:30.215","Text":"which is 1 of the conditions for the Leibniz theorem,"},{"Start":"05:30.215 ","End":"05:32.615","Text":"and also tends to 0,"},{"Start":"05:32.615 ","End":"05:36.325","Text":"which obviously it is."},{"Start":"05:36.325 ","End":"05:40.935","Text":"By the Leibniz alternating series theorem,"},{"Start":"05:40.935 ","End":"05:48.425","Text":"when we have alternating signs on a series which decreases to 0, then it\u0027s convergent."},{"Start":"05:48.425 ","End":"05:53.520","Text":"This is also a convergent point."},{"Start":"05:53.520 ","End":"05:55.455","Text":"If we want to take all the convergence,"},{"Start":"05:55.455 ","End":"06:00.060","Text":"it\u0027s between minus 1 and 1 not including the 1,"},{"Start":"06:00.060 ","End":"06:02.100","Text":"but including the minus 1."},{"Start":"06:02.100 ","End":"06:06.750","Text":"What we get is minus 1 less than or"},{"Start":"06:06.750 ","End":"06:13.515","Text":"equal to x strictly less than 1."},{"Start":"06:13.515 ","End":"06:19.385","Text":"This is the radius of convergence,"},{"Start":"06:19.385 ","End":"06:23.780","Text":"and this is the range of convergence."},{"Start":"06:23.780 ","End":"06:28.020","Text":"We\u0027ve got the radius, we\u0027ve got the range, and we\u0027re done."}],"ID":7973},{"Watched":false,"Name":"Exercise 3 part b","Duration":"2m 44s","ChapterTopicVideoID":7912,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this exercise, we have a series."},{"Start":"00:04.185 ","End":"00:07.350","Text":"We have to find the radius of convergence,"},{"Start":"00:07.350 ","End":"00:09.450","Text":"and also the range."},{"Start":"00:09.450 ","End":"00:16.490","Text":"Now, the general term a_n is equal to"},{"Start":"00:16.490 ","End":"00:23.955","Text":"minus 1 to the n over n factorial."},{"Start":"00:23.955 ","End":"00:33.630","Text":"What we can do is use the formula that the radius of convergence is the limit as n goes"},{"Start":"00:33.630 ","End":"00:43.600","Text":"to infinity of a_n over a_n plus 1 in absolute value."},{"Start":"00:43.600 ","End":"00:45.320","Text":"Before I take the limit,"},{"Start":"00:45.320 ","End":"00:47.045","Text":"let me just see what is this,"},{"Start":"00:47.045 ","End":"00:50.195","Text":"a_n over a_n plus 1."},{"Start":"00:50.195 ","End":"00:54.550","Text":"I substitute separately n an n plus 1 in here and divide,"},{"Start":"00:54.550 ","End":"00:59.135","Text":"but of course, I can throw out the minus 1 to the n because we\u0027re taking absolute value,"},{"Start":"00:59.135 ","End":"01:01.985","Text":"so we just get on the numerator,"},{"Start":"01:01.985 ","End":"01:08.045","Text":"1 over n factorial and on the denominator,"},{"Start":"01:08.045 ","End":"01:13.505","Text":"1 over n plus 1 factorial."},{"Start":"01:13.505 ","End":"01:21.615","Text":"This is equal to n plus 1 factorial over n factorial."},{"Start":"01:21.615 ","End":"01:24.405","Text":"Remember, we have a formula,"},{"Start":"01:24.405 ","End":"01:31.965","Text":"it\u0027s pretty obvious that n plus 1 factorial is n plus 1 times n factorial,"},{"Start":"01:31.965 ","End":"01:35.800","Text":"seen this before, the product of the numbers from 1-n,"},{"Start":"01:35.800 ","End":"01:40.570","Text":"and then times n plus 1 is just the product up to n minus 1."},{"Start":"01:40.570 ","End":"01:43.285","Text":"This over n factorial,"},{"Start":"01:43.285 ","End":"01:48.675","Text":"this cancels, so what we\u0027re left with is n plus 1."},{"Start":"01:48.675 ","End":"01:50.370","Text":"I\u0027m continuing here."},{"Start":"01:50.370 ","End":"01:59.419","Text":"This is the limit as n goes to infinity of n plus 1,"},{"Start":"01:59.419 ","End":"02:01.995","Text":"and that\u0027s clearly infinity,"},{"Start":"02:01.995 ","End":"02:06.865","Text":"so we have that the radius of convergence is infinity,"},{"Start":"02:06.865 ","End":"02:08.575","Text":"and when this happens,"},{"Start":"02:08.575 ","End":"02:12.180","Text":"that means that it converges for all x,"},{"Start":"02:12.180 ","End":"02:17.220","Text":"so that\u0027s the radius,"},{"Start":"02:17.220 ","End":"02:21.800","Text":"and we could say all x. Yeah,"},{"Start":"02:21.800 ","End":"02:26.940","Text":"sometimes we just say minus infinity, less than x,"},{"Start":"02:26.940 ","End":"02:33.520","Text":"less than infinity, that\u0027s the range of convergence."},{"Start":"02:40.700 ","End":"02:43.810","Text":"That\u0027s it, we\u0027re done."}],"ID":7974},{"Watched":false,"Name":"Exercise 3 part c","Duration":"6m 43s","ChapterTopicVideoID":7913,"CourseChapterTopicPlaylistID":4250,"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 to find the radius of"},{"Start":"00:02.730 ","End":"00:06.690","Text":"convergence and the range of convergence for this series."},{"Start":"00:06.690 ","End":"00:09.989","Text":"The general term a_n,"},{"Start":"00:09.989 ","End":"00:18.720","Text":"I mean the coefficient a_n here is 5_n over n squared."},{"Start":"00:18.720 ","End":"00:23.265","Text":"There\u0027s 2 main formulas to use to find the radius of convergence."},{"Start":"00:23.265 ","End":"00:26.145","Text":"One of them is a ratio, one of them is a root."},{"Start":"00:26.145 ","End":"00:28.815","Text":"Both will work. Let\u0027s try the one with the root."},{"Start":"00:28.815 ","End":"00:36.465","Text":"The formula would be that R is the limit as n goes to infinity of 1"},{"Start":"00:36.465 ","End":"00:45.005","Text":"over the nth root of a_n not quite in absolute value is what we have."},{"Start":"00:45.005 ","End":"00:49.700","Text":"Now, let me just compute the denominator here."},{"Start":"00:49.700 ","End":"00:57.860","Text":"In our case, the nth root of absolute value of a_n."},{"Start":"00:57.860 ","End":"01:00.275","Text":"Since a_n is positive,"},{"Start":"01:00.275 ","End":"01:06.275","Text":"I can basically just throw out the absolute value and I get the nth root"},{"Start":"01:06.275 ","End":"01:14.345","Text":"of 5_n over n squared."},{"Start":"01:14.345 ","End":"01:17.210","Text":"Take the route separately for the top and the bottom."},{"Start":"01:17.210 ","End":"01:21.905","Text":"For the top, I get 5 and for the bottom,"},{"Start":"01:21.905 ","End":"01:27.625","Text":"I get the nth root of n squared."},{"Start":"01:27.625 ","End":"01:31.770","Text":"What we want now is the limit."},{"Start":"01:31.770 ","End":"01:41.190","Text":"We have that R is the limit as n goes to infinity 1 over this."},{"Start":"01:41.190 ","End":"01:44.380","Text":"If it\u0027s 1 over I can just,"},{"Start":"01:47.630 ","End":"01:54.925","Text":"nth root of n squared over 5."},{"Start":"01:54.925 ","End":"02:03.995","Text":"Now, I\u0027d like to point out a result that you should be familiar with is that the limit"},{"Start":"02:03.995 ","End":"02:14.240","Text":"as n goes to infinity of the nth root of n is 1 and in fact,"},{"Start":"02:14.240 ","End":"02:19.790","Text":"we can even put any value k here because that limit would be 1 to the k,"},{"Start":"02:19.790 ","End":"02:24.020","Text":"which is also 1, and this is a result worth remembering."},{"Start":"02:24.020 ","End":"02:29.310","Text":"If we apply it in our case with k equals 2,"},{"Start":"02:29.340 ","End":"02:33.770","Text":"then we get that this is 1/5."},{"Start":"02:34.510 ","End":"02:39.800","Text":"I borrowed a sketch from a previous exercise,"},{"Start":"02:39.800 ","End":"02:45.350","Text":"where this now is 1/5 and this is minus a 1/5."},{"Start":"02:45.350 ","End":"02:47.690","Text":"The radius of convergence is about 0."},{"Start":"02:47.690 ","End":"02:52.130","Text":"We take plus and minus a 1/5 and then we know that inside"},{"Start":"02:52.130 ","End":"02:57.665","Text":"this radius we have convergence and outside we have divergence."},{"Start":"02:57.665 ","End":"03:03.860","Text":"But we have to separately check for the actual borders for 1/5,"},{"Start":"03:03.860 ","End":"03:06.380","Text":"and for minus a 1/5."},{"Start":"03:06.380 ","End":"03:10.610","Text":"If x is plus 1/5,"},{"Start":"03:10.610 ","End":"03:13.520","Text":"what we get is the series,"},{"Start":"03:13.520 ","End":"03:18.025","Text":"the sum from 1 to infinity."},{"Start":"03:18.025 ","End":"03:24.330","Text":"Now, x_n, 5_n will just be 1."},{"Start":"03:24.330 ","End":"03:26.715","Text":"Because if x is 1/5,"},{"Start":"03:26.715 ","End":"03:29.460","Text":"maybe I\u0027ll just write this out it\u0027ll be easier,"},{"Start":"03:29.460 ","End":"03:37.820","Text":"5_n over n squared and x is 1/5 to the power of n. Well,"},{"Start":"03:37.820 ","End":"03:41.675","Text":"this is like 5_n, on the denominator and it cancels with this."},{"Start":"03:41.675 ","End":"03:50.280","Text":"It\u0027s just the sum from 1 to infinity of 1 over n squared."},{"Start":"03:50.770 ","End":"03:55.850","Text":"This is a well-known convergent series."},{"Start":"03:55.850 ","End":"03:58.639","Text":"You probably seen it before many times,"},{"Start":"03:58.639 ","End":"04:03.845","Text":"but I can remind you why this is convergent or give you a reason."},{"Start":"04:03.845 ","End":"04:09.360","Text":"One of the ways is that this is a p-series."},{"Start":"04:09.460 ","End":"04:13.550","Text":"P-series is a generalized harmonic series,"},{"Start":"04:13.550 ","End":"04:17.820","Text":"is the sum of 1 over n_p."},{"Start":"04:18.590 ","End":"04:23.315","Text":"We know that this is convergent."},{"Start":"04:23.315 ","End":"04:31.430","Text":"It converges for p bigger than 1 and in our case,"},{"Start":"04:31.430 ","End":"04:34.435","Text":"p equals 2, so it is bigger than 1."},{"Start":"04:34.435 ","End":"04:37.020","Text":"This is okay."},{"Start":"04:37.020 ","End":"04:41.150","Text":"Now if we take x equals minus a 1/5,"},{"Start":"04:41.150 ","End":"04:42.710","Text":"the difference will be,"},{"Start":"04:42.710 ","End":"04:46.609","Text":"is that we\u0027ll get an extra minus in here,"},{"Start":"04:46.609 ","End":"04:54.800","Text":"so that the series will be the sum of n goes from 1 to infinity."},{"Start":"04:54.800 ","End":"04:56.180","Text":"We\u0027ll just get the same as this,"},{"Start":"04:56.180 ","End":"05:02.610","Text":"but with a minus 1 to the n over n squared."},{"Start":"05:03.400 ","End":"05:06.710","Text":"I\u0027m claiming that this is also convergent."},{"Start":"05:06.710 ","End":"05:08.450","Text":"There are several ways of doing it,"},{"Start":"05:08.450 ","End":"05:13.190","Text":"you could use the Leibniz alternating series test,"},{"Start":"05:13.190 ","End":"05:19.910","Text":"but there\u0027s something even easier because if I take the absolute value of this,"},{"Start":"05:19.910 ","End":"05:23.870","Text":"then I\u0027ll get the same as this series."},{"Start":"05:23.870 ","End":"05:30.810","Text":"This 1 is absolutely convergent"},{"Start":"05:31.030 ","End":"05:34.400","Text":"because the absolute value of the terms converge"},{"Start":"05:34.400 ","End":"05:38.915","Text":"and it is well-known that if something\u0027s absolutely convergent,"},{"Start":"05:38.915 ","End":"05:44.220","Text":"then that implies that it\u0027s also just plain convergent."},{"Start":"05:46.190 ","End":"05:50.435","Text":"I have no doubt about the question marks now."},{"Start":"05:50.435 ","End":"05:53.725","Text":"I can erase those."},{"Start":"05:53.725 ","End":"05:56.240","Text":"Well, don\u0027t have room for convergent,"},{"Start":"05:56.240 ","End":"05:58.910","Text":"but I\u0027ll write c for convergent,"},{"Start":"05:58.910 ","End":"06:03.260","Text":"and c for convergent here, converges there."},{"Start":"06:03.260 ","End":"06:07.840","Text":"Altogether I have that,"},{"Start":"06:07.840 ","End":"06:11.435","Text":"it converges including the borders."},{"Start":"06:11.435 ","End":"06:20.040","Text":"We have minus 1/5 less than or equal to x,"},{"Start":"06:20.040 ","End":"06:22.470","Text":"less than or equal to 1/5."},{"Start":"06:22.470 ","End":"06:28.040","Text":"This is the range of"},{"Start":"06:28.040 ","End":"06:34.255","Text":"convergence and this is the radius of convergence."},{"Start":"06:34.255 ","End":"06:36.440","Text":"We\u0027ve answered both questions,"},{"Start":"06:36.440 ","End":"06:43.290","Text":"the range of convergence and the radius of convergence and we are done."}],"ID":7975},{"Watched":false,"Name":"Exercise 3 part d","Duration":"11m 11s","ChapterTopicVideoID":7914,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we need to find the radius of"},{"Start":"00:04.050 ","End":"00:09.210","Text":"convergence and also the range of convergence for this series."},{"Start":"00:09.210 ","End":"00:11.954","Text":"For this series, which is a power series,"},{"Start":"00:11.954 ","End":"00:16.605","Text":"the coefficients are a_n equals"},{"Start":"00:16.605 ","End":"00:23.910","Text":"sine squared of 1."},{"Start":"00:23.910 ","End":"00:27.240","Text":"One of the formulas for radius of convergence R"},{"Start":"00:27.240 ","End":"00:31.050","Text":"is that it\u0027s equal to the limit as n goes to"},{"Start":"00:31.050 ","End":"00:39.810","Text":"infinity of the absolute value of a_n over a_n plus 1."},{"Start":"00:39.810 ","End":"00:42.500","Text":"Let\u0027s see if we can compute this."},{"Start":"00:42.500 ","End":"00:51.675","Text":"First of all, let\u0027s see what is the absolute value of a_n over a_n plus 1."},{"Start":"00:51.675 ","End":"00:54.385","Text":"What is this equal to?"},{"Start":"00:54.385 ","End":"01:01.200","Text":"It\u0027s equal to, we don\u0027t need the absolute value because everything\u0027s positive."},{"Start":"01:01.810 ","End":"01:07.235","Text":"We just get sine squared of"},{"Start":"01:07.235 ","End":"01:16.154","Text":"1 over sine squared 1 plus 1,"},{"Start":"01:16.154 ","End":"01:18.905","Text":"and if I want to,"},{"Start":"01:18.905 ","End":"01:20.435","Text":"and I might want to,"},{"Start":"01:20.435 ","End":"01:25.050","Text":"I could write this as something squared sine of"},{"Start":"01:25.050 ","End":"01:33.060","Text":"1 over sine of 1 plus 1 squared."},{"Start":"01:33.060 ","End":"01:35.880","Text":"I\u0027ll use this or this. Let\u0027s see."},{"Start":"01:35.880 ","End":"01:43.815","Text":"Our radius is the limit as n goes to infinity."},{"Start":"01:43.815 ","End":"01:53.495","Text":"I\u0027ll use this form because if I have the limit of something squared,"},{"Start":"01:53.495 ","End":"01:59.510","Text":"I can take the whole limit of whatever it was and make that squared,"},{"Start":"01:59.510 ","End":"02:03.620","Text":"so I don\u0027t need to deal with the squared till the end."},{"Start":"02:03.620 ","End":"02:13.130","Text":"I can just put here n goes to infinity of just this bit,"},{"Start":"02:13.130 ","End":"02:21.385","Text":"sine of 1 over sine of 1 plus 1."},{"Start":"02:21.385 ","End":"02:27.680","Text":"First of all, I\u0027ll deal with this limit and at the end, we\u0027ll square it."},{"Start":"02:27.680 ","End":"02:31.055","Text":"There are many ways to tackle this."},{"Start":"02:31.055 ","End":"02:35.270","Text":"I\u0027m going to tackle it using the famous formula"},{"Start":"02:35.270 ","End":"02:44.990","Text":"that the limit as"},{"Start":"02:44.990 ","End":"02:51.275","Text":"x goes to 0 of sine x over x is equal to 1."},{"Start":"02:51.275 ","End":"02:57.530","Text":"In our case, when n goes to infinity,"},{"Start":"02:57.530 ","End":"03:04.645","Text":"we have that 1 tends to 0 and also 1 plus 1 tends to 0,"},{"Start":"03:04.645 ","End":"03:07.985","Text":"so we can use this property,"},{"Start":"03:07.985 ","End":"03:10.820","Text":"but we don\u0027t have like sine x over x,"},{"Start":"03:10.820 ","End":"03:12.950","Text":"so we do some algebra."},{"Start":"03:12.950 ","End":"03:15.350","Text":"Let me call this bit asterisk,"},{"Start":"03:15.350 ","End":"03:17.030","Text":"and I\u0027ll do this over here."},{"Start":"03:17.030 ","End":"03:24.200","Text":"What we get is the limit as n goes to infinity of."},{"Start":"03:24.200 ","End":"03:26.675","Text":"Now, to make it look like this,"},{"Start":"03:26.675 ","End":"03:32.390","Text":"I can write sine of 1 over 1."},{"Start":"03:32.390 ","End":"03:35.780","Text":"I\u0027m thinking of 1 here as the x here."},{"Start":"03:35.780 ","End":"03:38.210","Text":"Big dividing line."},{"Start":"03:38.210 ","End":"03:43.730","Text":"Sine of 1 plus"},{"Start":"03:43.730 ","End":"03:49.530","Text":"1 over 1 plus 1."},{"Start":"03:49.530 ","End":"03:54.780","Text":"But now I have to correct this because I\u0027ve changed the exercise."},{"Start":"03:54.780 ","End":"04:03.605","Text":"So to compensate, I multiply the numerator by a 1 so that 1/ n and 1 cancel,"},{"Start":"04:03.605 ","End":"04:07.770","Text":"and here, 1 plus 1."},{"Start":"04:09.640 ","End":"04:13.640","Text":"What I get is"},{"Start":"04:13.640 ","End":"04:21.590","Text":"actually the numerator will go to 1,"},{"Start":"04:21.590 ","End":"04:24.870","Text":"this and the denominator will go to 1."},{"Start":"04:25.180 ","End":"04:30.029","Text":"This, let me do this one separately at the side."},{"Start":"04:32.830 ","End":"04:43.725","Text":"1 over 1 plus 1 is the same as n plus 1,"},{"Start":"04:43.725 ","End":"04:50.190","Text":"which is 1 plus 1."},{"Start":"04:50.190 ","End":"04:54.050","Text":"What I\u0027m saying is that this bit tends to 1"},{"Start":"04:54.050 ","End":"04:58.370","Text":"as n goes to infinity because 1 is like x here,"},{"Start":"04:58.370 ","End":"05:00.740","Text":"and here, 1 plus 1 is like x."},{"Start":"05:00.740 ","End":"05:05.370","Text":"So this bit also tends to 1."},{"Start":"05:05.680 ","End":"05:10.055","Text":"This is like 1 plus 1."},{"Start":"05:10.055 ","End":"05:15.050","Text":"This also tends to 1,"},{"Start":"05:15.050 ","End":"05:17.335","Text":"because then goes to infinity,"},{"Start":"05:17.335 ","End":"05:20.150","Text":"1/ n goes to 0, and this goes to 1."},{"Start":"05:20.150 ","End":"05:24.440","Text":"All together, this is equal."},{"Start":"05:24.440 ","End":"05:26.210","Text":"The asterisk came out to be 1,"},{"Start":"05:26.210 ","End":"05:27.754","Text":"so this is 1 squared,"},{"Start":"05:27.754 ","End":"05:29.990","Text":"so this is 1."},{"Start":"05:29.990 ","End":"05:34.390","Text":"Now that we\u0027ve done that, let me put in a little picture."},{"Start":"05:34.390 ","End":"05:38.015","Text":"If 1 is the radius of convergence,"},{"Start":"05:38.015 ","End":"05:41.780","Text":"then I take from 0 plus 1 and minus 1,"},{"Start":"05:41.780 ","End":"05:43.400","Text":"1 in both directions,"},{"Start":"05:43.400 ","End":"05:45.455","Text":"and it converges here."},{"Start":"05:45.455 ","End":"05:48.170","Text":"I also know that outside it diverges."},{"Start":"05:48.170 ","End":"05:50.480","Text":"But in the case of power series,"},{"Start":"05:50.480 ","End":"05:52.115","Text":"when we have the radius of convergence,"},{"Start":"05:52.115 ","End":"05:59.554","Text":"we don\u0027t know what happens actually on the border at exactly a distance of 1 from 0,"},{"Start":"05:59.554 ","End":"06:03.030","Text":"and we have to check these manually separately."},{"Start":"06:03.030 ","End":"06:11.530","Text":"But before I forget, let me just highlight the answer for the radius of convergence."},{"Start":"06:11.530 ","End":"06:17.110","Text":"Now we just have to do the region of convergence,"},{"Start":"06:17.110 ","End":"06:19.810","Text":"which will be from minus 1 to 1,"},{"Start":"06:19.810 ","End":"06:21.565","Text":"possibly with the endpoints."},{"Start":"06:21.565 ","End":"06:24.835","Text":"Let\u0027s check each one separately."},{"Start":"06:24.835 ","End":"06:27.700","Text":"If x is equal to 1,"},{"Start":"06:27.700 ","End":"06:38.200","Text":"our series becomes the sum from 1 to infinity of,"},{"Start":"06:38.200 ","End":"06:43.750","Text":"it\u0027s always a_n, x_n even though I\u0027ve lost the original series,"},{"Start":"06:43.750 ","End":"06:48.850","Text":"its sine squared 1."},{"Start":"06:48.850 ","End":"06:50.690","Text":"There was an x to the n here,"},{"Start":"06:50.690 ","End":"06:53.210","Text":"but that\u0027s just 1."},{"Start":"06:53.210 ","End":"06:56.345","Text":"I\u0027ll go back up and show you."},{"Start":"06:56.345 ","End":"06:59.960","Text":"What I\u0027m saying is that if I let x equals 1 here,"},{"Start":"06:59.960 ","End":"07:02.450","Text":"the x_n is 1 and that disappears,"},{"Start":"07:02.450 ","End":"07:08.630","Text":"so it\u0027s just the sine squared 1 like here."},{"Start":"07:08.630 ","End":"07:14.000","Text":"I see I\u0027m going to want another fact to introduce,"},{"Start":"07:14.000 ","End":"07:18.290","Text":"and it\u0027s similar to this in some ways."},{"Start":"07:18.290 ","End":"07:23.795","Text":"The fact is that sine x is less than x,"},{"Start":"07:23.795 ","End":"07:27.090","Text":"and that\u0027s actually for all x."},{"Start":"07:33.830 ","End":"07:38.050","Text":"Well, for all positive x anyway,"},{"Start":"07:38.050 ","End":"07:41.760","Text":"yeah, actually it only works for positive x."},{"Start":"07:41.760 ","End":"07:47.250","Text":"It\u0027s usually written with non-strict inequalities."},{"Start":"07:47.250 ","End":"07:50.495","Text":"I\u0027m going to apply it here because in our case,"},{"Start":"07:50.495 ","End":"07:53.090","Text":"the x from here will be 1,"},{"Start":"07:53.090 ","End":"07:56.520","Text":"which is certainly positive."},{"Start":"08:00.470 ","End":"08:06.530","Text":"What I can conclude is that sine squared of"},{"Start":"08:06.530 ","End":"08:10.670","Text":"1 is less than or"},{"Start":"08:10.670 ","End":"08:17.040","Text":"equal to 1 squared,"},{"Start":"08:17.040 ","End":"08:21.555","Text":"and I can also put an absolute value here."},{"Start":"08:21.555 ","End":"08:24.665","Text":"The reason I\u0027m doing that is that I can then use the theorem"},{"Start":"08:24.665 ","End":"08:28.130","Text":"that if I have a series where the absolute value of"},{"Start":"08:28.130 ","End":"08:35.675","Text":"each term is less than or equal to the terms from another series,"},{"Start":"08:35.675 ","End":"08:37.175","Text":"and the other series,"},{"Start":"08:37.175 ","End":"08:44.880","Text":"and I\u0027m talking about n goes from 1 to infinity of 1 squared,"},{"Start":"08:45.970 ","End":"08:51.530","Text":"this non-negative series, if it converges, then this converges."},{"Start":"08:51.530 ","End":"08:54.110","Text":"There\u0027s a theorem that if the absolute value"},{"Start":"08:54.110 ","End":"08:58.265","Text":"1 theory is less than or equal to another and the other is convergent."},{"Start":"08:58.265 ","End":"09:00.995","Text":"Now, we know this is convergent,"},{"Start":"09:00.995 ","End":"09:03.920","Text":"we\u0027ve seen it many times before."},{"Start":"09:03.920 ","End":"09:07.880","Text":"One way of looking at it is a P series with p equals 2,"},{"Start":"09:07.880 ","End":"09:09.965","Text":"and when p is bigger than 1, it\u0027s convergent."},{"Start":"09:09.965 ","End":"09:12.110","Text":"We\u0027ve seen this many times before."},{"Start":"09:12.110 ","End":"09:14.355","Text":"This is convergent."},{"Start":"09:14.355 ","End":"09:17.990","Text":"This is convergent too,"},{"Start":"09:17.990 ","End":"09:23.485","Text":"and so I can write that at x equals 1,"},{"Start":"09:23.485 ","End":"09:25.470","Text":"I no longer have a question mark,"},{"Start":"09:25.470 ","End":"09:26.610","Text":"I know it\u0027s convergent,"},{"Start":"09:26.610 ","End":"09:29.030","Text":"I\u0027ll just write c for convergent."},{"Start":"09:29.030 ","End":"09:32.520","Text":"Now, what happens at minus 1?"},{"Start":"09:32.770 ","End":"09:37.835","Text":"Well, when x is minus 1,"},{"Start":"09:37.835 ","End":"09:41.840","Text":"what I get if I plug into the original series,"},{"Start":"09:41.840 ","End":"09:44.555","Text":"remember there was like an x_n here."},{"Start":"09:44.555 ","End":"09:50.640","Text":"I get the sum from 1 to infinity,"},{"Start":"09:50.640 ","End":"09:54.360","Text":"the minus 1_n was there."},{"Start":"09:54.360 ","End":"09:56.040","Text":"I\u0027ll just write it in front,"},{"Start":"09:56.040 ","End":"10:04.260","Text":"minus 1_n sine squared of 1."},{"Start":"10:04.260 ","End":"10:11.255","Text":"I\u0027m claimed that this actually more than converges, it converges absolutely."},{"Start":"10:11.255 ","End":"10:13.970","Text":"What else converges absolutely mean?"},{"Start":"10:13.970 ","End":"10:20.090","Text":"It means that if I put absolute value around each term, then that converges."},{"Start":"10:20.090 ","End":"10:24.425","Text":"Of course, if I put an absolute value here,"},{"Start":"10:24.425 ","End":"10:27.185","Text":"what I would get is just this series."},{"Start":"10:27.185 ","End":"10:29.645","Text":"We\u0027ve already shown this converges."},{"Start":"10:29.645 ","End":"10:32.275","Text":"This converges absolutely."},{"Start":"10:32.275 ","End":"10:35.090","Text":"When something converges absolutely,"},{"Start":"10:35.090 ","End":"10:39.340","Text":"it also converges; well-known theorem."},{"Start":"10:39.470 ","End":"10:48.080","Text":"This mystery is dispelled also and I can write that it converges here too."},{"Start":"10:48.130 ","End":"10:51.530","Text":"Together, when I collect all the converges,"},{"Start":"10:51.530 ","End":"10:58.105","Text":"I get that x can be between minus 1 and 1 inclusive."},{"Start":"10:58.105 ","End":"11:00.435","Text":"I\u0027ll just highlight this,"},{"Start":"11:00.435 ","End":"11:04.875","Text":"and this here is the range of convergence,"},{"Start":"11:04.875 ","End":"11:08.585","Text":"and we\u0027ve already found the radius of convergence."},{"Start":"11:08.585 ","End":"11:12.060","Text":"We\u0027re done with this exercise."}],"ID":7976},{"Watched":false,"Name":"Exercise 3 part e","Duration":"9m 19s","ChapterTopicVideoID":7915,"CourseChapterTopicPlaylistID":4250,"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 want to find the radius of"},{"Start":"00:03.480 ","End":"00:08.820","Text":"convergence and the range of convergence for this series."},{"Start":"00:08.820 ","End":"00:10.650","Text":"Just from time to time,"},{"Start":"00:10.650 ","End":"00:13.320","Text":"I\u0027d like to mention that the range of convergence"},{"Start":"00:13.320 ","End":"00:16.335","Text":"can sometimes be called the region of convergence."},{"Start":"00:16.335 ","End":"00:20.160","Text":"We\u0027re even seeing interval of convergence."},{"Start":"00:20.160 ","End":"00:24.240","Text":"Just various ways if you like it better with region,"},{"Start":"00:24.240 ","End":"00:26.174","Text":"then call it region."},{"Start":"00:26.174 ","End":"00:32.730","Text":"We have the sum of something slightly different than we usually have."},{"Start":"00:32.730 ","End":"00:34.125","Text":"Usually, we have x^n."},{"Start":"00:34.125 ","End":"00:37.230","Text":"Here, it\u0027s x plus 2^n."},{"Start":"00:37.230 ","End":"00:43.890","Text":"What we do in this case is we make a substitution."},{"Start":"00:44.630 ","End":"00:52.220","Text":"Let\u0027s let t equal x plus 2 or any letter except x."},{"Start":"00:52.220 ","End":"00:55.219","Text":"What we have is more familiar."},{"Start":"00:55.219 ","End":"01:01.010","Text":"We have the sum from 1 to infinity minus 1 to the end."},{"Start":"01:01.010 ","End":"01:02.915","Text":"We don\u0027t have the plus 2 now,"},{"Start":"01:02.915 ","End":"01:07.685","Text":"as we have t^n over root n,"},{"Start":"01:07.685 ","End":"01:10.410","Text":"except now we have t instead of x."},{"Start":"01:10.780 ","End":"01:18.620","Text":"Our an, the coefficient of t^n, this with this,"},{"Start":"01:18.620 ","End":"01:24.440","Text":"it\u0027s minus 1^n over square root"},{"Start":"01:24.440 ","End":"01:30.125","Text":"of n. We want to find the radius of convergence,"},{"Start":"01:30.125 ","End":"01:36.075","Text":"even though it\u0027s the series with t. We\u0027ll see how to get back to x later."},{"Start":"01:36.075 ","End":"01:39.720","Text":"We take the 2 main formulas for radius."},{"Start":"01:39.720 ","End":"01:41.340","Text":"1 involving a ratio,"},{"Start":"01:41.340 ","End":"01:42.900","Text":"1 involving a root,"},{"Start":"01:42.900 ","End":"01:44.835","Text":"is that the radius,"},{"Start":"01:44.835 ","End":"01:46.780","Text":"I\u0027ll take the 1 with the ratio,"},{"Start":"01:46.780 ","End":"01:55.110","Text":"is the limit as n goes to infinity of an over an plus 1,"},{"Start":"01:55.110 ","End":"01:57.610","Text":"but in absolute value."},{"Start":"01:59.000 ","End":"02:02.745","Text":"Let\u0027s do that."},{"Start":"02:02.745 ","End":"02:07.054","Text":"Let\u0027s first of all see what is the absolute value"},{"Start":"02:07.054 ","End":"02:11.565","Text":"of an over an plus 1 then we\u0027ll take the limit."},{"Start":"02:11.565 ","End":"02:13.970","Text":"This is equal to, obviously,"},{"Start":"02:13.970 ","End":"02:16.070","Text":"I can throw out the plus or minus 1,"},{"Start":"02:16.070 ","End":"02:19.955","Text":"this alternating sign because we\u0027re in absolute value."},{"Start":"02:19.955 ","End":"02:23.970","Text":"In the numerator, we have 1 over square root of n,"},{"Start":"02:23.970 ","End":"02:26.435","Text":"then on the denominator,"},{"Start":"02:26.435 ","End":"02:31.010","Text":"we have 1 over the square root of n plus 1."},{"Start":"02:31.010 ","End":"02:32.720","Text":"If you just do a bit of work with fractions,"},{"Start":"02:32.720 ","End":"02:39.115","Text":"we get the square root of n plus 1 over square root of n."},{"Start":"02:39.115 ","End":"02:46.345","Text":"We get that r equals the limit of this thing as n goes to infinity."},{"Start":"02:46.345 ","End":"02:52.730","Text":"We can rearrange this to say it\u0027s the square root of,"},{"Start":"02:52.730 ","End":"03:01.970","Text":"and I can also do the division 1 plus 1 over n. When n goes to infinity,"},{"Start":"03:01.970 ","End":"03:03.935","Text":"this is going to go to 0."},{"Start":"03:03.935 ","End":"03:09.320","Text":"I mean, I could write it symbolically as the square root of 1 plus 1 over infinity."},{"Start":"03:09.320 ","End":"03:11.540","Text":"But 1 over infinity is 0,"},{"Start":"03:11.540 ","End":"03:14.165","Text":"1 plus 0 is 1 square root of 1."},{"Start":"03:14.165 ","End":"03:16.950","Text":"This is equal to 1."},{"Start":"03:17.200 ","End":"03:21.485","Text":"It brought in a picture from a previous exercise."},{"Start":"03:21.485 ","End":"03:23.480","Text":"In the previous exercise,"},{"Start":"03:23.480 ","End":"03:30.455","Text":"it was x and here it\u0027s t. We take a radius of 1 around the 0,"},{"Start":"03:30.455 ","End":"03:32.630","Text":"we get to plus and minus 1."},{"Start":"03:32.630 ","End":"03:37.400","Text":"We know that inside here it converges, outside, it diverges."},{"Start":"03:37.400 ","End":"03:39.275","Text":"We just don\u0027t know here and here."},{"Start":"03:39.275 ","End":"03:42.305","Text":"But the thing is that we want to get from t back to x."},{"Start":"03:42.305 ","End":"03:46.085","Text":"Now if t is x plus 2 and I reverse that,"},{"Start":"03:46.085 ","End":"03:50.315","Text":"then I can say that x is t minus 2."},{"Start":"03:50.315 ","End":"03:54.830","Text":"If I want to change t to x,"},{"Start":"03:54.830 ","End":"03:56.705","Text":"let me erase this."},{"Start":"03:56.705 ","End":"04:00.155","Text":"I just subtract 2 from everything,"},{"Start":"04:00.155 ","End":"04:06.260","Text":"and I\u0027ll get where we had 0 minus 2,"},{"Start":"04:06.260 ","End":"04:08.374","Text":"doesn\u0027t really matter, but that\u0027s the center."},{"Start":"04:08.374 ","End":"04:14.075","Text":"Then we had a radius of 1 minus 1, and minus 3."},{"Start":"04:14.075 ","End":"04:21.075","Text":"Now it\u0027s x and not t. You could have just said,"},{"Start":"04:21.075 ","End":"04:24.435","Text":"x plus 2 is 0 when x is minus 2."},{"Start":"04:24.435 ","End":"04:30.870","Text":"Put on the minus 2 and then taken the radius plus and minus from this point."},{"Start":"04:31.000 ","End":"04:35.915","Text":"We pretty much know where it converges except for these 2 special points."},{"Start":"04:35.915 ","End":"04:39.620","Text":"Let\u0027s deal with these separately."},{"Start":"04:39.620 ","End":"04:45.660","Text":"Let\u0027s take x equals minus 1."},{"Start":"04:47.010 ","End":"04:50.920","Text":"Then we get if x is minus 1,"},{"Start":"04:50.920 ","End":"04:54.360","Text":"then x plus 2 is 1."},{"Start":"04:54.360 ","End":"04:56.970","Text":"We get the series,"},{"Start":"04:56.970 ","End":"05:04.680","Text":"the sum from 1 to infinity minus 1 to the n minus 1."},{"Start":"05:04.680 ","End":"05:12.775","Text":"Then 1 to the n is just 1 over the square root of n. Now,"},{"Start":"05:12.775 ","End":"05:21.010","Text":"I claim that this converges because it\u0027s a Leibniz alternating series."},{"Start":"05:21.010 ","End":"05:23.455","Text":"If we apply the test,"},{"Start":"05:23.455 ","End":"05:25.450","Text":"what we have is that,"},{"Start":"05:25.450 ","End":"05:28.145","Text":"without this alternating sign,"},{"Start":"05:28.145 ","End":"05:32.160","Text":"this 1 over square root of n is positive."},{"Start":"05:32.560 ","End":"05:38.000","Text":"If I just take the 1 over square root of n sequence,"},{"Start":"05:38.000 ","End":"05:42.935","Text":"it\u0027s positive, it\u0027s decreasing,"},{"Start":"05:42.935 ","End":"05:44.870","Text":"and it tends to 0."},{"Start":"05:44.870 ","End":"05:47.240","Text":"I wrote that in code and brief."},{"Start":"05:47.240 ","End":"05:51.065","Text":"Because this positive decreasing tends to 0,"},{"Start":"05:51.065 ","End":"05:57.025","Text":"the alternating sum of these converges by the Newton-Leibniz test."},{"Start":"05:57.025 ","End":"06:05.155","Text":"That\u0027s check. Now, what about x equals minus 3?"},{"Start":"06:05.155 ","End":"06:07.790","Text":"If x is minus 3,"},{"Start":"06:07.790 ","End":"06:11.160","Text":"then here we get minus 1."},{"Start":"06:11.160 ","End":"06:20.310","Text":"We have the sum from 1 to infinity of,"},{"Start":"06:20.310 ","End":"06:27.880","Text":"we have minus 1 to the n minus 1 and here, minus 1^n."},{"Start":"06:27.880 ","End":"06:31.355","Text":"I just collected the minuses together and on the denominator,"},{"Start":"06:31.355 ","End":"06:35.495","Text":"square root of n. Now notice,"},{"Start":"06:35.495 ","End":"06:43.940","Text":"I\u0027ll do this at the side that this numerator is minus 1^n minus 1 plus n,"},{"Start":"06:43.940 ","End":"06:50.850","Text":"which is minus 1^2n minus 1."},{"Start":"06:50.850 ","End":"06:57.015","Text":"This is an odd number because 2n is always even."},{"Start":"06:57.015 ","End":"06:58.470","Text":"2n minus 1 is odd,"},{"Start":"06:58.470 ","End":"07:02.595","Text":"and minus 1 to an odd power is minus 1."},{"Start":"07:02.595 ","End":"07:03.720","Text":"To an even power, it\u0027s 1."},{"Start":"07:03.720 ","End":"07:05.670","Text":"To an odd power is minus 1."},{"Start":"07:05.670 ","End":"07:09.570","Text":"Here this is equal to,"},{"Start":"07:09.570 ","End":"07:11.385","Text":"I can just put the minus in front,"},{"Start":"07:11.385 ","End":"07:17.210","Text":"and I\u0027ve got the sum n equals 1 to infinity of 1"},{"Start":"07:17.210 ","End":"07:23.660","Text":"over the square root of n. I want to write the square root of n as n^0.5."},{"Start":"07:23.660 ","End":"07:29.385","Text":"The reason I want to do this is because this is now a p-series."},{"Start":"07:29.385 ","End":"07:35.265","Text":"The sum n from 1 to infinity of 1 over n^p."},{"Start":"07:35.265 ","End":"07:39.470","Text":"What we know is that for p-series,"},{"Start":"07:39.470 ","End":"07:44.095","Text":"if p is bigger than 1, it converges."},{"Start":"07:44.095 ","End":"07:50.445","Text":"Otherwise, if it\u0027s less than or equal to 1, then it diverges."},{"Start":"07:50.445 ","End":"07:53.310","Text":"Now, in our case,"},{"Start":"07:53.310 ","End":"07:58.155","Text":"p is equal to 1.5."},{"Start":"07:58.155 ","End":"08:00.945","Text":"A 0.5 Is less than or equal to 1."},{"Start":"08:00.945 ","End":"08:05.470","Text":"We\u0027re in the diverges case."},{"Start":"08:06.710 ","End":"08:16.625","Text":"The answer to these 2 is that here it converges and here it diverges."},{"Start":"08:16.625 ","End":"08:22.880","Text":"I\u0027ll just replace the question marks with c for converges and d for diverges."},{"Start":"08:22.880 ","End":"08:27.635","Text":"Now we\u0027re looking for the region or range of convergence."},{"Start":"08:27.635 ","End":"08:31.565","Text":"It converges from minus 3 to minus 1,"},{"Start":"08:31.565 ","End":"08:33.950","Text":"including the minus 1, not including this."},{"Start":"08:33.950 ","End":"08:36.430","Text":"In other words, I can write that as"},{"Start":"08:36.430 ","End":"08:45.125","Text":"x less than or equal to minus 1,"},{"Start":"08:45.125 ","End":"08:49.650","Text":"but strictly greater than minus 3."},{"Start":"08:49.650 ","End":"08:53.240","Text":"I want to do some highlighting."},{"Start":"08:53.240 ","End":"09:02.575","Text":"This is the radius of convergence and this is the range of convergence."},{"Start":"09:02.575 ","End":"09:08.955","Text":"It\u0027s right that that\u0027s the radius of convergence."},{"Start":"09:08.955 ","End":"09:16.520","Text":"This is the range or region or interval of convergence and so we\u0027ve answered both parts,"},{"Start":"09:16.520 ","End":"09:19.590","Text":"and we are done."}],"ID":7977},{"Watched":false,"Name":"Exercise 3 part f","Duration":"8m 4s","ChapterTopicVideoID":7916,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.925","Text":"In this exercise, we have to find the radius of convergence and the range of convergence,"},{"Start":"00:05.925 ","End":"00:11.835","Text":"also sometimes called the region or interval of convergence for this series."},{"Start":"00:11.835 ","End":"00:15.120","Text":"There\u0027s only a slightly different about this series."},{"Start":"00:15.120 ","End":"00:17.430","Text":"Notice that there\u0027s a 2 there."},{"Start":"00:17.430 ","End":"00:22.080","Text":"Now, the normal formula for finding the radius,"},{"Start":"00:22.080 ","End":"00:32.220","Text":"which was the limit as n goes to infinity of absolute value of an/an plus 1 will"},{"Start":"00:32.220 ","End":"00:37.080","Text":"not exactly work here because this assumed that the original series was of"},{"Start":"00:37.080 ","End":"00:42.210","Text":"the form an x^n."},{"Start":"00:42.210 ","End":"00:48.585","Text":"But here we don\u0027t have x^n, we have x^2n."},{"Start":"00:48.585 ","End":"00:51.179","Text":"We\u0027ll use our little trick."},{"Start":"00:51.179 ","End":"01:00.000","Text":"Just note that x^2n is x squared to the power of n,"},{"Start":"01:00.000 ","End":"01:09.815","Text":"and this gives me the idea that if we let t be equal to x squared as a substitution,"},{"Start":"01:09.815 ","End":"01:14.010","Text":"then x^2n is just t^n."},{"Start":"01:14.010 ","End":"01:23.820","Text":"Here we have t. So this series can be written as the sum"},{"Start":"01:23.820 ","End":"01:30.445","Text":"from 1 to infinity of n plus 1 to the fifth"},{"Start":"01:30.445 ","End":"01:38.690","Text":"over 2n plus 1 times t^n."},{"Start":"01:38.690 ","End":"01:43.795","Text":"Now we can use our standard formulas because our an"},{"Start":"01:43.795 ","End":"01:49.080","Text":"is n plus 1 to the fifth over 2n plus 1."},{"Start":"01:49.080 ","End":"01:50.250","Text":"It is something to the n,"},{"Start":"01:50.250 ","End":"01:51.930","Text":"it doesn\u0027t matter if it\u0027s t instead of x,"},{"Start":"01:51.930 ","End":"01:53.580","Text":"it\u0027s just the letter."},{"Start":"01:53.580 ","End":"01:58.035","Text":"Now we can calculate R,"},{"Start":"01:58.035 ","End":"01:59.940","Text":"but for this series,"},{"Start":"01:59.940 ","End":"02:07.330","Text":"as the limit, as n goes to infinity."},{"Start":"02:07.330 ","End":"02:10.795","Text":"Now let\u0027s see what do we have inside this absolute value."},{"Start":"02:10.795 ","End":"02:13.255","Text":"An just as is,"},{"Start":"02:13.255 ","End":"02:17.835","Text":"n plus 1 to the fifth over"},{"Start":"02:17.835 ","End":"02:24.985","Text":"2n plus 1 divided by same thing with n plus 1."},{"Start":"02:24.985 ","End":"02:33.610","Text":"So it\u0027s n plus 2 to the fifth over,"},{"Start":"02:33.610 ","End":"02:36.100","Text":"now, if I put n plus 1 instead of n,"},{"Start":"02:36.100 ","End":"02:40.910","Text":"I get 2n plus 3."},{"Start":"02:40.910 ","End":"02:44.440","Text":"Everything\u0027s positive, so I don\u0027t need the absolute value,"},{"Start":"02:44.440 ","End":"02:46.640","Text":"but I would like to rewrite it,"},{"Start":"02:46.640 ","End":"02:49.140","Text":"and what we can do as for 3,"},{"Start":"02:49.140 ","End":"02:51.965","Text":"I\u0027ll say, drop the absolute value and then"},{"Start":"02:51.965 ","End":"02:57.960","Text":"put n plus 1 to the fifth over n plus 2 to the fifth."},{"Start":"03:01.550 ","End":"03:12.765","Text":"Then this goes to the top times 2n plus 3 over 2n plus 1."},{"Start":"03:12.765 ","End":"03:21.315","Text":"Now, I claim that this thing tends to 1 and so does this."},{"Start":"03:21.315 ","End":"03:24.300","Text":"I\u0027ll do some computations at the side."},{"Start":"03:24.300 ","End":"03:27.680","Text":"The limit as n goes to infinity, if it wasn\u0027t a 5,"},{"Start":"03:27.680 ","End":"03:31.595","Text":"if it was just n plus 1 over n plus 2,"},{"Start":"03:31.595 ","End":"03:36.050","Text":"we would just take the leading coefficients as we do with polynomial over polynomial,"},{"Start":"03:36.050 ","End":"03:39.420","Text":"we\u0027d get n over n is 1."},{"Start":"03:39.650 ","End":"03:45.680","Text":"The limit as n goes to infinity of the same thing,"},{"Start":"03:45.680 ","End":"03:51.070","Text":"but to the power of 5 is going to be 1^5,"},{"Start":"03:51.070 ","End":"03:53.835","Text":"or still equals to 1."},{"Start":"03:53.835 ","End":"04:02.385","Text":"As for the other 1, the limit as n goes to infinity of 2n plus 3 over 2n plus 1."},{"Start":"04:02.385 ","End":"04:04.580","Text":"Also, because it\u0027s polynomial over polynomial,"},{"Start":"04:04.580 ","End":"04:06.425","Text":"take the leading term is 2n,"},{"Start":"04:06.425 ","End":"04:08.555","Text":"the leading term here is 2n,"},{"Start":"04:08.555 ","End":"04:10.790","Text":"so it\u0027s also equal to 1."},{"Start":"04:10.790 ","End":"04:13.115","Text":"Everything here goes to 1,"},{"Start":"04:13.115 ","End":"04:23.740","Text":"so the answer here is just 1 and that\u0027s part of the question."},{"Start":"04:24.460 ","End":"04:30.415","Text":"But wait, this is the radius for t and we want for x."},{"Start":"04:30.415 ","End":"04:34.720","Text":"Now, this implies that"},{"Start":"04:45.260 ","End":"04:50.620","Text":"between minus 1 and 1, we have convergence."},{"Start":"04:50.710 ","End":"04:59.195","Text":"Now actually t is non-negative because it\u0027s equal to x squared."},{"Start":"04:59.195 ","End":"05:06.630","Text":"But anyway, when x squared is between minus 1 and 1,"},{"Start":"05:06.630 ","End":"05:08.330","Text":"it\u0027s never going to be minus 1,"},{"Start":"05:08.330 ","End":"05:11.915","Text":"it\u0027s always going to be between 0 inclusive and 1."},{"Start":"05:11.915 ","End":"05:18.215","Text":"But x itself will still be between minus 1 and 1."},{"Start":"05:18.215 ","End":"05:23.630","Text":"It all works out in the end that x has"},{"Start":"05:23.630 ","End":"05:31.370","Text":"a radius of convergence because the radius of convergence now would be 1."},{"Start":"05:31.370 ","End":"05:38.240","Text":"Perhaps I\u0027ll just write that the radius of"},{"Start":"05:38.240 ","End":"05:44.720","Text":"convergence for x is 1 because this is symmetrically placed around 0."},{"Start":"05:44.720 ","End":"05:46.969","Text":"It\u0027s 0 plus or minus 1,"},{"Start":"05:46.969 ","End":"05:48.635","Text":"so the radius is 1."},{"Start":"05:48.635 ","End":"05:50.840","Text":"Now, as for the region,"},{"Start":"05:50.840 ","End":"05:53.960","Text":"we have to also check the end points."},{"Start":"05:53.960 ","End":"05:59.045","Text":"We know it diverges outside for bigger than 1 and for less than minus 1,"},{"Start":"05:59.045 ","End":"06:00.635","Text":"but at the end points,"},{"Start":"06:00.635 ","End":"06:03.650","Text":"precisely, we have to check each one."},{"Start":"06:03.650 ","End":"06:10.595","Text":"Let\u0027s check what happens when x is equal to 1."},{"Start":"06:10.595 ","End":"06:20.650","Text":"The series becomes the sum n plus 1 to the fifth over 2n plus 1."},{"Start":"06:20.650 ","End":"06:23.150","Text":"Here we had x^2n,"},{"Start":"06:23.150 ","End":"06:26.535","Text":"but when x is 1, 1 to the anything is just 1."},{"Start":"06:26.535 ","End":"06:31.320","Text":"This is all we get when x equals 1."},{"Start":"06:31.320 ","End":"06:36.260","Text":"We\u0027ll check what happens with the convergence of this."},{"Start":"06:36.300 ","End":"06:40.930","Text":"Now this series is obviously divergent."},{"Start":"06:40.930 ","End":"06:43.355","Text":"I say obviously, because"},{"Start":"06:43.355 ","End":"06:48.445","Text":"1 basic test of divergence is that the general term doesn\u0027t go to 0."},{"Start":"06:48.445 ","End":"06:51.560","Text":"This goes to infinity."},{"Start":"06:53.180 ","End":"07:04.835","Text":"The general term, it\u0027s a polynomial of higher power here than here, goes to infinity."},{"Start":"07:04.835 ","End":"07:08.520","Text":"When x is minus 1,"},{"Start":"07:08.720 ","End":"07:18.150","Text":"then we get exactly the same because when we take minus 1^2n,"},{"Start":"07:18.150 ","End":"07:21.280","Text":"it\u0027s the same as 1^2n."},{"Start":"07:21.740 ","End":"07:23.940","Text":"When we plug in the x,"},{"Start":"07:23.940 ","End":"07:25.505","Text":"we\u0027re going to get the same series,"},{"Start":"07:25.505 ","End":"07:27.905","Text":"and it\u0027s also going to be divergent."},{"Start":"07:27.905 ","End":"07:31.265","Text":"When we summarize, we don\u0027t get anything at the end points,"},{"Start":"07:31.265 ","End":"07:39.840","Text":"and this becomes a range of convergence."},{"Start":"07:41.170 ","End":"07:45.570","Text":"Well, not this 1, but this 1."},{"Start":"07:45.790 ","End":"07:51.695","Text":"This is the range of convergence and this is"},{"Start":"07:51.695 ","End":"07:57.935","Text":"the radius of convergence."},{"Start":"07:57.935 ","End":"08:00.050","Text":"We\u0027ve answered the question,"},{"Start":"08:00.050 ","End":"08:03.540","Text":"both parts of it. We\u0027re done."}],"ID":7978},{"Watched":false,"Name":"Exercise 3 part g","Duration":"4m 9s","ChapterTopicVideoID":7917,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.660","Text":"In this exercise, we want to find the radius of convergence"},{"Start":"00:04.660 ","End":"00:09.695","Text":"and the range of convergence of this series."},{"Start":"00:09.695 ","End":"00:16.660","Text":"What we usually have for a power series is x to the n. Here we have x"},{"Start":"00:16.660 ","End":"00:23.500","Text":"minus 1 to the n. We can fix that or adapt to that."},{"Start":"00:23.500 ","End":"00:28.975","Text":"What we can do is we can let t equals x minus 1."},{"Start":"00:28.975 ","End":"00:34.570","Text":"Then the series becomes the sum from"},{"Start":"00:34.570 ","End":"00:42.645","Text":"0 to infinity of n factorial over 3 to the n,"},{"Start":"00:42.645 ","End":"00:50.945","Text":"and then it\u0027s t to the n. Our an now can be n factorial"},{"Start":"00:50.945 ","End":"00:55.230","Text":"over 3 to the n. Then we\u0027ll find"},{"Start":"00:55.230 ","End":"01:00.800","Text":"what\u0027s needed for t and then later we\u0027ll worry how to get back from t to x."},{"Start":"01:00.800 ","End":"01:07.670","Text":"Now, the radius of convergence for t is given by, well,"},{"Start":"01:07.670 ","End":"01:10.160","Text":"there are several formulas but I\u0027m going to use the one with"},{"Start":"01:10.160 ","End":"01:14.195","Text":"the ratio is the limit n goes to infinity."},{"Start":"01:14.195 ","End":"01:20.555","Text":"In general, the absolute value of an over an plus 1."},{"Start":"01:20.555 ","End":"01:24.809","Text":"Let\u0027s see what that comes out to in our case."},{"Start":"01:26.210 ","End":"01:30.695","Text":"We don\u0027t need the absolute value because everything\u0027s positive."},{"Start":"01:30.695 ","End":"01:32.900","Text":"So we\u0027ll take an,"},{"Start":"01:32.900 ","End":"01:39.260","Text":"which is n factorial over 3 to the n. Instead of putting it in the denominator,"},{"Start":"01:39.260 ","End":"01:42.320","Text":"I\u0027ll just multiply by the reverse of it."},{"Start":"01:42.320 ","End":"01:48.605","Text":"Now n plus 1 here means that we have 3 to the n plus 1 and here,"},{"Start":"01:48.605 ","End":"01:52.530","Text":"n plus 1 factorial."},{"Start":"01:52.730 ","End":"01:58.565","Text":"We can do some canceling because I could rewrite this."},{"Start":"01:58.565 ","End":"02:05.300","Text":"I\u0027ll just write the limit first and then here as is but 3 to"},{"Start":"02:05.300 ","End":"02:12.220","Text":"the n plus 1 is the same as 3 times 3 to the n. I\u0027m doing this so I can cancel stuff."},{"Start":"02:12.220 ","End":"02:16.850","Text":"I can rewrite this also because n plus 1 factorial,"},{"Start":"02:16.850 ","End":"02:18.590","Text":"and we\u0027ve seen this many times,"},{"Start":"02:18.590 ","End":"02:23.745","Text":"is n plus 1 times n factorial."},{"Start":"02:23.745 ","End":"02:27.990","Text":"Now we can cancel 3 to the n with 3 to the n,"},{"Start":"02:27.990 ","End":"02:33.260","Text":"n factorial with n factorial and so what we\u0027re left with"},{"Start":"02:33.260 ","End":"02:38.960","Text":"is the limit of just the 3 over n plus 1."},{"Start":"02:38.960 ","End":"02:43.490","Text":"Could write it symbolically as 3 over infinity plus 1,"},{"Start":"02:43.490 ","End":"02:46.445","Text":"and that is 0."},{"Start":"02:46.445 ","End":"02:49.939","Text":"Now what do we do when the radius is 0?"},{"Start":"02:49.939 ","End":"02:55.490","Text":"It means that the center is always 0."},{"Start":"02:55.490 ","End":"02:58.820","Text":"But if I go plus or minus 0, I\u0027m still at 0."},{"Start":"02:58.820 ","End":"03:02.375","Text":"So it can converge at most for t equals 0,"},{"Start":"03:02.375 ","End":"03:05.870","Text":"maybe not even t equals 0 because it\u0027s like the end points."},{"Start":"03:05.870 ","End":"03:07.775","Text":"But at t equals 0,"},{"Start":"03:07.775 ","End":"03:14.755","Text":"there is no problem because it\u0027s a series of zeros so it\u0027s t equals 0 only."},{"Start":"03:14.755 ","End":"03:19.260","Text":"There\u0027s no range, it\u0027s just a point."},{"Start":"03:19.260 ","End":"03:23.565","Text":"T equals 0 means that x minus 1 is 0."},{"Start":"03:23.565 ","End":"03:28.185","Text":"That gives us that x equals 1 only."},{"Start":"03:28.185 ","End":"03:30.260","Text":"Ultimately, I mean at the end,"},{"Start":"03:30.260 ","End":"03:37.880","Text":"I\u0027ll write that the radius of convergence is 0."},{"Start":"03:37.880 ","End":"03:42.840","Text":"The range, there is no range,"},{"Start":"03:42.840 ","End":"03:48.305","Text":"but let\u0027s just say that it\u0027s the interval from 0 to 0,"},{"Start":"03:48.305 ","End":"03:52.990","Text":"it\u0027s just from 1 to 1."},{"Start":"03:52.990 ","End":"04:00.290","Text":"Anyway, it\u0027s x equals 1 and anything with a distance of 0 from it. That\u0027s all."},{"Start":"04:00.290 ","End":"04:03.140","Text":"Just the only converges for the single point,"},{"Start":"04:03.140 ","End":"04:09.030","Text":"x equals 1 and there\u0027s no radius. We\u0027re done."}],"ID":7979},{"Watched":false,"Name":"Exercise 3 part h","Duration":"5m 22s","ChapterTopicVideoID":7918,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"Here, we have to find the radius of convergence and also"},{"Start":"00:03.270 ","End":"00:08.430","Text":"the range for convergence for this series."},{"Start":"00:08.430 ","End":"00:12.250","Text":"It\u0027s x^n here, so we\u0027re in our usual situation."},{"Start":"00:12.250 ","End":"00:17.685","Text":"This bit here is what we call a_n and that\u0027s equal to,"},{"Start":"00:17.685 ","End":"00:19.950","Text":"what does this mean with the dot, dot, dot?"},{"Start":"00:19.950 ","End":"00:22.785","Text":"Means the product 1 times 3 times 5,"},{"Start":"00:22.785 ","End":"00:25.470","Text":"you keep going up in steps of 2, in other words,"},{"Start":"00:25.470 ","End":"00:26.745","Text":"all the odd numbers,"},{"Start":"00:26.745 ","End":"00:33.105","Text":"and you finally finish when you get to 2n minus 1."},{"Start":"00:33.105 ","End":"00:41.650","Text":"If n was 4, then you would end up at twice 4 minus 1 is 7."},{"Start":"00:41.650 ","End":"00:45.890","Text":"Actually that before terms because you\u0027d get 1,3,5,7,"},{"Start":"00:45.890 ","End":"00:48.290","Text":"so it turns out that\u0027s the way it is."},{"Start":"00:48.290 ","End":"00:57.680","Text":"The denominator is 2n minus 2 factorial."},{"Start":"00:57.680 ","End":"01:01.745","Text":"Now, I\u0027m going to use the formula for the radius of convergence,"},{"Start":"01:01.745 ","End":"01:10.355","Text":"that it\u0027s the limit as n goes to infinity of a_n over a_n plus 1,"},{"Start":"01:10.355 ","End":"01:13.015","Text":"this bit in absolute value."},{"Start":"01:13.015 ","End":"01:15.275","Text":"I\u0027m going to apply it here."},{"Start":"01:15.275 ","End":"01:16.300","Text":"Let me first of all see,"},{"Start":"01:16.300 ","End":"01:22.610","Text":"what is a_n plus 1 because it\u0027s not so obvious with all this dot, dot, dot."},{"Start":"01:22.610 ","End":"01:25.150","Text":"Well, this is going to equal,"},{"Start":"01:25.150 ","End":"01:28.049","Text":"we\u0027re going to start off the same."},{"Start":"01:28.049 ","End":"01:33.050","Text":"We\u0027re just going to end a bit further along."},{"Start":"01:33.050 ","End":"01:35.915","Text":"If I replace n by n plus 1,"},{"Start":"01:35.915 ","End":"01:44.910","Text":"then this becomes 2n plus 1 because I get 2n plus 2 minus 1,"},{"Start":"01:44.910 ","End":"01:46.560","Text":"so it\u0027s 2n plus 1."},{"Start":"01:46.560 ","End":"01:54.255","Text":"But notice that we still get the 2n minus 1 before it,"},{"Start":"01:54.255 ","End":"01:56.820","Text":"we just end 1 later."},{"Start":"01:56.820 ","End":"01:59.360","Text":"The reason I\u0027m writing the previous one,"},{"Start":"01:59.360 ","End":"02:01.475","Text":"is because we\u0027re going to want to cancel."},{"Start":"02:01.475 ","End":"02:09.945","Text":"On the denominator, if I replace n by n plus 1 with 2n plus 2 minus 2,"},{"Start":"02:09.945 ","End":"02:12.845","Text":"we\u0027ll get 2n factorial."},{"Start":"02:12.845 ","End":"02:18.549","Text":"Now, r, therefore, will be the limit as n goes to infinity."},{"Start":"02:18.549 ","End":"02:22.045","Text":"There\u0027s no need for absolute value because everything\u0027s positive."},{"Start":"02:22.045 ","End":"02:25.910","Text":"But what we will get if I do this divided by this,"},{"Start":"02:25.910 ","End":"02:28.600","Text":"it\u0027s like multiplying by the inverse."},{"Start":"02:28.600 ","End":"02:36.695","Text":"I\u0027ll get 1 times 3 times 5 up to 2n minus 1."},{"Start":"02:36.695 ","End":"02:40.840","Text":"Then this will be on the denominator,"},{"Start":"02:40.840 ","End":"02:44.935","Text":"1 times 3 times 5, and so on,"},{"Start":"02:44.935 ","End":"02:49.270","Text":"up to 2n minus 1, 2n plus 1."},{"Start":"02:49.270 ","End":"02:53.585","Text":"See why I wanted to write the previous next to last term."},{"Start":"02:53.585 ","End":"02:59.090","Text":"Then times this goes up to the numerator,"},{"Start":"02:59.090 ","End":"03:04.245","Text":"2n factorial, and this stays in the denominator,"},{"Start":"03:04.245 ","End":"03:09.810","Text":"2n minus 2 factorial."},{"Start":"03:09.810 ","End":"03:12.150","Text":"Here, I can see how to cancel,"},{"Start":"03:12.150 ","End":"03:13.490","Text":"but what about here?"},{"Start":"03:13.490 ","End":"03:17.705","Text":"The side, I\u0027ll note that 2n factorial,"},{"Start":"03:17.705 ","End":"03:20.270","Text":"I want to somehow express it in terms of this,"},{"Start":"03:20.270 ","End":"03:25.280","Text":"it\u0027s just when I count down from 2n to 1 and take the products, I got 2n."},{"Start":"03:25.280 ","End":"03:28.450","Text":"The next term is 2n minus 1,"},{"Start":"03:28.450 ","End":"03:31.815","Text":"and then 2n minus 2,"},{"Start":"03:31.815 ","End":"03:34.035","Text":"all the way down to 1."},{"Start":"03:34.035 ","End":"03:36.719","Text":"But this bit here,"},{"Start":"03:36.719 ","End":"03:41.055","Text":"is just 2n minus 2 factorial."},{"Start":"03:41.055 ","End":"03:46.400","Text":"If I replace this with what\u0027s here, after we cancel,"},{"Start":"03:46.400 ","End":"03:51.490","Text":"we\u0027ll get the limit as n goes to infinity."},{"Start":"03:51.490 ","End":"03:57.840","Text":"Well, here we\u0027ll just get 1 over 2n plus 1, and from here,"},{"Start":"03:57.840 ","End":"04:06.215","Text":"we\u0027ll get this 2 times 2n times 2n minus 1."},{"Start":"04:06.215 ","End":"04:12.819","Text":"Continuing, I can write this as the limit as n goes to infinity."},{"Start":"04:12.819 ","End":"04:14.155","Text":"If I open the brackets,"},{"Start":"04:14.155 ","End":"04:24.360","Text":"I\u0027ve got 4n squared minus 2n over 2n plus 1."},{"Start":"04:24.360 ","End":"04:27.610","Text":"Now, when you have a polynomial over polynomial,"},{"Start":"04:27.610 ","End":"04:30.370","Text":"it\u0027s enough to just look at the leading terms,"},{"Start":"04:30.370 ","End":"04:32.990","Text":"here it\u0027s 4n squared, here it\u0027s the 2n,"},{"Start":"04:32.990 ","End":"04:38.050","Text":"and the limit of this is like the limit of 2n."},{"Start":"04:40.040 ","End":"04:47.184","Text":"This is the same as the limit of 2n as n goes to infinity,"},{"Start":"04:47.184 ","End":"04:49.870","Text":"and this, of course, is infinity."},{"Start":"04:49.870 ","End":"04:58.750","Text":"When this happens, when you get the radius of convergence to be infinity,"},{"Start":"04:58.750 ","End":"05:01.360","Text":"it means it converges for all x."},{"Start":"05:01.360 ","End":"05:08.740","Text":"You could say that the range is just everything,"},{"Start":"05:08.740 ","End":"05:11.485","Text":"but we can write everything in this way."},{"Start":"05:11.485 ","End":"05:14.410","Text":"X between minus infinity, infinity,"},{"Start":"05:14.410 ","End":"05:19.435","Text":"instead of saying just all x, it converges everywhere."},{"Start":"05:19.435 ","End":"05:22.400","Text":"We\u0027re done with this one."}],"ID":7980},{"Watched":false,"Name":"Exercise 3 part i","Duration":"10m 17s","ChapterTopicVideoID":7919,"CourseChapterTopicPlaylistID":4250,"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, you want to find the range of convergence and"},{"Start":"00:03.570 ","End":"00:07.935","Text":"the radius of convergence of this series."},{"Start":"00:07.935 ","End":"00:10.680","Text":"Now, note that it\u0027s not x to the n,"},{"Start":"00:10.680 ","End":"00:14.550","Text":"it\u0027s x plus 1 to the n. Our formulas work with x to the n"},{"Start":"00:14.550 ","End":"00:19.770","Text":"so we do the usual thing of making a substitution that say t"},{"Start":"00:19.770 ","End":"00:24.660","Text":"equals x plus 1 and then it will be of the form that we"},{"Start":"00:24.660 ","End":"00:30.630","Text":"like a_n x to the n only it\u0027ll be a_n t to the n,"},{"Start":"00:30.630 ","End":"00:32.700","Text":"where in our case,"},{"Start":"00:32.700 ","End":"00:40.545","Text":"a_n is equal to the coefficient will be minus 1 to the n plus 1 over"},{"Start":"00:40.545 ","End":"00:50.395","Text":"n times 4 to the n. Now we have 2 main formulas for finding the radius of convergence,"},{"Start":"00:50.395 ","End":"00:54.015","Text":"1 involving a ratio and 1 involving a root."},{"Start":"00:54.015 ","End":"00:58.165","Text":"Both will work and I\u0027m going to use the 1 involving a root,"},{"Start":"00:58.165 ","End":"01:02.090","Text":"but I know I\u0027m going to need a certain result and I\u0027ll write it up front."},{"Start":"01:02.090 ","End":"01:08.960","Text":"That is that the limit as n goes to infinity of the nth root of n,"},{"Start":"01:08.960 ","End":"01:10.430","Text":"we\u0027ve seen this before."},{"Start":"01:10.430 ","End":"01:12.725","Text":"This is equal to 1."},{"Start":"01:12.725 ","End":"01:16.955","Text":"In fact, it works more generally if I put any k here,"},{"Start":"01:16.955 ","End":"01:18.110","Text":"it will also work,"},{"Start":"01:18.110 ","End":"01:20.360","Text":"but you can take k equals 1."},{"Start":"01:20.360 ","End":"01:21.650","Text":"I\u0027m going to use this."},{"Start":"01:21.650 ","End":"01:23.270","Text":"If you don\u0027t like this,"},{"Start":"01:23.270 ","End":"01:29.650","Text":"then use the ratio formula,"},{"Start":"01:30.170 ","End":"01:32.745","Text":"just reminder of both."},{"Start":"01:32.745 ","End":"01:38.090","Text":"If you use the root,"},{"Start":"01:38.090 ","End":"01:42.950","Text":"then R is the limit as n goes to infinity of"},{"Start":"01:42.950 ","End":"01:50.535","Text":"1 over the nth root of absolute value of a_n."},{"Start":"01:50.535 ","End":"01:56.060","Text":"The other formula using the ratio is the limit as n goes to"},{"Start":"01:56.060 ","End":"02:01.335","Text":"infinity of an plus 1 over a_n."},{"Start":"02:01.335 ","End":"02:04.715","Text":"Like I said, we\u0027re going to use this one here,"},{"Start":"02:04.715 ","End":"02:06.335","Text":"but we\u0027ll need this result,"},{"Start":"02:06.335 ","End":"02:08.900","Text":"and if you use this, you don\u0027t need this result."},{"Start":"02:08.900 ","End":"02:12.470","Text":"But it\u0027s easier with the root,"},{"Start":"02:12.470 ","End":"02:20.135","Text":"so we get that R is the limit as n goes to infinity."},{"Start":"02:20.135 ","End":"02:23.600","Text":"Now, the absolute value of a_n means"},{"Start":"02:23.600 ","End":"02:26.960","Text":"that we can throw out this minus 1 to the power of something."},{"Start":"02:26.960 ","End":"02:28.610","Text":"This is either plus or minus 1,"},{"Start":"02:28.610 ","End":"02:30.320","Text":"and either way with the absolute value,"},{"Start":"02:30.320 ","End":"02:31.630","Text":"we don\u0027t need it,"},{"Start":"02:31.630 ","End":"02:39.965","Text":"so what we get is 1 over the nth root of"},{"Start":"02:39.965 ","End":"02:46.270","Text":"the absolute value of just 1 over"},{"Start":"02:46.270 ","End":"02:56.150","Text":"n times 4 to the n. Now we can get rid of the absolute value,"},{"Start":"02:56.150 ","End":"03:05.435","Text":"and I can also put the nth root inside the, I mean,"},{"Start":"03:05.435 ","End":"03:08.570","Text":"I can do it for top and bottom separately, and in fact,"},{"Start":"03:08.570 ","End":"03:18.540","Text":"I\u0027ll do it on each factor, do it strictly."},{"Start":"03:18.540 ","End":"03:24.815","Text":"The nth root of 1 over the nth root of"},{"Start":"03:24.815 ","End":"03:34.380","Text":"n times the nth root of 4 to the n. All this is the big dividing line."},{"Start":"03:34.460 ","End":"03:37.860","Text":"Oh, I forgot to write the limit."},{"Start":"03:37.860 ","End":"03:42.465","Text":"Here we are, limit as n goes to infinity."},{"Start":"03:42.465 ","End":"03:45.840","Text":"Now nth root of 1 is 1, it\u0027s 1 over,"},{"Start":"03:45.840 ","End":"03:50.055","Text":"1 over, so I can put all this into the numerator,"},{"Start":"03:50.055 ","End":"03:55.535","Text":"and I get the limit as n goes to infinity"},{"Start":"03:55.535 ","End":"04:03.600","Text":"of the nth root of n times,"},{"Start":"04:03.600 ","End":"04:06.390","Text":"this becomes just 4."},{"Start":"04:06.390 ","End":"04:09.885","Text":"The nth root of 4 to the n is just 4,"},{"Start":"04:09.885 ","End":"04:13.729","Text":"and since this thing tends to 1,"},{"Start":"04:13.729 ","End":"04:15.265","Text":"like we wrote here,"},{"Start":"04:15.265 ","End":"04:18.550","Text":"this will just equal 4,"},{"Start":"04:18.550 ","End":"04:20.665","Text":"1 times 4 is 4."},{"Start":"04:20.665 ","End":"04:28.690","Text":"That\u0027s our radius, but it applies to t. We get this sketch for which"},{"Start":"04:28.690 ","End":"04:33.730","Text":"values of t it converges and diverges between plus and minus"},{"Start":"04:33.730 ","End":"04:38.230","Text":"4 then it converges outside, it diverges."},{"Start":"04:38.230 ","End":"04:41.035","Text":"We just don\u0027t know at these 2 points,"},{"Start":"04:41.035 ","End":"04:49.440","Text":"but we would like to have x and not t. What values of x correspond?"},{"Start":"04:49.440 ","End":"04:51.690","Text":"We\u0027ll look, t is x plus 1,"},{"Start":"04:51.690 ","End":"04:56.810","Text":"so x is t minus 1,"},{"Start":"04:56.810 ","End":"05:00.710","Text":"so if this is for t, I subtract 1, I\u0027ll get x\u0027s."},{"Start":"05:00.710 ","End":"05:02.180","Text":"This is less important,"},{"Start":"05:02.180 ","End":"05:03.730","Text":"the center minus 1,"},{"Start":"05:03.730 ","End":"05:06.725","Text":"but subtract 1 from here and I\u0027ve got 3,"},{"Start":"05:06.725 ","End":"05:09.340","Text":"subtract 1, I\u0027ve got minus 5."},{"Start":"05:09.340 ","End":"05:14.630","Text":"We\u0027re going to have to check separately what happens when x equals"},{"Start":"05:14.630 ","End":"05:22.505","Text":"3 and what happens when x equals minus 5."},{"Start":"05:22.505 ","End":"05:28.320","Text":"At the 3, what we get is,"},{"Start":"05:29.170 ","End":"05:34.100","Text":"I can\u0027t see the original series at the top,"},{"Start":"05:34.100 ","End":"05:36.660","Text":"but it was just like this."},{"Start":"05:36.680 ","End":"05:40.550","Text":"There was an x plus 1 to the n there."},{"Start":"05:40.550 ","End":"05:41.780","Text":"I don\u0027t want to scroll back up."},{"Start":"05:41.780 ","End":"05:46.350","Text":"This is how it was in Sigma,"},{"Start":"05:48.440 ","End":"05:51.680","Text":"Oh, I just copied it from above here."},{"Start":"05:51.680 ","End":"05:53.329","Text":"This is our series,"},{"Start":"05:53.329 ","End":"05:56.180","Text":"so x equals 3."},{"Start":"05:56.180 ","End":"06:02.735","Text":"What we get is the sum of"},{"Start":"06:02.735 ","End":"06:09.285","Text":"minus 1 to the power of n plus 1 from 1 to infinity."},{"Start":"06:09.285 ","End":"06:13.410","Text":"Now, 3 plus 1 is 4,"},{"Start":"06:13.410 ","End":"06:23.370","Text":"so we get 4 to the n over 4 to the n will cancel and we\u0027ll"},{"Start":"06:23.370 ","End":"06:27.510","Text":"just get 1 over"},{"Start":"06:27.510 ","End":"06:34.500","Text":"n. I could mention that we add a 4 to the n from the 3 plus 1 over 4 to the n,"},{"Start":"06:34.500 ","End":"06:37.980","Text":"but this whole thing is equal to 1,"},{"Start":"06:37.980 ","End":"06:39.720","Text":"so it says if it\u0027s not there."},{"Start":"06:39.720 ","End":"06:43.100","Text":"Now this series of 1 over n with"},{"Start":"06:43.100 ","End":"06:50.280","Text":"alternating signs it\u0027s the alternating harmonic,"},{"Start":"06:50.280 ","End":"06:59.975","Text":"it\u0027s a Newton Leibnitz series because the sequence of 1 over n has 3 properties."},{"Start":"06:59.975 ","End":"07:04.780","Text":"It\u0027s positive, it\u0027s decreasing,"},{"Start":"07:04.780 ","End":"07:11.360","Text":"I\u0027ll just write this symbolically and tends to 0, it satisfies these."},{"Start":"07:11.360 ","End":"07:18.540","Text":"Then the alternating sum of these satisfies the Leibnitz convergence test,"},{"Start":"07:18.540 ","End":"07:21.795","Text":"and so this converges,"},{"Start":"07:21.795 ","End":"07:26.500","Text":"so the 3 is in."},{"Start":"07:26.720 ","End":"07:35.050","Text":"For minus 5, we get the sum from 1 to infinity."},{"Start":"07:36.740 ","End":"07:39.330","Text":"If x is minus 5,"},{"Start":"07:39.330 ","End":"07:42.615","Text":"minus 5 plus 1 is minus 4,"},{"Start":"07:42.615 ","End":"07:47.400","Text":"so we get minus 1 to the n plus 1,"},{"Start":"07:47.400 ","End":"07:50.070","Text":"1 over n, this time,"},{"Start":"07:50.070 ","End":"07:58.185","Text":"minus 4 to the n over 4 to the n. It doesn\u0027t immediately cancel,"},{"Start":"07:58.185 ","End":"08:02.910","Text":"but if I just take care of the minuses at the side,"},{"Start":"08:02.910 ","End":"08:09.090","Text":"this term here is minus 1 to the n,"},{"Start":"08:09.090 ","End":"08:14.090","Text":"4 to the n. I combine the minus"},{"Start":"08:14.090 ","End":"08:21.180","Text":"1 to the n plus 1 with the minus 1 to the n here,"},{"Start":"08:21.180 ","End":"08:28.410","Text":"and we get minus 1 to the power of 2 n plus 1."},{"Start":"08:28.410 ","End":"08:31.970","Text":"Now this is an odd number and because this is an odd number,"},{"Start":"08:31.970 ","End":"08:35.580","Text":"minus 1 to an odd number is minus 1."},{"Start":"08:37.970 ","End":"08:40.880","Text":"The minus 1, I\u0027m also going to bring up front,"},{"Start":"08:40.880 ","End":"08:50.565","Text":"so I get minus the sum from 1 to infinity,"},{"Start":"08:50.565 ","End":"08:57.860","Text":"and then all I\u0027m left with is 1 over n because this now cancels 4 to"},{"Start":"08:57.860 ","End":"09:03.530","Text":"the n and then the minus 1 we took care of"},{"Start":"09:03.530 ","End":"09:09.290","Text":"and we\u0027re just left with 1 over n. Now this is the classic harmonic series,"},{"Start":"09:09.290 ","End":"09:12.605","Text":"and we know that this diverges,"},{"Start":"09:12.605 ","End":"09:15.229","Text":"seen it many times before."},{"Start":"09:15.229 ","End":"09:18.110","Text":"It can also be done as a p series."},{"Start":"09:18.110 ","End":"09:21.670","Text":"Anyway we\u0027ve seen it enough times to know that this divergence,"},{"Start":"09:21.670 ","End":"09:25.080","Text":"so here we\u0027ve solved the 2 question marks."},{"Start":"09:25.080 ","End":"09:27.230","Text":"That is that here it converges,"},{"Start":"09:27.230 ","End":"09:30.050","Text":"I\u0027ll just write c, but here it diverges,"},{"Start":"09:30.050 ","End":"09:34.700","Text":"I\u0027ll write d. So it converges altogether."},{"Start":"09:34.700 ","End":"09:40.075","Text":"The values of x are minus 5,"},{"Start":"09:40.075 ","End":"09:46.605","Text":"less than x, less than or equal to 3."},{"Start":"09:46.605 ","End":"09:51.495","Text":"As for the radius, it\u0027s still 4 because from here to here is 4,"},{"Start":"09:51.495 ","End":"09:53.925","Text":"and from here to here is 4."},{"Start":"09:53.925 ","End":"09:55.560","Text":"We still got 4 either way,"},{"Start":"09:55.560 ","End":"09:59.080","Text":"just the center is minus 1."},{"Start":"10:00.860 ","End":"10:08.375","Text":"Perhaps I\u0027ll summarize that the range of convergence is this,"},{"Start":"10:08.375 ","End":"10:18.790","Text":"and the radius is 4 and that\u0027s the answer, and we\u0027re done."}],"ID":7981},{"Watched":false,"Name":"Exercise 3 part j","Duration":"4m 38s","ChapterTopicVideoID":7920,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this exercise, as usual,"},{"Start":"00:02.460 ","End":"00:06.990","Text":"we have to find the range of convergence and the radius of convergence."},{"Start":"00:06.990 ","End":"00:10.695","Text":"But this series, if you look at it,"},{"Start":"00:10.695 ","End":"00:13.290","Text":"is actually a geometric series,"},{"Start":"00:13.290 ","End":"00:15.660","Text":"so we can take some shortcuts here."},{"Start":"00:15.660 ","End":"00:19.050","Text":"I see it has the same power of n here and here,"},{"Start":"00:19.050 ","End":"00:29.895","Text":"so I can actually write this series as 3/4 times x plus 5,"},{"Start":"00:29.895 ","End":"00:32.970","Text":"all this to the power of n,"},{"Start":"00:32.970 ","End":"00:35.655","Text":"and goes from 1 to infinity."},{"Start":"00:35.655 ","End":"00:41.460","Text":"Now, this looks very much like the sum from 1 to infinity,"},{"Start":"00:41.460 ","End":"00:46.250","Text":"of sum q to the power of n. This is a typical geometric series,"},{"Start":"00:46.250 ","End":"00:48.420","Text":"q stands for quotient."},{"Start":"00:50.410 ","End":"00:56.184","Text":"If we take this to be our q,"},{"Start":"00:56.184 ","End":"01:00.185","Text":"then we can apply the theory of this series."},{"Start":"01:00.185 ","End":"01:03.440","Text":"We know that when this converges,"},{"Start":"01:03.440 ","End":"01:05.645","Text":"there\u0027s only 2 cases."},{"Start":"01:05.645 ","End":"01:08.845","Text":"If q is less than 1,"},{"Start":"01:08.845 ","End":"01:14.010","Text":"then it converges, sorry,"},{"Start":"01:14.010 ","End":"01:15.975","Text":"absolute value of q."},{"Start":"01:15.975 ","End":"01:23.410","Text":"If absolute value of q is bigger or equal to 1, then it diverges."},{"Start":"01:24.460 ","End":"01:28.190","Text":"Of course, we\u0027re interested in the convergence part,"},{"Start":"01:28.190 ","End":"01:30.575","Text":"so if we just apply this rule here,"},{"Start":"01:30.575 ","End":"01:38.990","Text":"we get that the absolute value of 3/4 x plus 5 is less than 1."},{"Start":"01:38.990 ","End":"01:43.100","Text":"Now, when we have an absolute value inequality like this,"},{"Start":"01:43.100 ","End":"01:50.410","Text":"this translates to something being less than 1 and greater than minus 1,"},{"Start":"01:50.410 ","End":"01:55.040","Text":"this expression here, 3/4 x plus 5."},{"Start":"01:55.040 ","End":"01:56.600","Text":"It\u0027s a double inequality,"},{"Start":"01:56.600 ","End":"02:00.570","Text":"we can work with all 3 sides,"},{"Start":"02:00.570 ","End":"02:03.070","Text":"so to speak, at once."},{"Start":"02:06.530 ","End":"02:12.485","Text":"No, let me multiply by 4/3,"},{"Start":"02:12.485 ","End":"02:14.045","Text":"and then I\u0027ll get rid of this."},{"Start":"02:14.045 ","End":"02:22.330","Text":"Minus 4/3 is less than x plus 5 is less than 4/3."},{"Start":"02:22.330 ","End":"02:26.290","Text":"Then if I subtract 5,"},{"Start":"02:26.950 ","End":"02:29.510","Text":"well this is 1/3,"},{"Start":"02:29.510 ","End":"02:36.590","Text":"I\u0027ll get minus 5 and 1/3 is 6 and 1/3,"},{"Start":"02:36.590 ","End":"02:40.659","Text":"less than x less than,"},{"Start":"02:40.659 ","End":"02:44.390","Text":"and here I have to do 4/3 minus 5,"},{"Start":"02:44.390 ","End":"02:47.150","Text":"which is 1/3 minus 5,"},{"Start":"02:47.150 ","End":"02:57.060","Text":"which is minus 3 and 2/3."},{"Start":"03:00.040 ","End":"03:09.600","Text":"Actually, I\u0027ll just write that this was minus 5 plus 4/3,"},{"Start":"03:09.600 ","End":"03:14.315","Text":"and this we got from minus 5 minus 4/3."},{"Start":"03:14.315 ","End":"03:18.520","Text":"The point is that the center,"},{"Start":"03:18.520 ","End":"03:24.430","Text":"if I sketch this, say here,"},{"Start":"03:24.950 ","End":"03:29.615","Text":"the center is at minus 5,"},{"Start":"03:29.615 ","End":"03:35.510","Text":"but then I go plus 4/3 and minus 4/3,"},{"Start":"03:36.170 ","End":"03:40.935","Text":"and I get to these 2 points, which are,"},{"Start":"03:40.935 ","End":"03:45.195","Text":"like I said here, 3 and 2/3, sorry,"},{"Start":"03:45.195 ","End":"03:50.680","Text":"minus, and minus, it\u0027s a bit lopsided,"},{"Start":"03:50.680 ","End":"03:55.105","Text":"but never mind, minus 6 and 1/3."},{"Start":"03:55.105 ","End":"03:59.620","Text":"If I want the radius of convergence,"},{"Start":"03:59.620 ","End":"04:05.385","Text":"the radius, too lazy to write of convergence,"},{"Start":"04:05.385 ","End":"04:08.520","Text":"is equal to 4/3,"},{"Start":"04:08.520 ","End":"04:12.620","Text":"that\u0027s this distance each way from the center and the range of"},{"Start":"04:12.620 ","End":"04:18.380","Text":"convergence is between here and here, but not including."},{"Start":"04:18.380 ","End":"04:21.770","Text":"So minus 6 and 1/3,"},{"Start":"04:21.770 ","End":"04:26.660","Text":"less than x, less than minus 3 and 2/3,"},{"Start":"04:26.660 ","End":"04:28.825","Text":"just basically they copied from this line,"},{"Start":"04:28.825 ","End":"04:35.345","Text":"and this is our answer."},{"Start":"04:35.345 ","End":"04:38.250","Text":"We are done."}],"ID":7982},{"Watched":false,"Name":"Exercise 3 part k","Duration":"8m 20s","ChapterTopicVideoID":7921,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:06.150","Text":"In this exercise, we\u0027re going to find the radius of convergence,"},{"Start":"00:06.150 ","End":"00:10.215","Text":"and also the range of convergence of this series."},{"Start":"00:10.215 ","End":"00:15.480","Text":"Now it\u0027s not in the usual form of x_n."},{"Start":"00:15.480 ","End":"00:17.670","Text":"But we\u0027ve seen this thing before."},{"Start":"00:17.670 ","End":"00:19.875","Text":"What we do is the substitution."},{"Start":"00:19.875 ","End":"00:25.320","Text":"Let\u0027s let t equal x minus 1 squared"},{"Start":"00:25.320 ","End":"00:32.370","Text":"and then this series will be the sum n goes from 1 to infinity."},{"Start":"00:32.370 ","End":"00:38.070","Text":"This is x minus 1 squared to the power of n. It\u0027s like there was brackets here."},{"Start":"00:38.070 ","End":"00:47.530","Text":"This will be t^n over n^4 times 100^n."},{"Start":"00:49.910 ","End":"00:58.210","Text":"This is more familiar to regular power series just in t not in x."},{"Start":"00:58.550 ","End":"01:03.570","Text":"It\u0027s in the form the sum of AN, t^n,"},{"Start":"01:03.570 ","End":"01:08.145","Text":"where An is the coefficient,"},{"Start":"01:08.145 ","End":"01:14.290","Text":"is 1 over n^4 times 100^n."},{"Start":"01:15.190 ","End":"01:18.845","Text":"To get the radius of conversion for t,"},{"Start":"01:18.845 ","End":"01:21.500","Text":"there\u0027s 2 formulas we can use ratio or root."},{"Start":"01:21.500 ","End":"01:23.470","Text":"I\u0027ll go with root."},{"Start":"01:23.470 ","End":"01:27.110","Text":"What we get in general,"},{"Start":"01:27.110 ","End":"01:34.070","Text":"I\u0027ll just write the general rule is that R is the limit as n goes to infinity"},{"Start":"01:34.070 ","End":"01:41.555","Text":"of 1 over the nth root of An not quite,"},{"Start":"01:41.555 ","End":"01:43.625","Text":"this has to be an absolute value."},{"Start":"01:43.625 ","End":"01:52.250","Text":"In our case, we get that R is the limit as n goes to infinity of"},{"Start":"01:52.250 ","End":"02:02.420","Text":"1 over the nth root of."},{"Start":"02:02.420 ","End":"02:05.840","Text":"We don\u0027t need the absolute value because it\u0027s positive."},{"Start":"02:05.840 ","End":"02:10.385","Text":"But we need the nth root of 1"},{"Start":"02:10.385 ","End":"02:17.430","Text":"over n^4 times 100^n."},{"Start":"02:18.970 ","End":"02:25.339","Text":"Now this is the same as the limit as n goes to infinity."},{"Start":"02:25.339 ","End":"02:30.800","Text":"I can put the nth root inside and the nth root of 1 is 1."},{"Start":"02:30.800 ","End":"02:33.845","Text":"Basically we\u0027re just bringing this up to the numerator."},{"Start":"02:33.845 ","End":"02:43.080","Text":"We have the nth root of n^4 times 100^n."},{"Start":"02:45.290 ","End":"02:49.515","Text":"This also I can break up into 2 bits."},{"Start":"02:49.515 ","End":"02:59.030","Text":"Actually, this is the limit as n goes to infinity."},{"Start":"02:59.030 ","End":"03:05.360","Text":"Like this is the nth root of n^4"},{"Start":"03:05.360 ","End":"03:11.480","Text":"and the nth root of 100^n."},{"Start":"03:11.480 ","End":"03:13.774","Text":"This thing is just 100."},{"Start":"03:13.774 ","End":"03:19.280","Text":"This thing equals 100 and this thing tends to 1."},{"Start":"03:19.280 ","End":"03:25.570","Text":"There\u0027s a general rule that the limit as n goes to"},{"Start":"03:25.570 ","End":"03:34.060","Text":"infinity of the nth root of n to do any power k could be 4 is equal to 1."},{"Start":"03:34.060 ","End":"03:36.445","Text":"In our case, k is equal to 4 here,"},{"Start":"03:36.445 ","End":"03:39.350","Text":"but this goes to 1, this is a 100."},{"Start":"03:39.630 ","End":"03:45.270","Text":"This is equal to 100."},{"Start":"03:45.270 ","End":"03:48.840","Text":"Which means that this"},{"Start":"03:48.840 ","End":"03:56.280","Text":"converges when t is"},{"Start":"03:56.280 ","End":"04:02.655","Text":"between minus a 100 and 100."},{"Start":"04:02.655 ","End":"04:05.760","Text":"For this we have convergence."},{"Start":"04:05.760 ","End":"04:10.510","Text":"Then we have to check separately at a 100 and minus a 100."},{"Start":"04:10.510 ","End":"04:12.880","Text":"Then we know it diverges outside."},{"Start":"04:12.880 ","End":"04:16.780","Text":"Now, because t is something squared,"},{"Start":"04:16.780 ","End":"04:25.050","Text":"actually we can say that this part is irrelevant because t is non-negative."},{"Start":"04:25.050 ","End":"04:30.510","Text":"It\u0027s really just this bit here."},{"Start":"04:30.510 ","End":"04:34.390","Text":"I can write this as,"},{"Start":"04:34.390 ","End":"04:37.730","Text":"let\u0027s see, t is x minus 1 squared,"},{"Start":"04:37.730 ","End":"04:47.425","Text":"so I get x minus 1 squared is less than 100."},{"Start":"04:47.425 ","End":"04:52.180","Text":"Because this is t. Actually we only have 1 endpoint to check."},{"Start":"04:52.180 ","End":"04:54.175","Text":"Like I said, this is not applicable."},{"Start":"04:54.175 ","End":"04:58.030","Text":"But let\u0027s see what happens when t equals a 100."},{"Start":"04:58.030 ","End":"05:01.315","Text":"Of course I could go back to x."},{"Start":"05:01.315 ","End":"05:04.525","Text":"Let\u0027s just stick with t for the moment and then we\u0027ll get on to this."},{"Start":"05:04.525 ","End":"05:06.909","Text":"What happens when t is a 100?"},{"Start":"05:06.909 ","End":"05:10.135","Text":"If t equals 100,"},{"Start":"05:10.135 ","End":"05:13.510","Text":"then I get from this series,"},{"Start":"05:13.510 ","End":"05:19.340","Text":"the sum n goes from 1 to infinity of"},{"Start":"05:19.340 ","End":"05:29.540","Text":"100^n over n^4 times 100^n."},{"Start":"05:31.050 ","End":"05:34.045","Text":"\u0027Cause this cancels with this."},{"Start":"05:34.045 ","End":"05:37.220","Text":"What we get is a P series,"},{"Start":"05:37.620 ","End":"05:42.080","Text":"the sum of 1 over n^4."},{"Start":"05:42.830 ","End":"05:51.485","Text":"In general, if we have the sum from 1 to infinity of 1 over n^p,"},{"Start":"05:51.485 ","End":"06:01.820","Text":"this converges for p greater than 1,"},{"Start":"06:01.820 ","End":"06:05.700","Text":"which it is because in our case, p equals 4."},{"Start":"06:05.950 ","End":"06:10.410","Text":"For t equals a 100, it converges."},{"Start":"06:10.430 ","End":"06:13.110","Text":"I\u0027ll just add this, it converges."},{"Start":"06:13.110 ","End":"06:16.830","Text":"Also for t equals a 100, we\u0027re okay."},{"Start":"06:16.830 ","End":"06:19.425","Text":"Really it\u0027s less than or equal to."},{"Start":"06:19.425 ","End":"06:23.440","Text":"Our convergence for x is this."},{"Start":"06:23.540 ","End":"06:26.570","Text":"Now I can translate this."},{"Start":"06:26.570 ","End":"06:30.515","Text":"If something squared is less than or equal to a 100,"},{"Start":"06:30.515 ","End":"06:34.690","Text":"then it\u0027s got to be between plus or minus the square root of this."},{"Start":"06:34.690 ","End":"06:39.410","Text":"X minus 1 has got to be between minus"},{"Start":"06:39.410 ","End":"06:44.465","Text":"10 and 10 in order for it squared to be less than or equal to a 100."},{"Start":"06:44.465 ","End":"06:48.720","Text":"Just you can investigate a parabola."},{"Start":"06:50.350 ","End":"06:53.870","Text":"It\u0027s fairly clear that this is true."},{"Start":"06:53.870 ","End":"06:59.410","Text":"Or you could take the square root and say absolute value of x minus 1 is less than 10,"},{"Start":"06:59.410 ","End":"07:02.540","Text":"and so x minus 1 is between 10 and minus 10."},{"Start":"07:02.540 ","End":"07:05.270","Text":"Then finally just adding 1,"},{"Start":"07:05.270 ","End":"07:11.615","Text":"we get that minus 9 less than or equal to x,"},{"Start":"07:11.615 ","End":"07:14.809","Text":"less than or equal to 11."},{"Start":"07:14.809 ","End":"07:25.745","Text":"Now, the radius of convergence stays 10 even if I\u0027ve shifted it, it\u0027s still 10."},{"Start":"07:25.745 ","End":"07:29.435","Text":"Just because I center it round."},{"Start":"07:29.435 ","End":"07:33.000","Text":"It would be centered on 1."},{"Start":"07:33.860 ","End":"07:40.655","Text":"But because here I have 11 and here I have"},{"Start":"07:40.655 ","End":"07:47.915","Text":"minus 9 and this distance is 10 and this distance is 10."},{"Start":"07:47.915 ","End":"07:56.390","Text":"What I can say to answer the question that the radius of convergence,"},{"Start":"07:56.390 ","End":"08:01.159","Text":"I\u0027m just lazy to write off convergence is 10."},{"Start":"08:01.159 ","End":"08:10.830","Text":"The range or region or interval of convergence is what we have here,"},{"Start":"08:10.830 ","End":"08:13.670","Text":"minus 9 less than or equal to x,"},{"Start":"08:13.670 ","End":"08:17.525","Text":"less than or equal to 11 including the endpoints."},{"Start":"08:17.525 ","End":"08:20.280","Text":"We are done."}],"ID":7983},{"Watched":false,"Name":"Exercise 3 part l","Duration":"8m 13s","ChapterTopicVideoID":7922,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to find the range of"},{"Start":"00:03.270 ","End":"00:06.750","Text":"convergence and radius of convergence of this series."},{"Start":"00:06.750 ","End":"00:12.550","Text":"This is a bit different than we\u0027ve had previously."},{"Start":"00:13.610 ","End":"00:17.430","Text":"We usually have something to the power of n."},{"Start":"00:17.430 ","End":"00:21.645","Text":"We\u0027re used to having x plus 5 instead of x and that sort of thing."},{"Start":"00:21.645 ","End":"00:24.645","Text":"But usually if we have to the power of n,"},{"Start":"00:24.645 ","End":"00:26.880","Text":"or if we have to the power of 2n,"},{"Start":"00:26.880 ","End":"00:29.730","Text":"We now to make that something squared,"},{"Start":"00:29.730 ","End":"00:32.400","Text":"the 2n plus 1 is not so good."},{"Start":"00:32.400 ","End":"00:35.715","Text":"What I\u0027m going to do is"},{"Start":"00:35.715 ","End":"00:40.850","Text":"rewrite that this x plus 5 to the n. Let me just do a side exercise."},{"Start":"00:40.850 ","End":"00:51.250","Text":"x plus 5, throw it to the 2n plus 1 is x plus 5 times x plus"},{"Start":"00:52.070 ","End":"01:03.455","Text":"5 to the power of 2n and this is x plus 5 times x plus 5 squared"},{"Start":"01:03.455 ","End":"01:07.940","Text":"to the power of n. What I propose is that we make"},{"Start":"01:07.940 ","End":"01:13.710","Text":"a substitution that x plus 5 squared is t. Maybe I\u0027ll"},{"Start":"01:13.710 ","End":"01:24.759","Text":"just write that t equals x plus 5 squared and now I can write my series as"},{"Start":"01:24.759 ","End":"01:32.940","Text":"sigma sum n from 1 to infinity of x plus"},{"Start":"01:32.940 ","End":"01:39.900","Text":"5 times t to"},{"Start":"01:39.900 ","End":"01:44.330","Text":"the n over same as what it was before,"},{"Start":"01:44.330 ","End":"01:49.975","Text":"n times 2 to the 2n plus 1."},{"Start":"01:49.975 ","End":"01:53.805","Text":"Now, x as far as t goes is a constant."},{"Start":"01:53.805 ","End":"02:01.815","Text":"I can take the x plus 5 in front and say that this is x plus 5 outside the sigma."},{"Start":"02:01.815 ","End":"02:08.370","Text":"You know what, I\u0027d like to have this to be 2n instead of 2n plus"},{"Start":"02:08.370 ","End":"02:14.980","Text":"1 because I might take the nth root of this if I do the route computation."},{"Start":"02:14.980 ","End":"02:17.705","Text":"Let me take a 2 out of here also."},{"Start":"02:17.705 ","End":"02:19.640","Text":"That will be a bit nicer."},{"Start":"02:19.640 ","End":"02:25.905","Text":"I\u0027ll get t to the n over n times 2 to the 2n,"},{"Start":"02:25.905 ","End":"02:28.745","Text":"sum from 1 to infinity."},{"Start":"02:28.745 ","End":"02:32.765","Text":"This is going to come out nicer when I take the root test."},{"Start":"02:32.765 ","End":"02:35.645","Text":"As far as the series in t goes,"},{"Start":"02:35.645 ","End":"02:44.690","Text":"my a_n is just 1 over n times 2 to the power of 2n."},{"Start":"02:44.690 ","End":"02:49.955","Text":"The R, if I use the root test, is in general,"},{"Start":"02:49.955 ","End":"02:54.965","Text":"the limit as n goes to infinity of 1 over"},{"Start":"02:54.965 ","End":"03:01.730","Text":"the nth root of a_n, absolute value."},{"Start":"03:01.730 ","End":"03:03.875","Text":"But in our case,"},{"Start":"03:03.875 ","End":"03:14.110","Text":"we get that r is equal to 1 over the nth root of,"},{"Start":"03:15.860 ","End":"03:20.505","Text":"sorry, make this a bit larger,"},{"Start":"03:20.505 ","End":"03:29.700","Text":"absolute value of 1 over n times 2 to the power of 2n."},{"Start":"03:29.700 ","End":"03:32.000","Text":"This looks so messy. Why don\u0027t I just delete"},{"Start":"03:32.000 ","End":"03:34.955","Text":"the absolute value because everything\u0027s positive?"},{"Start":"03:34.955 ","End":"03:38.289","Text":"I forgot to write the limit."},{"Start":"03:38.289 ","End":"03:43.170","Text":"Limit as n goes to infinity and as usual,"},{"Start":"03:43.170 ","End":"03:45.380","Text":"the 1 over and the 1 over,"},{"Start":"03:45.380 ","End":"03:47.450","Text":"bring this to the numerator,"},{"Start":"03:47.450 ","End":"03:51.365","Text":"so we get the limit as n goes to infinity."},{"Start":"03:51.365 ","End":"03:54.095","Text":"In fact, if I take the nth root of each bit separately,"},{"Start":"03:54.095 ","End":"04:00.004","Text":"I\u0027ll get the nth root of n times"},{"Start":"04:00.004 ","End":"04:10.680","Text":"the nth root of 2 to the power of 2n and we know that this limit is 1."},{"Start":"04:12.170 ","End":"04:18.225","Text":"This is actually 2 squared to the power of n."},{"Start":"04:18.225 ","End":"04:25.070","Text":"What I\u0027ll get is that this is equal to 1 times 2 squared,"},{"Start":"04:25.070 ","End":"04:26.595","Text":"because 2 squared to the n,"},{"Start":"04:26.595 ","End":"04:28.065","Text":"then the nth root."},{"Start":"04:28.065 ","End":"04:31.150","Text":"In other words, it is 4."},{"Start":"04:31.190 ","End":"04:42.005","Text":"This means that we have convergence for t between 4 and minus 4."},{"Start":"04:42.005 ","End":"04:46.265","Text":"We have to check separately at the end points."},{"Start":"04:46.265 ","End":"04:50.495","Text":"But in our case, t is non-negative."},{"Start":"04:50.495 ","End":"04:53.640","Text":"It\u0027s bigger or equal to 0."},{"Start":"04:53.960 ","End":"04:57.270","Text":"If t is bigger or equal to 0,"},{"Start":"04:57.270 ","End":"05:01.470","Text":"I don\u0027t have to check at this end,"},{"Start":"05:01.470 ","End":"05:04.420","Text":"I only have to check at 4."},{"Start":"05:05.900 ","End":"05:09.050","Text":"Instead of putting it in the original series,"},{"Start":"05:09.050 ","End":"05:10.730","Text":"I\u0027ll put it in here."},{"Start":"05:10.730 ","End":"05:14.480","Text":"If I put here t equals 4,"},{"Start":"05:14.480 ","End":"05:18.695","Text":"then I will get, well,"},{"Start":"05:18.695 ","End":"05:21.950","Text":"this bit\u0027s not interesting as far as convergence is just a constant,"},{"Start":"05:21.950 ","End":"05:27.800","Text":"I just need to look at the sigma part from 1 to infinity of 4 to"},{"Start":"05:27.800 ","End":"05:33.930","Text":"the n over n times 2 to the 2n is 2 squared to the end,"},{"Start":"05:33.930 ","End":"05:35.790","Text":"just like before, 2 squared is 4,"},{"Start":"05:35.790 ","End":"05:38.435","Text":"so this cancels with this,"},{"Start":"05:38.435 ","End":"05:46.030","Text":"and we just get the sum from 1 to infinity of 1."},{"Start":"05:46.030 ","End":"05:51.710","Text":"This is the classic harmonic series and it diverges."},{"Start":"05:51.710 ","End":"05:56.615","Text":"We know this. We just have convergence for t less than 4."},{"Start":"05:56.615 ","End":"05:58.490","Text":"Actually, as I said,"},{"Start":"05:58.490 ","End":"06:01.235","Text":"because t is something squared,"},{"Start":"06:01.235 ","End":"06:10.465","Text":"I could rewrite this as t between 0 and 4."},{"Start":"06:10.465 ","End":"06:12.110","Text":"I don\u0027t even need that."},{"Start":"06:12.110 ","End":"06:16.165","Text":"I just have to say that t is less than 4."},{"Start":"06:16.165 ","End":"06:20.585","Text":"Just this bit is the interesting because then if I substitute back,"},{"Start":"06:20.585 ","End":"06:22.745","Text":"what t is, I get,"},{"Start":"06:22.745 ","End":"06:24.395","Text":"and I\u0027ll continue over here,"},{"Start":"06:24.395 ","End":"06:31.350","Text":"x plus 5 squared is less than 4."},{"Start":"06:31.350 ","End":"06:33.600","Text":"We\u0027ve done this thing before."},{"Start":"06:33.600 ","End":"06:38.520","Text":"We take this 4 is 2 squared and this gives us that x plus"},{"Start":"06:38.520 ","End":"06:43.910","Text":"5 is between the square root of 4 and minus the square root of 4,"},{"Start":"06:43.910 ","End":"06:47.390","Text":"between minus 2 and 2 not including the endpoints"},{"Start":"06:47.390 ","End":"06:51.530","Text":"because we ruled 4 out, here diverges 4."},{"Start":"06:51.530 ","End":"06:59.315","Text":"Then we just subtract 5 from each of the 3 bits to get x."},{"Start":"06:59.315 ","End":"07:04.410","Text":"But notice that from here we can see that the radius of"},{"Start":"07:04.410 ","End":"07:10.830","Text":"convergence is 2 because it\u0027s 2 either way."},{"Start":"07:10.830 ","End":"07:16.260","Text":"I subtract 5, and so I\u0027ve got minus"},{"Start":"07:16.260 ","End":"07:22.560","Text":"7 and minus 3."},{"Start":"07:22.560 ","End":"07:26.910","Text":"I mean, the radius is still 2 if I drew it on the number line,"},{"Start":"07:26.910 ","End":"07:31.755","Text":"I\u0027ll get here, minus 7, here minus 3."},{"Start":"07:31.755 ","End":"07:37.790","Text":"This minus 5 is always in the middle and the 1 where t is 0,"},{"Start":"07:37.790 ","End":"07:43.800","Text":"x is minus 5 and from here to here it is 2."},{"Start":"07:43.800 ","End":"07:46.110","Text":"From here to here is 2."},{"Start":"07:46.110 ","End":"07:50.475","Text":"The radius is 2 and this bit here,"},{"Start":"07:50.475 ","End":"08:00.700","Text":"x between minus 7 and 3 is the region or range of convergence."},{"Start":"08:03.860 ","End":"08:07.880","Text":"This bit here answers the questions for the radius and"},{"Start":"08:07.880 ","End":"08:12.630","Text":"the range of convergence and we are done."}],"ID":7984},{"Watched":false,"Name":"Exercise 4 part a","Duration":"2m 35s","ChapterTopicVideoID":7923,"CourseChapterTopicPlaylistID":4250,"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 want to expand the function,"},{"Start":"00:05.100 ","End":"00:08.760","Text":"f of x is 1 over 1 plus x as a power series,"},{"Start":"00:08.760 ","End":"00:11.040","Text":"and to find the range of convergence,"},{"Start":"00:11.040 ","End":"00:15.190","Text":"sometimes called region or interval of convergence."},{"Start":"00:15.950 ","End":"00:25.260","Text":"Now, this is very similar to a familiar expansion."},{"Start":"00:25.260 ","End":"00:30.015","Text":"We know that 1 over 1 minus x,"},{"Start":"00:30.015 ","End":"00:36.830","Text":"is 1 plus x plus x squared plus x cubed and so on,"},{"Start":"00:36.830 ","End":"00:44.960","Text":"which we usually write in Sigma form as sum of n equals 0"},{"Start":"00:44.960 ","End":"00:50.060","Text":"to infinity of x to the n or 1 x to the n."},{"Start":"00:50.060 ","End":"00:57.035","Text":"The range of convergence for this is absolute value of x less than 1."},{"Start":"00:57.035 ","End":"00:59.150","Text":"How we write it, if you prefer,"},{"Start":"00:59.150 ","End":"01:03.510","Text":"you could write it as this form."},{"Start":"01:03.510 ","End":"01:10.760","Text":"Now, we want to use this to help ourselves with this 1,"},{"Start":"01:10.760 ","End":"01:15.755","Text":"so what I\u0027m going to do is just rewrite this."},{"Start":"01:15.755 ","End":"01:18.020","Text":"In order to be able to use this,"},{"Start":"01:18.020 ","End":"01:26.625","Text":"I can write 1 over 1 plus x as 1 over 1 minus, minus x."},{"Start":"01:26.625 ","End":"01:31.955","Text":"Now, if I take minus x and substitute it here,"},{"Start":"01:31.955 ","End":"01:39.195","Text":"then what I\u0027ll get is that this is equal to the sum,"},{"Start":"01:39.195 ","End":"01:41.310","Text":"n equals 0 to infinity."},{"Start":"01:41.310 ","End":"01:47.910","Text":"Instead of x, I\u0027ll put minus x to the power of n"},{"Start":"01:47.910 ","End":"01:55.975","Text":"and the range of convergence will be the absolute value of minus x is less than 1."},{"Start":"01:55.975 ","End":"02:00.110","Text":"I can simplify or rewrite this and this."},{"Start":"02:00.110 ","End":"02:04.040","Text":"This is minus 1 times x."},{"Start":"02:04.040 ","End":"02:14.675","Text":"So this is equal to the sum from 0 to infinity of minus 1 to the power of n,"},{"Start":"02:14.675 ","End":"02:16.650","Text":"x to the power of n."},{"Start":"02:16.650 ","End":"02:19.880","Text":"The absolute value of minus x is the same as"},{"Start":"02:19.880 ","End":"02:21.500","Text":"the absolute value of x."},{"Start":"02:21.500 ","End":"02:26.990","Text":"I can just rewrite this as absolute value of x less than 1."},{"Start":"02:26.990 ","End":"02:29.930","Text":"So this is the power series expansion"},{"Start":"02:29.930 ","End":"02:32.515","Text":"and this is the range of convergence,"},{"Start":"02:32.515 ","End":"02:35.840","Text":"and that\u0027s all there is to it."}],"ID":7985},{"Watched":false,"Name":"Exercise 4 part b","Duration":"3m 29s","ChapterTopicVideoID":7924,"CourseChapterTopicPlaylistID":4250,"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":"In this exercise, we want to expand this function here as a power series,"},{"Start":"00:06.180 ","End":"00:09.580","Text":"and to find the range of convergence."},{"Start":"00:10.280 ","End":"00:13.890","Text":"We\u0027ve done this kind of thing before,"},{"Start":"00:13.890 ","End":"00:17.880","Text":"where we\u0027ve used a familiar series and then taken a variation of it."},{"Start":"00:17.880 ","End":"00:24.810","Text":"The familiar series I\u0027m talking about is the series 1 over 1 minus x."},{"Start":"00:24.810 ","End":"00:31.860","Text":"Most famous power series function."},{"Start":"00:31.860 ","End":"00:34.335","Text":"This is equal to the sum."},{"Start":"00:34.335 ","End":"00:39.120","Text":"It\u0027s from 0 to infinity of just x to the n."},{"Start":"00:39.120 ","End":"00:44.690","Text":"We\u0027re going to use this to help us with this"},{"Start":"00:44.690 ","End":"00:46.670","Text":"because we see it\u0027s got the general shape."},{"Start":"00:46.670 ","End":"00:48.275","Text":"It\u0027s 1 minus something,"},{"Start":"00:48.275 ","End":"00:50.855","Text":"and the constant doesn\u0027t change much."},{"Start":"00:50.855 ","End":"01:01.980","Text":"What we can do in this series is simply to replace x by x to the fourth"},{"Start":"01:01.980 ","End":"01:05.385","Text":"and then we\u0027ll get this function."},{"Start":"01:05.385 ","End":"01:16.180","Text":"About the range, the range for this one is absolute value of x less than 1."},{"Start":"01:16.180 ","End":"01:19.190","Text":"Sometimes, you can write it as minus 1,"},{"Start":"01:19.190 ","End":"01:21.900","Text":"less than x, less than 1, whatever."},{"Start":"01:22.110 ","End":"01:27.575","Text":"What we get is that f of x."},{"Start":"01:27.575 ","End":"01:36.855","Text":"First of all, I can see it as 3 times 1 over 1 minus x to the fourth,"},{"Start":"01:36.855 ","End":"01:41.170","Text":"and now, I can do the substitution here."},{"Start":"01:41.170 ","End":"01:56.010","Text":"This is 3 times the sum from 0 to infinity of this with x to the fourth replacing x."},{"Start":"01:56.010 ","End":"02:02.220","Text":"So I get x to the fourth to the power of n."},{"Start":"02:02.220 ","End":"02:11.505","Text":"Now, all I have to do is do this exponent to simplify it,"},{"Start":"02:11.505 ","End":"02:14.460","Text":"x to the 4 to the n is x to the 4n."},{"Start":"02:14.460 ","End":"02:20.210","Text":"So the answer I get is the 3,"},{"Start":"02:20.210 ","End":"02:24.095","Text":"I can leave in front, or I can stick it inside."},{"Start":"02:24.095 ","End":"02:25.645","Text":"I\u0027ll put it inside."},{"Start":"02:25.645 ","End":"02:28.250","Text":"It\u0027s just a power series with nothing else."},{"Start":"02:28.250 ","End":"02:33.470","Text":"It\u0027s 3 times, and then x to the 4 to the n,"},{"Start":"02:33.470 ","End":"02:36.660","Text":"we said was x to the 4n."},{"Start":"02:36.740 ","End":"02:41.955","Text":"This is the expansion, rewrite f of x."},{"Start":"02:41.955 ","End":"02:45.910","Text":"As for the range,"},{"Start":"02:47.240 ","End":"02:52.670","Text":"we really have x to the fourth less than 1"},{"Start":"02:52.670 ","End":"02:55.790","Text":"because we replaced x with x to the fourth."},{"Start":"02:55.790 ","End":"02:59.720","Text":"But the absolute value of x to the fourth"},{"Start":"02:59.720 ","End":"03:03.510","Text":"is the same as the absolute value of x to the fourth."},{"Start":"03:04.210 ","End":"03:11.570","Text":"This just implies that the absolute value of x is less than the fourth root of 1."},{"Start":"03:11.570 ","End":"03:14.855","Text":"We can see straight away that the answer is going to be 1."},{"Start":"03:14.855 ","End":"03:18.330","Text":"It\u0027s the same, x less than 1."},{"Start":"03:18.330 ","End":"03:20.315","Text":"I\u0027ll just rewrite that here."},{"Start":"03:20.315 ","End":"03:23.140","Text":"The absolute value of x is less than 1."},{"Start":"03:23.140 ","End":"03:25.835","Text":"The power series and the range."},{"Start":"03:25.835 ","End":"03:28.830","Text":"We\u0027ve answered the question, and we\u0027re done."}],"ID":7986},{"Watched":false,"Name":"Exercise 4 part c","Duration":"3m 17s","ChapterTopicVideoID":7925,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this exercise, we want to expand this function as a power series"},{"Start":"00:04.830 ","End":"00:07.050","Text":"and to find the range of convergence."},{"Start":"00:07.050 ","End":"00:08.850","Text":"What we\u0027re going to do,"},{"Start":"00:08.850 ","End":"00:12.195","Text":"is start with a familiar power series,"},{"Start":"00:12.195 ","End":"00:15.705","Text":"the most familiar power series that there is,"},{"Start":"00:15.705 ","End":"00:25.995","Text":"which is, and I\u0027ll write it,1 over 1 minus x is the sum from 0 to infinity of x to the n."},{"Start":"00:25.995 ","End":"00:31.590","Text":"I\u0027m going to modify it so that we can get to our f of x as 2 things."},{"Start":"00:31.590 ","End":"00:36.705","Text":"Instead of x, there\u0027s 9x squared and instead of a plus, there\u0027s a minus."},{"Start":"00:36.705 ","End":"00:40.230","Text":"What I\u0027m going to do, is I\u0027m going to rewrite my f of x."},{"Start":"00:40.230 ","End":"00:42.785","Text":"First of all, take care of the minus."},{"Start":"00:42.785 ","End":"00:51.650","Text":"I have 1 over 1 minus minus 9x squared."},{"Start":"00:51.650 ","End":"00:55.554","Text":"Now, if I take my,"},{"Start":"00:55.554 ","End":"00:58.204","Text":"I\u0027ll call it the template series,"},{"Start":"00:58.204 ","End":"01:02.150","Text":"the model series, the most famous one,"},{"Start":"01:02.150 ","End":"01:10.380","Text":"and replace x by minus 9x squared then I can get to f of x."},{"Start":"01:10.380 ","End":"01:13.485","Text":"I forgot to mention there\u0027s the range."},{"Start":"01:13.485 ","End":"01:19.460","Text":"For this series, the convergence is absolute value of x less than 1."},{"Start":"01:19.460 ","End":"01:22.430","Text":"Sometimes you can write this as x between minus 1 and 1,"},{"Start":"01:22.430 ","End":"01:24.330","Text":"I mean if you want to."},{"Start":"01:26.810 ","End":"01:29.975","Text":"If I do this substitution,"},{"Start":"01:29.975 ","End":"01:35.255","Text":"the sum from 0 to infinity, instead of x,"},{"Start":"01:35.255 ","End":"01:41.150","Text":"I\u0027ll write minus 9x squared to the n."},{"Start":"01:41.150 ","End":"01:44.960","Text":"I\u0027ll return later to the matter of the range."},{"Start":"01:44.960 ","End":"01:47.495","Text":"Let\u0027s just simplify this."},{"Start":"01:47.495 ","End":"01:49.970","Text":"This is equal to the sum."},{"Start":"01:49.970 ","End":"01:56.395","Text":"N goes from 0 to infinity minus 9."},{"Start":"01:56.395 ","End":"01:59.870","Text":"I can write as minus 1 times 9."},{"Start":"01:59.870 ","End":"02:03.360","Text":"I prefer to separate it and write it minus 1 to the n,"},{"Start":"02:03.460 ","End":"02:16.450","Text":"and then 9 to the power n and x squared to the power of n is x to the 2n."},{"Start":"02:16.450 ","End":"02:18.395","Text":"As for the range,"},{"Start":"02:18.395 ","End":"02:22.605","Text":"also substitute instead of x minus 9x squared here,"},{"Start":"02:22.605 ","End":"02:29.515","Text":"so the range would be minus 9x squared in absolute value is less than 1."},{"Start":"02:29.515 ","End":"02:36.730","Text":"This is the same as absolute value of minus 9 is just 9."},{"Start":"02:36.730 ","End":"02:44.375","Text":"Then absolute value of x squared is less than 1."},{"Start":"02:44.375 ","End":"02:47.920","Text":"Now, I\u0027m going to divide both sides by 9,"},{"Start":"02:47.920 ","End":"02:50.185","Text":"so I\u0027ll get here 1/9,"},{"Start":"02:50.185 ","End":"02:56.910","Text":"and also I can write this as absolute value of x squared."},{"Start":"02:56.910 ","End":"03:00.230","Text":"I can take the square root of both sides now"},{"Start":"03:00.230 ","End":"03:05.380","Text":"and get absolute value of x is less than 1/3."},{"Start":"03:05.380 ","End":"03:10.335","Text":"Optionally, you could write this as minus 1/3 less than x,"},{"Start":"03:10.335 ","End":"03:12.735","Text":"less than 1/3, whatever."},{"Start":"03:12.735 ","End":"03:14.990","Text":"The answer is in this last line,"},{"Start":"03:14.990 ","End":"03:17.370","Text":"and we are done."}],"ID":7987},{"Watched":false,"Name":"Exercise 4 part d","Duration":"3m 48s","ChapterTopicVideoID":7926,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we want to expand this function"},{"Start":"00:04.530 ","End":"00:09.285","Text":"as a power series and find the range of convergence."},{"Start":"00:09.285 ","End":"00:12.360","Text":"As in several previous exercises,"},{"Start":"00:12.360 ","End":"00:19.125","Text":"we want to base it on something similar or maybe not so similar."},{"Start":"00:19.125 ","End":"00:26.190","Text":"1 over 1 minus x is a template or model series."},{"Start":"00:26.190 ","End":"00:33.000","Text":"We know that this is equal to the sum from 0 to infinity of x^n."},{"Start":"00:33.000 ","End":"00:38.460","Text":"The question is, how do we get f of x to somehow look like this or be based on this?"},{"Start":"00:38.460 ","End":"00:40.365","Text":"Well, I\u0027ll show you how."},{"Start":"00:40.365 ","End":"00:46.040","Text":"One thing we can do to this 1 over x minus 5 is,"},{"Start":"00:46.040 ","End":"00:51.170","Text":"first of all, rearrange the order because we want the number to come before the x."},{"Start":"00:51.170 ","End":"00:57.410","Text":"So this is equal to minus 1 over 5 minus x."},{"Start":"00:57.410 ","End":"00:59.330","Text":"If I reverse the order of the subtraction,"},{"Start":"00:59.330 ","End":"01:01.234","Text":"I can throw in a minus."},{"Start":"01:01.234 ","End":"01:10.455","Text":"Now, I can take 5 out of the denominator and get minus 1 over 5 times 1 minus,"},{"Start":"01:10.455 ","End":"01:14.610","Text":"and then here I\u0027m left with x over 5."},{"Start":"01:14.610 ","End":"01:21.165","Text":"I can split it up as minus 1/5"},{"Start":"01:21.165 ","End":"01:29.230","Text":"times 1 over 1 minus x over 5."},{"Start":"01:29.780 ","End":"01:33.620","Text":"I should have said also the range of convergence for"},{"Start":"01:33.620 ","End":"01:39.180","Text":"this series is absolute value of x less than 1."},{"Start":"01:39.310 ","End":"01:44.205","Text":"Back to this, what I can say, therefore,"},{"Start":"01:44.205 ","End":"01:53.290","Text":"is that this f"},{"Start":"01:53.290 ","End":"01:55.025","Text":"of x is equal to,"},{"Start":"01:55.025 ","End":"01:59.420","Text":"looking over here, I can write the minus 1/5 and I\u0027ll just"},{"Start":"01:59.420 ","End":"02:05.050","Text":"expand this by looking at this and instead of x,"},{"Start":"02:05.050 ","End":"02:09.570","Text":"putting x over 5."},{"Start":"02:09.570 ","End":"02:17.610","Text":"So we get the sum from 0 to infinity just like this,"},{"Start":"02:17.610 ","End":"02:21.970","Text":"but with x over 5 in place of x."},{"Start":"02:23.090 ","End":"02:27.820","Text":"Now, I\u0027d like to simplify it so there\u0027s no extra bits,"},{"Start":"02:27.820 ","End":"02:30.159","Text":"just the power series."},{"Start":"02:30.159 ","End":"02:36.200","Text":"I can say this is the sum from 0 to infinity."},{"Start":"02:36.200 ","End":"02:42.669","Text":"I want to put the x^n separately here,"},{"Start":"02:42.669 ","End":"02:48.120","Text":"then I\u0027ll get the minus here,"},{"Start":"02:48.120 ","End":"02:56.655","Text":"and I need to put 5 times 5^n which is 5^n plus 1."},{"Start":"02:56.655 ","End":"02:59.010","Text":"I\u0027ll just show you that again."},{"Start":"02:59.010 ","End":"03:07.650","Text":"Minus 1/5 times 1 over 5^n is single minus,"},{"Start":"03:07.650 ","End":"03:11.590","Text":"and 5 times 5^n is 5^n plus 1."},{"Start":"03:12.260 ","End":"03:18.390","Text":"This would be our answer for the power series."},{"Start":"03:18.390 ","End":"03:21.055","Text":"You just have to relate to the range of convergence."},{"Start":"03:21.055 ","End":"03:27.475","Text":"This converges for x over 5 less than 1."},{"Start":"03:27.475 ","End":"03:29.630","Text":"If I multiply by 5,"},{"Start":"03:29.630 ","End":"03:31.980","Text":"I can get the range for x,"},{"Start":"03:31.980 ","End":"03:34.880","Text":"it\u0027s x less than 5."},{"Start":"03:34.880 ","End":"03:38.735","Text":"If you prefer, you can write this as minus 5 less than x,"},{"Start":"03:38.735 ","End":"03:41.035","Text":"less than 5, whatever."},{"Start":"03:41.035 ","End":"03:46.040","Text":"This is our expansion and this is the range of convergence,"},{"Start":"03:46.040 ","End":"03:48.810","Text":"and so we are done."}],"ID":7988},{"Watched":false,"Name":"Exercise 4 part e","Duration":"3m 19s","ChapterTopicVideoID":7898,"CourseChapterTopicPlaylistID":4250,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here we want to expand this function as a power series"},{"Start":"00:03.150 ","End":"00:05.190","Text":"and to find the range of convergence."},{"Start":"00:05.190 ","End":"00:07.665","Text":"As we\u0027ve done several times before,"},{"Start":"00:07.665 ","End":"00:13.530","Text":"we want to base it on the power series for 1 over 1 minus x,"},{"Start":"00:13.530 ","End":"00:15.825","Text":"which you probably remember,"},{"Start":"00:15.825 ","End":"00:20.730","Text":"is equal to sum from 0 to infinity of x to the n."},{"Start":"00:20.730 ","End":"00:26.250","Text":"What we\u0027ll do is modify our f of x to get it to look somewhat like this."},{"Start":"00:26.250 ","End":"00:30.555","Text":"First thing I can do is take x aside."},{"Start":"00:30.555 ","End":"00:37.965","Text":"I\u0027ll write this just in the opposite order,"},{"Start":"00:37.965 ","End":"00:41.855","Text":"I wanted to keep getting closer and closer to that."},{"Start":"00:41.855 ","End":"00:46.585","Text":"That\u0027s a bit better but I need a minus here,"},{"Start":"00:46.585 ","End":"00:56.130","Text":"so I\u0027ll write it as x times 1 over 1 minus, minus 4x."},{"Start":"00:56.130 ","End":"01:05.950","Text":"Now if I take the x here and replace it by minus 4x,"},{"Start":"01:07.070 ","End":"01:10.750","Text":"then I can expand this."},{"Start":"01:10.750 ","End":"01:15.200","Text":"I forgot to mention that this converges,"},{"Start":"01:15.200 ","End":"01:20.135","Text":"when absolute value of x is less than 1."},{"Start":"01:20.135 ","End":"01:25.520","Text":"We\u0027ll see what that means as far as this function."},{"Start":"01:25.520 ","End":"01:28.535","Text":"Let\u0027s first of all work on the expansion."},{"Start":"01:28.535 ","End":"01:33.170","Text":"This is equal to x times."},{"Start":"01:33.170 ","End":"01:42.695","Text":"Now, here I\u0027ll take the sum from 0 to infinity of this but replacing x with minus 4x,"},{"Start":"01:42.695 ","End":"01:48.725","Text":"so I\u0027ve got minus 4x instead of x to the power n,"},{"Start":"01:48.725 ","End":"02:04.075","Text":"and this is equal to x times the sum 0 to infinity of minus 4 to the n."},{"Start":"02:04.075 ","End":"02:06.800","Text":"I\u0027d rather write it as minus 1 to the n"},{"Start":"02:06.800 ","End":"02:13.670","Text":"and then the 4 to the n separately, and then x to the n."},{"Start":"02:13.670 ","End":"02:16.400","Text":"Finally I\u0027ll throw this x inside,"},{"Start":"02:16.400 ","End":"02:24.950","Text":"and so I get the sum from 0 to infinity minus 1 to the n, 4 to the n"},{"Start":"02:24.950 ","End":"02:27.755","Text":"and the x I can combine with x to the n,"},{"Start":"02:27.755 ","End":"02:32.580","Text":"and I\u0027ll get x to the n plus 1."},{"Start":"02:32.840 ","End":"02:37.595","Text":"That\u0027s the expansion of f of x as a power series."},{"Start":"02:37.595 ","End":"02:40.100","Text":"What about the range of convergence?"},{"Start":"02:40.100 ","End":"02:43.100","Text":"Well, we replaced x by minus 4x."},{"Start":"02:43.100 ","End":"02:50.695","Text":"For this, we have absolute value of minus 4x is less than 1,"},{"Start":"02:50.695 ","End":"02:55.035","Text":"and absolute value of minus 4 is 4,"},{"Start":"02:55.035 ","End":"02:58.715","Text":"so I\u0027ve got 4 absolute value of x less than 1,"},{"Start":"02:58.715 ","End":"03:04.570","Text":"and then dividing by 4, absolute value of x is less than 1/4."},{"Start":"03:04.570 ","End":"03:11.710","Text":"If you prefer, you could write this as minus 1/4 less than x, less than 1/4."},{"Start":"03:11.710 ","End":"03:13.895","Text":"This is the power series,"},{"Start":"03:13.895 ","End":"03:16.655","Text":"and this is its range of convergence,"},{"Start":"03:16.655 ","End":"03:20.010","Text":"and so we are done."}],"ID":7989}],"Thumbnail":null,"ID":4250}]

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