Taylor and Maclaurin Series
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Basic Exercises with Maclaurin Series
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Expansions about General Point
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Finding Nonzero Terms in Expansions
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Sum of Series Using Taylor and Maclaurin Expansions
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Finding Limits Using Expansions
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Computations with Taylor Series
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- Computations with Taylor Series and a given precision
- Exercise 1 Part a
- Exercise 1 Part b
- Exercise 1 Part c
- Exercise 2 Part a
- Exercise 2 Part b
- Exercise 2 Part c
- Exercise 3 Part a
- Exercise 3 Part b
- Exercise 3 Part c
- Exercise 4 Part a
- Exercise 4 Part b
- Exercise 5 Part a
- Exercise 5 Part b
- Exercise 5 Part c

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[{"Name":"Taylor and Maclaurin Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Sigma notation for summation","Duration":"6m 55s","ChapterTopicVideoID":10115,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10115.jpeg","UploadDate":"2020-01-16T13:03:31.9970000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In this clip, I\u0027m going to introduce"},{"Start":"00:03.045 ","End":"00:08.055","Text":"the Sigma notation for summation in case you haven\u0027t seen it already."},{"Start":"00:08.055 ","End":"00:11.475","Text":"This is the Greek letter Sigma."},{"Start":"00:11.475 ","End":"00:13.095","Text":"It\u0027s capital Sigma."},{"Start":"00:13.095 ","End":"00:18.790","Text":"Actually, there\u0027s also a small sigma used in statistics, for example."},{"Start":"00:18.800 ","End":"00:21.105","Text":"What is this thing?"},{"Start":"00:21.105 ","End":"00:22.470","Text":"I\u0027ll give you an example."},{"Start":"00:22.470 ","End":"00:27.000","Text":"Suppose you have a sum of several terms of the form,"},{"Start":"00:27.000 ","End":"00:31.590","Text":"1 squared plus 2 squared plus 3 squared plus"},{"Start":"00:31.590 ","End":"00:36.760","Text":"4 squared plus 5 squared plus 6 squared plus 7 squared,"},{"Start":"00:36.760 ","End":"00:39.735","Text":"and I\u0027ll stop at 7, but it could have even been longer."},{"Start":"00:39.735 ","End":"00:45.830","Text":"Now, this kind of a sum is quite tiresome to write out in full."},{"Start":"00:45.830 ","End":"00:48.890","Text":"We want a shorthand way of writing this,"},{"Start":"00:48.890 ","End":"00:54.035","Text":"and if you notice all of them are of the form n squared,"},{"Start":"00:54.035 ","End":"00:58.145","Text":"where n is some number,"},{"Start":"00:58.145 ","End":"01:00.605","Text":"1, 2, 3, 4, 5, 6, or 7."},{"Start":"01:00.605 ","End":"01:02.840","Text":"In fact, they\u0027re even in sequence."},{"Start":"01:02.840 ","End":"01:05.540","Text":"The way we write this is a convention."},{"Start":"01:05.540 ","End":"01:07.255","Text":"If we write it as Sigma,"},{"Start":"01:07.255 ","End":"01:08.810","Text":"and we say the sum,"},{"Start":"01:08.810 ","End":"01:11.655","Text":"the sum of n squared,"},{"Start":"01:11.655 ","End":"01:16.190","Text":"and here and here we write where n goes from and to,"},{"Start":"01:16.190 ","End":"01:20.840","Text":"from n equals 1 to n equals 7."},{"Start":"01:20.840 ","End":"01:23.330","Text":"But we don\u0027t write the n equals at the top."},{"Start":"01:23.330 ","End":"01:25.910","Text":"N goes from 1 to 7,"},{"Start":"01:25.910 ","End":"01:27.695","Text":"the sum of n squared,"},{"Start":"01:27.695 ","End":"01:29.840","Text":"which means you let n equals 1,"},{"Start":"01:29.840 ","End":"01:32.820","Text":"n equals 2, and so on."},{"Start":"01:33.560 ","End":"01:42.365","Text":"Now another example, 1 cubed plus 2 cubed plus 3 cubed."},{"Start":"01:42.365 ","End":"01:44.675","Text":"I think you\u0027ve probably got the idea."},{"Start":"01:44.675 ","End":"01:49.340","Text":"The general pattern is n cubed and n goes from 1 to 3."},{"Start":"01:49.340 ","End":"01:57.165","Text":"We write this as Sigma n equals 1 to 3 of n cubed."},{"Start":"01:57.165 ","End":"02:00.155","Text":"Now let\u0027s try a reverse example."},{"Start":"02:00.155 ","End":"02:05.485","Text":"We\u0027ll start with the Sigma n equals 4"},{"Start":"02:05.485 ","End":"02:13.170","Text":"to 8 of 2n."},{"Start":"02:13.170 ","End":"02:16.570","Text":"What we have to do is substitute n equals 4,"},{"Start":"02:16.570 ","End":"02:17.930","Text":"then 5, then 6, then 7,"},{"Start":"02:17.930 ","End":"02:19.990","Text":"then 8 in this expression."},{"Start":"02:19.990 ","End":"02:23.450","Text":"For each of these, we put an addition sign between"},{"Start":"02:23.450 ","End":"02:27.215","Text":"what I\u0027m saying is it\u0027s 2 times 4, where n equals 4,"},{"Start":"02:27.215 ","End":"02:29.620","Text":"then I let n equals 5,"},{"Start":"02:29.620 ","End":"02:33.155","Text":"all along I\u0027m adding the Sigma is the sum,"},{"Start":"02:33.155 ","End":"02:35.105","Text":"and then 2 times 6,"},{"Start":"02:35.105 ","End":"02:38.795","Text":"2 times 7, and 2 times 8."},{"Start":"02:38.795 ","End":"02:40.520","Text":"There also is a numerical answer,"},{"Start":"02:40.520 ","End":"02:43.760","Text":"but I\u0027m not bothered to actually compute it."},{"Start":"02:43.760 ","End":"02:50.040","Text":"Another example, the sum from n equals"},{"Start":"02:50.040 ","End":"02:59.430","Text":"2 to 5 of minus 1 to the power of n times n plus 1."},{"Start":"02:59.430 ","End":"03:02.395","Text":"Well, let\u0027s see what this equals,"},{"Start":"03:02.395 ","End":"03:10.540","Text":"I first let n equals 2 minus 1 squared is 1 and 2 plus 1 is 3,"},{"Start":"03:10.540 ","End":"03:12.830","Text":"so this becomes 3."},{"Start":"03:12.830 ","End":"03:19.085","Text":"Next term, n equals 3 minus 1 to the power of 3 is minus 1,"},{"Start":"03:19.085 ","End":"03:20.420","Text":"and here it\u0027s 4."},{"Start":"03:20.420 ","End":"03:22.745","Text":"So we get minus 4."},{"Start":"03:22.745 ","End":"03:25.280","Text":"Then when n is 4, again,"},{"Start":"03:25.280 ","End":"03:28.325","Text":"we get plus here and here we get 5."},{"Start":"03:28.325 ","End":"03:31.770","Text":"When n is 5,"},{"Start":"03:31.770 ","End":"03:34.835","Text":"we get minus 6."},{"Start":"03:34.835 ","End":"03:37.660","Text":"The thing that makes this go plus, minus, plus,"},{"Start":"03:37.660 ","End":"03:40.800","Text":"minus is the minus 1_n,"},{"Start":"03:40.800 ","End":"03:42.480","Text":"and we\u0027ll see this a lot."},{"Start":"03:42.480 ","End":"03:46.610","Text":"In fact, this kind of a thing is called an alternating series."},{"Start":"03:46.610 ","End":"03:48.880","Text":"In case you don\u0027t know what a series is,"},{"Start":"03:48.880 ","End":"03:50.830","Text":"each of these things is a series."},{"Start":"03:50.830 ","End":"03:52.705","Text":"It\u0027s a bunch of numbers,"},{"Start":"03:52.705 ","End":"03:55.120","Text":"quantity with pluses in the middle."},{"Start":"03:55.120 ","End":"03:57.520","Text":"If it\u0027s commas, it\u0027s a sequence,"},{"Start":"03:57.520 ","End":"03:59.680","Text":"and if it\u0027s pluses, it\u0027s a series."},{"Start":"03:59.680 ","End":"04:02.410","Text":"So this is a series, this is a series, this is a series,"},{"Start":"04:02.410 ","End":"04:05.350","Text":"this is a series and this 1 is alternating."},{"Start":"04:05.350 ","End":"04:09.725","Text":"Like you to note that a series is allowed to be infinite."},{"Start":"04:09.725 ","End":"04:16.460","Text":"For example, we could have the sum from n equals"},{"Start":"04:16.460 ","End":"04:23.960","Text":"1 to infinity of 1 over n. What this is equal to,"},{"Start":"04:23.960 ","End":"04:28.040","Text":"if n equals 1, then it\u0027s 1 over 1."},{"Start":"04:28.040 ","End":"04:31.715","Text":"If n equals 2, it\u0027s 1 over 2."},{"Start":"04:31.715 ","End":"04:37.035","Text":"If n is 3, it\u0027s 1 over 3,"},{"Start":"04:37.035 ","End":"04:41.830","Text":"1 over 4, and so on."},{"Start":"04:41.960 ","End":"04:44.930","Text":"Sometimes I put the general term,"},{"Start":"04:44.930 ","End":"04:48.995","Text":"1 over n dot, dot, dot."},{"Start":"04:48.995 ","End":"04:54.215","Text":"Now let\u0027s look at 1 final example."},{"Start":"04:54.215 ","End":"05:02.720","Text":"There is a mathematical theorem that says that e to the power of x is equal to"},{"Start":"05:02.720 ","End":"05:07.520","Text":"the sum n goes from 0 to"},{"Start":"05:07.520 ","End":"05:15.815","Text":"infinity of x to the power of n over n factorial."},{"Start":"05:15.815 ","End":"05:20.505","Text":"Let\u0027s expand this and see what this is."},{"Start":"05:20.505 ","End":"05:24.845","Text":"This will give us what we call the infinite series for e to the x."},{"Start":"05:24.845 ","End":"05:29.860","Text":"First n equals 0, x_0 is 1,"},{"Start":"05:29.860 ","End":"05:34.950","Text":"and 0 factorial by convention is 1 also,"},{"Start":"05:34.950 ","End":"05:37.230","Text":"this gives us 1."},{"Start":"05:37.230 ","End":"05:39.330","Text":"When n equals 1,"},{"Start":"05:39.330 ","End":"05:41.175","Text":"this gives us x here,"},{"Start":"05:41.175 ","End":"05:45.939","Text":"1 factorial is 1. So this is x."},{"Start":"05:46.620 ","End":"05:48.940","Text":"Then if n equals 2,"},{"Start":"05:48.940 ","End":"05:54.189","Text":"we get x squared over 2 factorial,"},{"Start":"05:54.189 ","End":"05:55.659","Text":"and I\u0027ll leave it as 2 factorial,"},{"Start":"05:55.659 ","End":"05:57.955","Text":"even though I could compute that,"},{"Start":"05:57.955 ","End":"06:04.175","Text":"and you get the idea x to the power of 3 over 3 factorial,"},{"Start":"06:04.175 ","End":"06:07.140","Text":"and so on and so on."},{"Start":"06:07.140 ","End":"06:12.009","Text":"The general term is x_n over n factorial."},{"Start":"06:12.009 ","End":"06:13.555","Text":"I\u0027ll just copy that from here."},{"Start":"06:13.555 ","End":"06:15.845","Text":"But it goes on forever."},{"Start":"06:15.845 ","End":"06:22.749","Text":"Some people even write e_x in this form instead of the short Sigma notation,"},{"Start":"06:22.749 ","End":"06:26.630","Text":"and this is somewhat cumbersome and tedious."},{"Start":"06:26.630 ","End":"06:33.970","Text":"For example, if you wanted e_4x plus a 1 plus 4 x plus 4 x all squared and so on."},{"Start":"06:33.970 ","End":"06:37.355","Text":"The next notation does have its advantages."},{"Start":"06:37.355 ","End":"06:41.270","Text":"But in general, we\u0027ll be using the Sigma notation,"},{"Start":"06:41.270 ","End":"06:46.610","Text":"which is widely accepted in mathematics everywhere in the world,"},{"Start":"06:46.610 ","End":"06:51.930","Text":"and now you know what it is."},{"Start":"06:51.930 ","End":"06:55.780","Text":"Done with this introduction."}],"ID":10337},{"Watched":false,"Name":"Taylor Series - Informally","Duration":"12m 50s","ChapterTopicVideoID":10116,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10116.jpeg","UploadDate":"2020-01-16T13:04:53.7500000","DurationForVideoObject":"PT12M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this clip, I\u0027m going to very informally talk about Taylor Series."},{"Start":"00:05.190 ","End":"00:07.650","Text":"In the manner of speaking, you could say they were invented"},{"Start":"00:07.650 ","End":"00:10.200","Text":"to help people on desert islands."},{"Start":"00:10.200 ","End":"00:12.480","Text":"What could I mean by that?"},{"Start":"00:12.480 ","End":"00:15.090","Text":"Well, suppose you are on"},{"Start":"00:15.090 ","End":"00:19.825","Text":"a desert island and the main thing is you don\u0027t have a calculator."},{"Start":"00:19.825 ","End":"00:23.590","Text":"That you could be anywhere without a calculator and can\u0027t get 1,"},{"Start":"00:23.590 ","End":"00:29.225","Text":"and you really have a burning desire to know how much is the square root of 5,"},{"Start":"00:29.225 ","End":"00:33.040","Text":"or how much is the natural log of 4?"},{"Start":"00:33.040 ","End":"00:38.130","Text":"Or yet again, how much is e squared equals?"},{"Start":"00:38.130 ","End":"00:43.650","Text":"Not only this question can be answered using Taylor Series,"},{"Start":"00:43.650 ","End":"00:46.604","Text":"but almost any function,"},{"Start":"00:46.604 ","End":"00:48.000","Text":"if you study trigonometry,"},{"Start":"00:48.000 ","End":"00:55.395","Text":"you might want to know what sine of 4 degrees equals or cosine of 20 degrees equals."},{"Start":"00:55.395 ","End":"00:59.040","Text":"Very likely to happen in your life that you\u0027d be stuck on"},{"Start":"00:59.040 ","End":"01:02.490","Text":"a desert island and really want to know what cosine 20 is."},{"Start":"01:02.490 ","End":"01:06.645","Text":"This is where Taylor Series will come to our aid."},{"Start":"01:06.645 ","End":"01:10.335","Text":"Before I tell you and show you what they actually are."},{"Start":"01:10.335 ","End":"01:13.635","Text":"Let\u0027s just look at this in a slightly different light,"},{"Start":"01:13.635 ","End":"01:16.155","Text":"if I want to know the square root of 5."},{"Start":"01:16.155 ","End":"01:22.260","Text":"It\u0027s like I have a function, f of x equal square root of x,"},{"Start":"01:22.260 ","End":"01:27.315","Text":"and I want to know what the value of the function is at any particular point."},{"Start":"01:27.315 ","End":"01:28.590","Text":"For example, in this case,"},{"Start":"01:28.590 ","End":"01:32.130","Text":"I\u0027m asking what is f of 5?"},{"Start":"01:32.130 ","End":"01:34.890","Text":"Sometimes I know, for example,"},{"Start":"01:34.890 ","End":"01:37.365","Text":"if someone said to me, What is f of 4?"},{"Start":"01:37.365 ","End":"01:39.600","Text":"F of 4 is the square root of 4 is 2."},{"Start":"01:39.600 ","End":"01:42.420","Text":"That\u0027s okay, and so on."},{"Start":"01:42.420 ","End":"01:44.370","Text":"For example, natural log of 4,"},{"Start":"01:44.370 ","End":"01:47.130","Text":"it\u0027s as if I\u0027ve got the function natural log of x,"},{"Start":"01:47.130 ","End":"01:50.200","Text":"but I don\u0027t know how to substitute values."},{"Start":"01:51.710 ","End":"01:58.020","Text":"I want to know what is f of 4 equal to?"},{"Start":"01:58.020 ","End":"02:01.935","Text":"If someone said to me, \"What is f of 1?\""},{"Start":"02:01.935 ","End":"02:05.200","Text":"I could know f of 1 is 0."},{"Start":"02:05.270 ","End":"02:07.620","Text":"But what is f of 4?"},{"Start":"02:07.620 ","End":"02:09.960","Text":"Well, the natural log of 4, I don\u0027t know."},{"Start":"02:09.960 ","End":"02:15.060","Text":"An e squared, I look at a function f of x equals e to"},{"Start":"02:15.060 ","End":"02:21.120","Text":"the x. I don\u0027t really know what is e squared."},{"Start":"02:21.120 ","End":"02:23.100","Text":"I know e to the 0 is 1,"},{"Start":"02:23.100 ","End":"02:28.560","Text":"but what is f of 2 equal to?"},{"Start":"02:28.560 ","End":"02:32.820","Text":"I want to look at it that way as not just the square root of 5,"},{"Start":"02:32.820 ","End":"02:35.295","Text":"but the square root of any value,"},{"Start":"02:35.295 ","End":"02:38.025","Text":"5 is just an example."},{"Start":"02:38.025 ","End":"02:41.250","Text":"Square root of 10, square root of 17,"},{"Start":"02:41.250 ","End":"02:43.140","Text":"square root of 105,"},{"Start":"02:43.140 ","End":"02:45.645","Text":"and so on, the natural log of anything."},{"Start":"02:45.645 ","End":"02:52.170","Text":"We agree that these 3 functions are not so easy to compute in general,"},{"Start":"02:52.170 ","End":"02:54.405","Text":"there are certain values that we couldn\u0027t know,"},{"Start":"02:54.405 ","End":"02:56.805","Text":"but in general, they\u0027re not easy."},{"Start":"02:56.805 ","End":"03:01.935","Text":"The question is which functions are easy to compute and are also very flexible?"},{"Start":"03:01.935 ","End":"03:05.055","Text":"Probably the answer is polynomials."},{"Start":"03:05.055 ","End":"03:09.390","Text":"Polynomials which are of the form f of x equals,"},{"Start":"03:09.390 ","End":"03:11.190","Text":"or depends on what degree."},{"Start":"03:11.190 ","End":"03:13.740","Text":"If it\u0027s constant, it\u0027s just a,"},{"Start":"03:13.740 ","End":"03:15.690","Text":"if it\u0027s linear, it\u0027s got to be x,"},{"Start":"03:15.690 ","End":"03:16.709","Text":"if it\u0027s a quadratic,"},{"Start":"03:16.709 ","End":"03:18.105","Text":"it has cx squared."},{"Start":"03:18.105 ","End":"03:21.810","Text":"Well, a certain number of these terms,"},{"Start":"03:21.810 ","End":"03:23.460","Text":"and then it stops somewhere,"},{"Start":"03:23.460 ","End":"03:27.060","Text":"not infinite. That\u0027s a polynomial."},{"Start":"03:27.060 ","End":"03:30.975","Text":"The polynomial is very easy to substitute into."},{"Start":"03:30.975 ","End":"03:40.110","Text":"For example, if I have f of x equals 1 plus 2x plus x squared."},{"Start":"03:40.110 ","End":"03:42.840","Text":"Say we want to substitute,"},{"Start":"03:42.840 ","End":"03:45.405","Text":"I don\u0027t know, x equals 4."},{"Start":"03:45.405 ","End":"03:52.515","Text":"F of 4 would equal 1 plus 2 times 4 plus 4 squared."},{"Start":"03:52.515 ","End":"03:54.945","Text":"Easy to see, this is 25."},{"Start":"03:54.945 ","End":"03:59.010","Text":"Polynomials in general are fairly easy to substitute into."},{"Start":"03:59.010 ","End":"04:06.585","Text":"Taylor\u0027s idea was that if f of x was equal to a polynomial,"},{"Start":"04:06.585 ","End":"04:09.390","Text":"if e to the x, but in general,"},{"Start":"04:09.390 ","End":"04:12.030","Text":"f of x would be equal to a polynomial."},{"Start":"04:12.030 ","End":"04:15.450","Text":"Let\u0027s say that this was equal to this."},{"Start":"04:15.450 ","End":"04:20.115","Text":"Then instead of substituting x in e to the x,"},{"Start":"04:20.115 ","End":"04:22.005","Text":"I could substitute it here,"},{"Start":"04:22.005 ","End":"04:24.480","Text":"and it would be much simpler."},{"Start":"04:24.480 ","End":"04:27.510","Text":"Now, to his dismay,"},{"Start":"04:27.510 ","End":"04:31.665","Text":"he realized that e to the x cannot be a polynomial,"},{"Start":"04:31.665 ","End":"04:33.735","Text":"but all is not lost."},{"Start":"04:33.735 ","End":"04:38.235","Text":"I\u0027ll show you why we can\u0027t expect e to the x to be a polynomial."},{"Start":"04:38.235 ","End":"04:42.600","Text":"There are many reasons why e to the x can\u0027t be a polynomial,"},{"Start":"04:42.600 ","End":"04:44.790","Text":"but let me show you one simple way."},{"Start":"04:44.790 ","End":"04:49.095","Text":"Suppose we had that e to the x was equal to a polynomial."},{"Start":"04:49.095 ","End":"04:50.460","Text":"Let\u0027s say it\u0027s this one,"},{"Start":"04:50.460 ","End":"04:54.930","Text":"or you could take any other polynomial and the same argument would work."},{"Start":"04:54.930 ","End":"04:58.050","Text":"Let\u0027s just say. Well, if 2 functions are equal,"},{"Start":"04:58.050 ","End":"05:00.285","Text":"then their derivatives are equal."},{"Start":"05:00.285 ","End":"05:03.750","Text":"I could differentiate this and get e to the x."},{"Start":"05:03.750 ","End":"05:09.525","Text":"That\u0027s the derivative is equal to 2 plus 2x."},{"Start":"05:09.525 ","End":"05:15.540","Text":"If these are equal, I can differentiate again and get e to the x equals 2."},{"Start":"05:15.540 ","End":"05:20.295","Text":"If these are equal, I can differentiate and get e to the x equals 0."},{"Start":"05:20.295 ","End":"05:25.155","Text":"This would work with any polynomial if you differentiated enough times."},{"Start":"05:25.155 ","End":"05:29.565","Text":"That\u0027s a contradiction, so e to the x cannot be a polynomial."},{"Start":"05:29.565 ","End":"05:31.095","Text":"This idea doesn\u0027t work,"},{"Start":"05:31.095 ","End":"05:33.135","Text":"but then he thought, Okay,"},{"Start":"05:33.135 ","End":"05:37.965","Text":"so f of x is not equal to a polynomial,"},{"Start":"05:37.965 ","End":"05:42.615","Text":"but maybe it\u0027s equal to something else."},{"Start":"05:42.615 ","End":"05:46.200","Text":"Maybe it\u0027s equal to an infinite polynomial,"},{"Start":"05:46.200 ","End":"05:53.850","Text":"like a plus bx plus cx squared plus dx cubed plus ex to the 4th,"},{"Start":"05:53.850 ","End":"05:57.645","Text":"plus, and so on and so on, not ending."},{"Start":"05:57.645 ","End":"05:59.640","Text":"This was the next idea."},{"Start":"05:59.640 ","End":"06:01.770","Text":"This is actually called a Power Series,"},{"Start":"06:01.770 ","End":"06:04.320","Text":"if it doesn\u0027t end, it\u0027s like an infinite polynomial."},{"Start":"06:04.320 ","End":"06:06.840","Text":"This was his later thinking."},{"Start":"06:06.840 ","End":"06:11.310","Text":"We need an infinite number of letters to write this infinite series."},{"Start":"06:11.310 ","End":"06:14.175","Text":"We only have 26 in the English alphabet."},{"Start":"06:14.175 ","End":"06:22.420","Text":"What we do is call this coefficient a_0, a_1, a_2, a_3."},{"Start":"06:22.880 ","End":"06:26.280","Text":"I can write it in Sigma notation."},{"Start":"06:26.280 ","End":"06:32.355","Text":"Sigma when n goes from 0 to infinity,"},{"Start":"06:32.355 ","End":"06:34.860","Text":"the n will be the exponent like here,"},{"Start":"06:34.860 ","End":"06:36.720","Text":"it will be 2 here it\u0027ll be 3."},{"Start":"06:36.720 ","End":"06:44.775","Text":"In general, we\u0027d get a_n times x to the n, and so on."},{"Start":"06:44.775 ","End":"06:47.790","Text":"We can write this as a_n x to the n,"},{"Start":"06:47.790 ","End":"06:49.335","Text":"and this is a_0, a_1,"},{"Start":"06:49.335 ","End":"06:54.345","Text":"a_2 just subscripts, Taylor wanted to apply this to e to the x."},{"Start":"06:54.345 ","End":"07:00.930","Text":"He said, okay, e to the x is not a polynomial but maybe e to the x equals Sigma."},{"Start":"07:00.930 ","End":"07:07.680","Text":"The sum from 0 to infinity of some coefficients times the power of x."},{"Start":"07:07.680 ","End":"07:11.790","Text":"All you\u0027d have to do is give us the formula for the coefficients."},{"Start":"07:11.790 ","End":"07:16.485","Text":"Then we have an infinite polynomial or Power Series for e to the x."},{"Start":"07:16.485 ","End":"07:20.580","Text":"Now how would you go ahead finding these coefficients?"},{"Start":"07:20.580 ","End":"07:23.775","Text":"Taylor tried."},{"Start":"07:23.775 ","End":"07:28.950","Text":"Let\u0027s see what would happens if we take a finite polynomial and regular polynomial."},{"Start":"07:28.950 ","End":"07:37.575","Text":"Let\u0027s say we take from here a plus bx plus cx squared plus dx cubed and stop there."},{"Start":"07:37.575 ","End":"07:41.550","Text":"Same technique will work later with the infinite polynomial."},{"Start":"07:41.550 ","End":"07:43.770","Text":"He said as follows,"},{"Start":"07:43.770 ","End":"07:47.520","Text":"\"If this is equal to this and it\u0027s not really inequality,"},{"Start":"07:47.520 ","End":"07:49.890","Text":"it\u0027s actually more of an identity.\""},{"Start":"07:49.890 ","End":"07:52.590","Text":"Often use 3 lines here,"},{"Start":"07:52.590 ","End":"07:55.950","Text":"which means it\u0027s not just an equation for a specific x."},{"Start":"07:55.950 ","End":"07:57.630","Text":"It means that it\u0027s always true,"},{"Start":"07:57.630 ","End":"08:01.920","Text":"but sometimes we just use the regular equal signs here as well."},{"Start":"08:01.920 ","End":"08:05.310","Text":"If these are equal for all x, in particular,"},{"Start":"08:05.310 ","End":"08:09.315","Text":"they\u0027re going to be equal for x equals 0."},{"Start":"08:09.315 ","End":"08:11.730","Text":"Then let x equals 0,"},{"Start":"08:11.730 ","End":"08:14.280","Text":"I get e to the 0, which is 1,"},{"Start":"08:14.280 ","End":"08:19.920","Text":"is equal to, and all these powers of x are going to be 0 if x is 0,"},{"Start":"08:19.920 ","End":"08:21.720","Text":"so 1 equals a."},{"Start":"08:21.720 ","End":"08:24.420","Text":"We found what a is."},{"Start":"08:24.420 ","End":"08:28.080","Text":"Then if we differentiate this, that\u0027s the next step,"},{"Start":"08:28.080 ","End":"08:32.670","Text":"e to the x is also equal to b plus"},{"Start":"08:32.670 ","End":"08:40.920","Text":"2cx plus 3dx squared."},{"Start":"08:40.920 ","End":"08:43.799","Text":"Now if we let x equals 0,"},{"Start":"08:43.799 ","End":"08:49.425","Text":"we get that b equals 1,"},{"Start":"08:49.425 ","End":"08:52.305","Text":"or 1 equals b we got."},{"Start":"08:52.305 ","End":"08:55.575","Text":"Continuing to differentiate."},{"Start":"08:55.575 ","End":"08:57.030","Text":"Next we get that e to"},{"Start":"08:57.030 ","End":"09:00.780","Text":"the x equals 2c"},{"Start":"09:00.780 ","End":"09:11.950","Text":"plus 6dx."},{"Start":"09:14.810 ","End":"09:18.690","Text":"Letting x equals 0,"},{"Start":"09:18.690 ","End":"09:24.090","Text":"we get 1 equals to c. Again"},{"Start":"09:24.090 ","End":"09:33.150","Text":"differentiate e to the x equals 6d."},{"Start":"09:33.150 ","End":"09:35.070","Text":"Letting x equals 0 again,"},{"Start":"09:35.070 ","End":"09:39.690","Text":"we get that 1 is equal to 6d."},{"Start":"09:39.690 ","End":"09:42.520","Text":"If we tried another one."},{"Start":"09:42.560 ","End":"09:46.770","Text":"See that a equals 1 from here,"},{"Start":"09:46.770 ","End":"09:48.660","Text":"I see that b equals 1."},{"Start":"09:48.660 ","End":"09:51.765","Text":"From here, c equals 1-half,"},{"Start":"09:51.765 ","End":"09:58.875","Text":"d equals 1-6th, and e equals 1-24th."},{"Start":"09:58.875 ","End":"10:03.570","Text":"That means that I can write e to the x."},{"Start":"10:03.570 ","End":"10:05.460","Text":"If it were a finite polynomial,"},{"Start":"10:05.460 ","End":"10:09.240","Text":"I would then say that e to the ax is equal to 1."},{"Start":"10:09.240 ","End":"10:11.505","Text":"That\u0027s the a plus bx,"},{"Start":"10:11.505 ","End":"10:16.650","Text":"that\u0027s just x plus 1-half x squared plus"},{"Start":"10:16.650 ","End":"10:24.180","Text":"1-6th x cubed plus 1 over 24 x to the 4th."},{"Start":"10:24.180 ","End":"10:28.380","Text":"If for example, I wanted to know what e was,"},{"Start":"10:28.380 ","End":"10:39.040","Text":"e to the power of 1 is just e. I would put x equals 1 and say that e equals 1 plus 1."},{"Start":"10:39.040 ","End":"10:41.075","Text":"All these powers would be one."},{"Start":"10:41.075 ","End":"10:48.390","Text":"I get plus a half plus a sixth plus 1 over 24."},{"Start":"10:49.180 ","End":"10:55.410","Text":"Now what we could see is that 1 is not a very close approximation to e,"},{"Start":"10:55.410 ","End":"10:57.930","Text":"is not equal to e, but if I take 2 terms,"},{"Start":"10:57.930 ","End":"11:01.560","Text":"1 plus 1 is 2, it\u0027s getting closer to e, e is,"},{"Start":"11:01.560 ","End":"11:06.990","Text":"as you remember is approximately 2.718 something."},{"Start":"11:06.990 ","End":"11:11.055","Text":"Here we have 1 and 1 is 2, not very good."},{"Start":"11:11.055 ","End":"11:14.475","Text":"2.5 is 2.5 is getting closer."},{"Start":"11:14.475 ","End":"11:17.490","Text":"If I add the sixth, I get 2 and 2-thirds,"},{"Start":"11:17.490 ","End":"11:22.590","Text":"which is 2.666 and so on,"},{"Start":"11:22.590 ","End":"11:23.955","Text":"is already getting very good."},{"Start":"11:23.955 ","End":"11:27.285","Text":"If I add the 1-24th, well,"},{"Start":"11:27.285 ","End":"11:31.080","Text":"I really don\u0027t have my calculator with me and perhaps I\u0027m on a desert island."},{"Start":"11:31.080 ","End":"11:33.900","Text":"We could easily do this manually and compute what"},{"Start":"11:33.900 ","End":"11:37.650","Text":"this is and you\u0027d get even closer to 2.718."},{"Start":"11:37.650 ","End":"11:39.840","Text":"This is the idea."},{"Start":"11:39.840 ","End":"11:41.610","Text":"Now the big idea is to,"},{"Start":"11:41.610 ","End":"11:44.325","Text":"in general not to take finite polynomials."},{"Start":"11:44.325 ","End":"11:47.910","Text":"Take infinite polynomials or Power Series,"},{"Start":"11:47.910 ","End":"11:50.880","Text":"where we will get plus something,"},{"Start":"11:50.880 ","End":"11:54.195","Text":"something, something and here plus something, something, something."},{"Start":"11:54.195 ","End":"11:57.435","Text":"Then when we have these infinite series,"},{"Start":"11:57.435 ","End":"12:02.160","Text":"we can approximate all these calculations that were above."},{"Start":"12:02.160 ","End":"12:04.620","Text":"For example, I could get the value of e closer and"},{"Start":"12:04.620 ","End":"12:07.770","Text":"closer and closer as I take successive terms,"},{"Start":"12:07.770 ","End":"12:12.210","Text":"the more terms I take the closer I can get with any accuracy I want,"},{"Start":"12:12.210 ","End":"12:13.950","Text":"I just have to work harder."},{"Start":"12:13.950 ","End":"12:18.780","Text":"This is what he\u0027s going to do for all these functions we had."},{"Start":"12:18.780 ","End":"12:21.645","Text":"Which functions we have beside e to the x,"},{"Start":"12:21.645 ","End":"12:23.460","Text":"natural log of x, square root of x."},{"Start":"12:23.460 ","End":"12:28.575","Text":"Each of these would have its infinite Taylor Series, it\u0027s called."},{"Start":"12:28.575 ","End":"12:36.675","Text":"By substituting x equals 5 or 4 or 2, we would get,"},{"Start":"12:36.675 ","End":"12:39.480","Text":"by taking enough terms in the series,"},{"Start":"12:39.480 ","End":"12:42.990","Text":"we get as close as we want to the actual answer,"},{"Start":"12:42.990 ","End":"12:46.095","Text":"but so far we would just informal."},{"Start":"12:46.095 ","End":"12:47.385","Text":"In the next clip,"},{"Start":"12:47.385 ","End":"12:51.880","Text":"I\u0027ll show you how we get these infinite Taylor Series."}],"ID":10338},{"Watched":false,"Name":"Formal Definition of Taylor Series","Duration":"10m 45s","ChapterTopicVideoID":10117,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10117.jpeg","UploadDate":"2020-01-16T13:04:37.1230000","DurationForVideoObject":"PT10M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"This clip is a continuation of the previous clip on"},{"Start":"00:03.330 ","End":"00:07.260","Text":"Taylor Series where we talked about it informally,"},{"Start":"00:07.260 ","End":"00:10.920","Text":"and now want to get a bit more serious."},{"Start":"00:10.920 ","End":"00:15.615","Text":"The idea is that given a function f of x,"},{"Start":"00:15.615 ","End":"00:18.600","Text":"for example, square root of x,"},{"Start":"00:18.600 ","End":"00:20.925","Text":"e^x, natural log of x,"},{"Start":"00:20.925 ","End":"00:25.575","Text":"I would like to find a power series, an infinite 1,"},{"Start":"00:25.575 ","End":"00:30.600","Text":"a_0 plus a_1x plus a_2 x squared"},{"Start":"00:30.600 ","End":"00:36.210","Text":"plus a_3x cubed plus a_4 x^4,"},{"Start":"00:36.210 ","End":"00:40.145","Text":"plus a_5 x^5, and so on,"},{"Start":"00:40.145 ","End":"00:44.270","Text":"which will equal the original function f of x."},{"Start":"00:44.270 ","End":"00:46.040","Text":"That\u0027s what I\u0027d like."},{"Start":"00:46.040 ","End":"00:51.950","Text":"I deliberately gave a lot of terms here because we\u0027ll be doing some development here,"},{"Start":"00:51.950 ","End":"00:54.890","Text":"and we\u0027ll need to recognize patterns and rules,"},{"Start":"00:54.890 ","End":"00:56.870","Text":"and I think it\u0027ll help if we use a lot of"},{"Start":"00:56.870 ","End":"00:59.015","Text":"terms even though it\u0027ll be a little bit more tedious."},{"Start":"00:59.015 ","End":"01:00.440","Text":"Please bear with me."},{"Start":"01:00.440 ","End":"01:03.095","Text":"Like before, we\u0027re going to find the coefficients."},{"Start":"01:03.095 ","End":"01:06.440","Text":"Because after all, if we know the coefficients we know the whole power series,"},{"Start":"01:06.440 ","End":"01:07.610","Text":"just like with a polynomial,"},{"Start":"01:07.610 ","End":"01:09.455","Text":"all you need to know is its coefficients."},{"Start":"01:09.455 ","End":"01:12.890","Text":"We\u0027re going to do this like in the previous clip by"},{"Start":"01:12.890 ","End":"01:17.300","Text":"constantly differentiating and assigning x to 0 repeatedly."},{"Start":"01:17.300 ","End":"01:19.580","Text":"But I\u0027m going to do a lot of differentiations first,"},{"Start":"01:19.580 ","End":"01:21.590","Text":"and then we\u0027ll do the substitutions."},{"Start":"01:21.590 ","End":"01:25.010","Text":"As I say, please bear with me, it\u0027s slightly tedious."},{"Start":"01:25.010 ","End":"01:35.775","Text":"f prime of x is equal to a_1 plus 2a_2 x"},{"Start":"01:35.775 ","End":"01:39.840","Text":"plus 3a_3 x squared plus"},{"Start":"01:39.840 ","End":"01:50.070","Text":"4a_4 x cubed, plus 5a_5 x^4,"},{"Start":"01:50.070 ","End":"01:52.920","Text":"and so on to infinity."},{"Start":"01:52.920 ","End":"02:01.860","Text":"f double prime of x second derivative will be 2a_2 plus 3 times"},{"Start":"02:01.860 ","End":"02:07.800","Text":"2 times a_3 x plus 4 times"},{"Start":"02:07.800 ","End":"02:17.420","Text":"3a_4 x squared plus 5 times 4 times a_5 x cubed."},{"Start":"02:17.420 ","End":"02:22.745","Text":"Now third derivative, 3 times"},{"Start":"02:22.745 ","End":"02:29.569","Text":"2a_3 plus 4 times 3 times 2a_4 x,"},{"Start":"02:29.569 ","End":"02:37.705","Text":"plus 5 times 4 times 3a_5 x squared, continuing."},{"Start":"02:37.705 ","End":"02:44.445","Text":"4th derivative is 4 times 3 times 2a_4,"},{"Start":"02:44.445 ","End":"02:51.340","Text":"plus 5 times 4 times 3 times 2a_5 x, and so on."},{"Start":"02:53.690 ","End":"02:59.580","Text":"f^5 of x is equal to just"},{"Start":"02:59.580 ","End":"03:07.770","Text":"5 times 4 times 3 times 2a_5, and so on."},{"Start":"03:07.770 ","End":"03:13.170","Text":"Next I\u0027m going to do the substituting, so let\u0027s see."},{"Start":"03:13.170 ","End":"03:19.700","Text":"If I assign x equals 0 here,"},{"Start":"03:19.700 ","End":"03:23.210","Text":"I get that this is equal to just a_0,"},{"Start":"03:23.210 ","End":"03:26.615","Text":"because all the powers of x are 0."},{"Start":"03:26.615 ","End":"03:30.760","Text":"If I substitute 0 in f prime,"},{"Start":"03:30.760 ","End":"03:40.990","Text":"I get just a_1 because all the other terms are 0. f double prime of 0 is 2a_2,"},{"Start":"03:40.990 ","End":"03:48.275","Text":"f triple prime of 0 is 3 times 2 times a_3."},{"Start":"03:48.275 ","End":"03:55.229","Text":"f quadruple prime or 4th derivative of 0,"},{"Start":"03:55.229 ","End":"04:02.654","Text":"if I substitute is 4 times 3 times 2a_4,"},{"Start":"04:02.654 ","End":"04:07.975","Text":"and finally the 5th derivative at 0"},{"Start":"04:07.975 ","End":"04:15.590","Text":"is 5 times 4 times 3 times 2a_5."},{"Start":"04:15.590 ","End":"04:21.275","Text":"Now let\u0027s look at this the other way from the point of view of the coefficients."},{"Start":"04:21.275 ","End":"04:25.895","Text":"So a_0 is f of 0,"},{"Start":"04:25.895 ","End":"04:30.180","Text":"and a_1 is f prime of 0,"},{"Start":"04:30.180 ","End":"04:37.374","Text":"but a_2 is f double prime of 0 over 2,"},{"Start":"04:37.374 ","End":"04:48.950","Text":"and a_3 is f triple prime of 0 over 3 times 2,"},{"Start":"04:48.950 ","End":"05:00.315","Text":"a_4 is f 4th derivative at 0 divided by 4 times 3 times 2,"},{"Start":"05:00.315 ","End":"05:11.810","Text":"and a_5 is the 5th derivative of f at 0 over 5 times 4 times 3 times 2."},{"Start":"05:13.140 ","End":"05:22.240","Text":"Now notice that this denominator here is 5 factorial because it\u0027s always times 1;"},{"Start":"05:22.240 ","End":"05:25.515","Text":"5 times 4 times 3 times 2 times 1, is 5 factorial."},{"Start":"05:25.515 ","End":"05:28.100","Text":"Let me just write that in another color."},{"Start":"05:28.100 ","End":"05:31.500","Text":"This would be the denominator,5 factorial."},{"Start":"05:31.500 ","End":"05:33.480","Text":"Here we have 4 factorial,"},{"Start":"05:33.480 ","End":"05:36.825","Text":"here we have 3 times 2 times 1 is 3 factorial,"},{"Start":"05:36.825 ","End":"05:39.450","Text":"here 2 is 2 factorial,"},{"Start":"05:39.450 ","End":"05:43.875","Text":"and this could be written as if it was over 1 factorial,"},{"Start":"05:43.875 ","End":"05:47.740","Text":"and this could also be written as over 0 factorial which is 1."},{"Start":"05:47.740 ","End":"05:51.465","Text":"Let me just rewrite this there."},{"Start":"05:51.465 ","End":"05:54.965","Text":"It looks like there\u0027s a definite pattern here,"},{"Start":"05:54.965 ","End":"06:00.650","Text":"and that pattern is a_n will be the nth derivative of"},{"Start":"06:00.650 ","End":"06:07.860","Text":"the function at 0 divided by n factorial."},{"Start":"06:07.860 ","End":"06:13.130","Text":"Now we can continue to our definition of the Taylor Series."},{"Start":"06:13.130 ","End":"06:20.580","Text":"Now we\u0027re ready to define the Taylor Series of a function f of x."},{"Start":"06:20.580 ","End":"06:28.400","Text":"We say that the Taylor Series of a function f of x is precisely the a_0 plus"},{"Start":"06:28.400 ","End":"06:36.605","Text":"a_1x plus a_2x squared plus a_3x cubed plus and so on as above,"},{"Start":"06:36.605 ","End":"06:43.010","Text":"where the coefficients are given by this formula which I\u0027ll highlight."},{"Start":"06:43.010 ","End":"06:46.520","Text":"When we take a_n to be this for any n,"},{"Start":"06:46.520 ","End":"06:48.590","Text":"and we take this series,"},{"Start":"06:48.590 ","End":"06:51.960","Text":"this is the Taylor series,"},{"Start":"06:51.960 ","End":"06:55.640","Text":"actually it\u0027s also called the"},{"Start":"06:55.640 ","End":"07:01.220","Text":"I need to go back a bit to explain about Maclaurin."},{"Start":"07:01.220 ","End":"07:05.975","Text":"If you noticed, we\u0027ve been using 0 as a reference point,"},{"Start":"07:05.975 ","End":"07:08.480","Text":"like the series is around 0."},{"Start":"07:08.480 ","End":"07:11.210","Text":"We\u0027ve been taking powers of x,"},{"Start":"07:11.210 ","End":"07:14.735","Text":"and we\u0027ve been substituting 0 each time,"},{"Start":"07:14.735 ","End":"07:17.584","Text":"and this is not the most general."},{"Start":"07:17.584 ","End":"07:20.420","Text":"We can actually center ourselves around a value,"},{"Start":"07:20.420 ","End":"07:27.020","Text":"say of 5, and then we get a different development of the coefficients."},{"Start":"07:27.020 ","End":"07:29.930","Text":"But instead of x^n,"},{"Start":"07:29.930 ","End":"07:35.780","Text":"we\u0027ll be getting x minus 5^n."},{"Start":"07:35.780 ","End":"07:41.765","Text":"Taylor explored the more general case and in the specific case of 0,"},{"Start":"07:41.765 ","End":"07:44.600","Text":"that was Maclaurin specialty."},{"Start":"07:44.600 ","End":"07:46.790","Text":"When you say Maclaurin series,"},{"Start":"07:46.790 ","End":"07:52.940","Text":"we mean the regular power series centered around 0 and Taylor for the more general case,"},{"Start":"07:52.940 ","End":"07:54.770","Text":"but including this case."},{"Start":"07:54.770 ","End":"07:56.930","Text":"That\u0027s a bit of background."},{"Start":"07:56.930 ","End":"07:59.240","Text":"Mostly we\u0027ll be using the Maclaurin series"},{"Start":"07:59.240 ","End":"08:02.405","Text":"actually where we just get regular powers of x."},{"Start":"08:02.405 ","End":"08:04.010","Text":"We know all the coefficients,"},{"Start":"08:04.010 ","End":"08:06.020","Text":"not just up to 3 or 5."},{"Start":"08:06.020 ","End":"08:10.145","Text":"For example, if we want to know what a_10 is, a_10 says,"},{"Start":"08:10.145 ","End":"08:12.800","Text":"take the 10th derivative of the function,"},{"Start":"08:12.800 ","End":"08:17.570","Text":"substitute 0 and divide by 10 factorial, and so on."},{"Start":"08:17.570 ","End":"08:20.930","Text":"Now most books don\u0027t write it this way"},{"Start":"08:20.930 ","End":"08:24.185","Text":"as just a_0 or a_1 and refer you to another formula,"},{"Start":"08:24.185 ","End":"08:26.570","Text":"they just put this formula straight in here,"},{"Start":"08:26.570 ","End":"08:35.715","Text":"and we can say that f of x is equal to a_0 is just f of 0."},{"Start":"08:35.715 ","End":"08:38.639","Text":"Yeah, the 0 factorial is 1, f of 0,"},{"Start":"08:38.639 ","End":"08:44.340","Text":"plus f prime of"},{"Start":"08:44.340 ","End":"08:51.285","Text":"0 over 1 factorial and times x,"},{"Start":"08:51.285 ","End":"09:00.320","Text":"missed that almost, plus f double prime of 0 over 2 factorial times x"},{"Start":"09:00.320 ","End":"09:10.260","Text":"squared plus f triple prime of 0 over 3 factorial times x cubed plus and so on."},{"Start":"09:10.610 ","End":"09:13.575","Text":"Let\u0027s throw in the general term."},{"Start":"09:13.575 ","End":"09:18.945","Text":"What we have is f^n,"},{"Start":"09:18.945 ","End":"09:25.905","Text":"I mean nth derivative at 0 over n factorial."},{"Start":"09:25.905 ","End":"09:28.770","Text":"That\u0027s 1 way of writing it."},{"Start":"09:28.770 ","End":"09:37.015","Text":"We could also just use the Sigma notation and write it as f of x equals the sum,"},{"Start":"09:37.015 ","End":"09:42.725","Text":"n goes from 0 up to infinity of the general term,"},{"Start":"09:42.725 ","End":"09:45.780","Text":"which is this here,"},{"Start":"09:45.780 ","End":"09:50.350","Text":"nth derivative of f at 0 over n factorial,"},{"Start":"09:50.350 ","End":"09:56.000","Text":"that\u0027s the co-efficient and the power is x^n."},{"Start":"09:56.000 ","End":"10:04.505","Text":"This is the Taylor Series but centered around 0 or the Maclaurin series,"},{"Start":"10:04.505 ","End":"10:07.590","Text":"both will be okay."},{"Start":"10:08.000 ","End":"10:11.870","Text":"We\u0027re basically done, we\u0027ve got the formula we\u0027re looking for,"},{"Start":"10:11.870 ","End":"10:15.830","Text":"but we still haven\u0027t done anything practical on the practical side"},{"Start":"10:15.830 ","End":"10:19.880","Text":"of how to compute these coefficients for a given function,"},{"Start":"10:19.880 ","End":"10:21.665","Text":"and that will be in the next clip."},{"Start":"10:21.665 ","End":"10:24.110","Text":"But don\u0027t worry, we won\u0027t have to do all this work"},{"Start":"10:24.110 ","End":"10:27.335","Text":"and differentiate 10 times and all that."},{"Start":"10:27.335 ","End":"10:33.500","Text":"Mostly there are lists of Maclaurin series of well-known functions,"},{"Start":"10:33.500 ","End":"10:40.860","Text":"and we use the existing stock to tweak it and get our particular function."},{"Start":"10:40.860 ","End":"10:42.970","Text":"Anyway, that\u0027ll be for next time,"},{"Start":"10:42.970 ","End":"10:46.160","Text":"and for this we\u0027re done."}],"ID":10339},{"Watched":false,"Name":"Worked Example 1 -Common Maclaurin Series","Duration":"9m 46s","ChapterTopicVideoID":10118,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10118.jpeg","UploadDate":"2020-01-16T13:03:44.5470000","DurationForVideoObject":"PT9M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.055","Text":"I\u0027d like to give an example of how to compute a Maclaurin series."},{"Start":"00:05.055 ","End":"00:08.520","Text":"Specifically, I\u0027d like to use the function f of"},{"Start":"00:08.520 ","End":"00:12.900","Text":"x equals e^x and compute its Maclaurin series,"},{"Start":"00:12.900 ","End":"00:16.169","Text":"which means the Taylor series centered at 0."},{"Start":"00:16.169 ","End":"00:19.530","Text":"What we have to do is to keep"},{"Start":"00:19.530 ","End":"00:24.195","Text":"differentiating the function until we find the recurring pattern,"},{"Start":"00:24.195 ","End":"00:28.230","Text":"until you know what is the 40th derivative, for example."},{"Start":"00:28.230 ","End":"00:32.220","Text":"I\u0027d like to mention that that\u0027s not the only way of doing a Maclaurin Series."},{"Start":"00:32.220 ","End":"00:38.255","Text":"Usually, we in fact don\u0027t keep doing these Derivatives and substitutions."},{"Start":"00:38.255 ","End":"00:41.615","Text":"We have a common stock of common Maclaurin series,"},{"Start":"00:41.615 ","End":"00:46.235","Text":"either the lecture I gave them or they\u0027re given in a formula sheet and we use"},{"Start":"00:46.235 ","End":"00:51.470","Text":"certain Maclaurin series to find other variations, adaptations."},{"Start":"00:51.470 ","End":"00:54.875","Text":"For now, we\u0027re going to do it from scratch."},{"Start":"00:54.875 ","End":"00:57.655","Text":"Let\u0027s start with the derivatives,"},{"Start":"00:57.655 ","End":"01:03.245","Text":"f prime of x will equal derivative of e^x is e^x,"},{"Start":"01:03.245 ","End":"01:06.830","Text":"f double prime of x is e^x."},{"Start":"01:06.830 ","End":"01:11.605","Text":"Differentiate this again and we have e^x."},{"Start":"01:11.605 ","End":"01:14.705","Text":"Is the pattern well-known? I think so."},{"Start":"01:14.705 ","End":"01:20.600","Text":"In general, every derivative is e^x and so if I substitute"},{"Start":"01:20.600 ","End":"01:28.670","Text":"now x plus 0 at each level I get that f of 0 is equal to 1,"},{"Start":"01:28.670 ","End":"01:32.315","Text":"f prime of 0 equals 1,"},{"Start":"01:32.315 ","End":"01:36.695","Text":"f double prime of 0 equals 1,"},{"Start":"01:36.695 ","End":"01:42.755","Text":"and f triple prime of 0 equals 1, and so on."},{"Start":"01:42.755 ","End":"01:46.220","Text":"Now according to the formula,"},{"Start":"01:46.220 ","End":"01:48.230","Text":"just in case you missed it,"},{"Start":"01:48.230 ","End":"01:50.000","Text":"the reason that all these are 1 is"},{"Start":"01:50.000 ","End":"01:54.650","Text":"because e^0 equals 1 and I\u0027m not going to write this every time."},{"Start":"01:54.650 ","End":"01:59.510","Text":"Now the formula, we get that f of x,"},{"Start":"01:59.510 ","End":"02:04.580","Text":"which in our case is e^x,"},{"Start":"02:04.580 ","End":"02:07.385","Text":"is equal to, we get the coefficients."},{"Start":"02:07.385 ","End":"02:17.790","Text":"This is 1 and then 1x and then 1x squared over 2 factorial."},{"Start":"02:17.790 ","End":"02:20.070","Text":"Well, I\u0027m not going to write the 1."},{"Start":"02:20.070 ","End":"02:26.385","Text":"Just write it as x squared over"},{"Start":"02:26.385 ","End":"02:33.960","Text":"2 factorial plus x cubed over 3 factorial plus and so on,"},{"Start":"02:33.960 ","End":"02:42.905","Text":"x^4th over 4 factorial do 1 pole and the general term is x^n over n factorial,"},{"Start":"02:42.905 ","End":"02:49.655","Text":"and so on and this is the Maclaurin series for e^x."},{"Start":"02:49.655 ","End":"02:51.980","Text":"According to the formula,"},{"Start":"02:51.980 ","End":"02:57.230","Text":"we get that f of x is equal"},{"Start":"02:57.230 ","End":"03:06.380","Text":"to f of 0 plus f prime of 0 x,"},{"Start":"03:06.380 ","End":"03:12.530","Text":"plus f double prime of 0 over 2 factorial x squared,"},{"Start":"03:12.530 ","End":"03:21.785","Text":"plus f triple prime of 0 over 3 factorial x cubed, and so on."},{"Start":"03:21.785 ","End":"03:28.160","Text":"In other words, f of x is e^x and"},{"Start":"03:28.160 ","End":"03:34.220","Text":"this is equal to this where I just replaced all the f of 0,"},{"Start":"03:34.220 ","End":"03:38.225","Text":"f prime of 0, all these derivatives at 0 or equal to 1."},{"Start":"03:38.225 ","End":"03:40.295","Text":"All the coefficients are 1,"},{"Start":"03:40.295 ","End":"03:42.200","Text":"which have just left with 1x,"},{"Start":"03:42.200 ","End":"03:44.900","Text":"x squared, x cubed."},{"Start":"03:44.900 ","End":"03:51.775","Text":"They\u0027re 1 over n factorial, of course,"},{"Start":"03:51.775 ","End":"03:56.390","Text":"and this gives us our Maclaurin series for"},{"Start":"03:56.390 ","End":"04:05.480","Text":"e^x and note that the general term is just x^n over n factorial,"},{"Start":"04:05.480 ","End":"04:10.235","Text":"but the coefficient is just 1 over n factorial."},{"Start":"04:10.235 ","End":"04:13.010","Text":"For e^x, it\u0027s probably the easiest function just"},{"Start":"04:13.010 ","End":"04:16.010","Text":"about for computing our Maclaurin series,"},{"Start":"04:16.010 ","End":"04:18.380","Text":"which is why we began with it."},{"Start":"04:18.380 ","End":"04:23.750","Text":"Now a typical exam question could be something like,"},{"Start":"04:23.750 ","End":"04:28.675","Text":"find the Maclaurin series not of e^x,"},{"Start":"04:28.675 ","End":"04:31.390","Text":"but e to the power"},{"Start":"04:31.390 ","End":"04:39.105","Text":"of 10x and maybe x^4th times that."},{"Start":"04:39.105 ","End":"04:41.204","Text":"This would be a good example."},{"Start":"04:41.204 ","End":"04:43.115","Text":"I\u0027m not going to start from scratch."},{"Start":"04:43.115 ","End":"04:44.840","Text":"If you start differentiating,"},{"Start":"04:44.840 ","End":"04:47.780","Text":"if I let this be f of x,"},{"Start":"04:47.780 ","End":"04:53.810","Text":"and I start calculating the derivatives successively using the product rule,"},{"Start":"04:53.810 ","End":"04:58.945","Text":"it will be a bit of a mess and may not be easy to see the pattern."},{"Start":"04:58.945 ","End":"05:01.140","Text":"We\u0027re not going to do that."},{"Start":"05:01.140 ","End":"05:04.165","Text":"Instead, we\u0027re going to build on something we already know."},{"Start":"05:04.165 ","End":"05:05.600","Text":"You will have learnt this,"},{"Start":"05:05.600 ","End":"05:06.980","Text":"a lecturer would have taught it."},{"Start":"05:06.980 ","End":"05:11.375","Text":"So e^x is like a given as far as this Maclaurin series goes,"},{"Start":"05:11.375 ","End":"05:14.570","Text":"we just have to adapt it to this variation."},{"Start":"05:14.570 ","End":"05:21.870","Text":"What we might say was that would be that each of the power of 10x using this is,"},{"Start":"05:21.870 ","End":"05:24.410","Text":"first of all, write the x^4th,"},{"Start":"05:24.410 ","End":"05:27.080","Text":"and then e^10x would be,"},{"Start":"05:27.080 ","End":"05:33.620","Text":"I\u0027m looking at this and replacing x by 10x, 1 plus 10x,"},{"Start":"05:33.620 ","End":"05:38.105","Text":"plus 10x squared over 2 factorial,"},{"Start":"05:38.105 ","End":"05:42.215","Text":"plus 10x cubed over 3 factorial,"},{"Start":"05:42.215 ","End":"05:48.425","Text":"plus 10x^4th over 4 factorial, etc."},{"Start":"05:48.425 ","End":"05:52.640","Text":"Which is equal to x^4th,"},{"Start":"05:52.640 ","End":"06:00.905","Text":"plus 10x times x^4th is 10x^5th plus,"},{"Start":"06:00.905 ","End":"06:04.075","Text":"this is a 100x squared."},{"Start":"06:04.075 ","End":"06:13.650","Text":"Yeah, a 100 x squared times x to the 4th is a 100x^6th,"},{"Start":"06:14.000 ","End":"06:23.495","Text":"plus1000x^7th and here I\u0027ve started to forget the factorials, 2 factorial,"},{"Start":"06:23.495 ","End":"06:27.420","Text":"3 factorial, and then"},{"Start":"06:28.510 ","End":"06:36.200","Text":"10,000x^8th over 4 factorial and so on."},{"Start":"06:36.200 ","End":"06:39.470","Text":"The general term would be,"},{"Start":"06:39.470 ","End":"06:45.940","Text":"let\u0027s see, if here the general term is x^4th,"},{"Start":"06:45.940 ","End":"06:50.720","Text":"10^x to the power of n over n factorial and this will just"},{"Start":"06:50.720 ","End":"06:55.700","Text":"be x to the power of n and then x to the power of 4."},{"Start":"06:55.700 ","End":"07:01.625","Text":"Altogether, x to the power of n plus 4,"},{"Start":"07:01.625 ","End":"07:11.105","Text":"n factorial and 10 to the power of n, and etc."},{"Start":"07:11.105 ","End":"07:14.330","Text":"Now, this whole thing is much easier with"},{"Start":"07:14.330 ","End":"07:18.595","Text":"the Sigma notation which I would recommend using."},{"Start":"07:18.595 ","End":"07:23.060","Text":"Let\u0027s start this from scratch this time with the Sigma method."},{"Start":"07:23.060 ","End":"07:28.775","Text":"What we have to start with is this e^x series and I write this"},{"Start":"07:28.775 ","End":"07:35.930","Text":"as e^x is Sigma n equals 0 to infinity of this series,"},{"Start":"07:35.930 ","End":"07:43.880","Text":"which is x^n over n factorial and then I say, okay,"},{"Start":"07:43.880 ","End":"07:49.445","Text":"so x^4th, e to the power of"},{"Start":"07:49.445 ","End":"07:56.110","Text":"10x is equal to its x^4th,"},{"Start":"07:56.110 ","End":"08:01.050","Text":"I\u0027ll put it outside the Sigma and n equals 0"},{"Start":"08:01.050 ","End":"08:06.165","Text":"to infinity and here I put 10x to the power of"},{"Start":"08:06.165 ","End":"08:11.670","Text":"n over n factorial and this is"},{"Start":"08:11.670 ","End":"08:17.445","Text":"equal to the Sigma n equals 0 to infinity."},{"Start":"08:17.445 ","End":"08:23.910","Text":"Now 10x to the power of n is 10 to"},{"Start":"08:23.910 ","End":"08:31.025","Text":"the power of n times x^n over n factorial."},{"Start":"08:31.025 ","End":"08:35.074","Text":"But I have to take care of this x^4th also,"},{"Start":"08:35.074 ","End":"08:42.065","Text":"x^4th is a constant in the sense of n. The summation goes on n as far as n is concerned,"},{"Start":"08:42.065 ","End":"08:44.855","Text":"x^4th is a constant, so I can put it inside."},{"Start":"08:44.855 ","End":"08:47.015","Text":"If I multiply by x^4th,"},{"Start":"08:47.015 ","End":"08:56.960","Text":"all I have to do is to add plus 4 here and this now gives me the answer for x^4th,"},{"Start":"08:56.960 ","End":"09:05.130","Text":"e^10x will be equal to this and this is exactly what we got here,"},{"Start":"09:05.130 ","End":"09:11.490","Text":"no difference, it\u0027s just much easier and we\u0027re basically done."},{"Start":"09:11.490 ","End":"09:15.890","Text":"In the next clip, we\u0027re going to do another 1 from definition,"},{"Start":"09:15.890 ","End":"09:17.300","Text":"which is what we did up here."},{"Start":"09:17.300 ","End":"09:20.315","Text":"We actually differentiated until we found the pattern."},{"Start":"09:20.315 ","End":"09:21.620","Text":"We\u0027ll do another 1 of those."},{"Start":"09:21.620 ","End":"09:23.435","Text":"Although in practice on the exams,"},{"Start":"09:23.435 ","End":"09:29.695","Text":"this is more common to use an existing function like we know e^x,"},{"Start":"09:29.695 ","End":"09:32.775","Text":"and from this we\u0027ll build on x^4th, e^10x."},{"Start":"09:32.775 ","End":"09:38.270","Text":"In the next clip, we\u0027ll do 1 from scratch and I\u0027m not sure maybe we\u0027ll do another"},{"Start":"09:38.270 ","End":"09:46.980","Text":"1 or more even of a series based on existing Maclaurin series."}],"ID":10340},{"Watched":false,"Name":"Worked Example 2 - Common Maclaurin Series","Duration":"18m 19s","ChapterTopicVideoID":10119,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10119.jpeg","UploadDate":"2020-01-16T13:04:09.1570000","DurationForVideoObject":"PT18M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"This clip continues the previous clip on Maclaurin Series."},{"Start":"00:05.310 ","End":"00:08.955","Text":"Last time, we took a simple example of e to the x."},{"Start":"00:08.955 ","End":"00:12.540","Text":"This time, we\u0027ll take a slightly more complicated example,"},{"Start":"00:12.540 ","End":"00:15.390","Text":"the natural log of 1 plus x."},{"Start":"00:15.390 ","End":"00:19.155","Text":"We want to find its Maclaurin Series."},{"Start":"00:19.155 ","End":"00:20.730","Text":"We\u0027ll do it from scratch."},{"Start":"00:20.730 ","End":"00:24.435","Text":"Although there are other more advanced techniques that are shorter,"},{"Start":"00:24.435 ","End":"00:27.789","Text":"you need the practice of doing it from scratch."},{"Start":"00:28.010 ","End":"00:31.200","Text":"Before we start, I\u0027d like to ask a question,"},{"Start":"00:31.200 ","End":"00:35.160","Text":"why do you think I asked for the natural log of 1 plus x?"},{"Start":"00:35.160 ","End":"00:39.960","Text":"Why not simply go for the natural log of x?"},{"Start":"00:39.960 ","End":"00:42.190","Text":"The answer is pretty straightforward."},{"Start":"00:42.190 ","End":"00:44.550","Text":"It\u0027s not defined there."},{"Start":"00:44.550 ","End":"00:47.480","Text":"Natural log is only defined for x bigger than 0."},{"Start":"00:47.480 ","End":"00:50.105","Text":"I have to start shifted a bit,"},{"Start":"00:50.105 ","End":"00:52.980","Text":"and that\u0027s the reason why."},{"Start":"00:52.980 ","End":"00:54.800","Text":"Let\u0027s get started."},{"Start":"00:54.800 ","End":"01:00.100","Text":"If you remember, we do a bunch of differentiations first until we find the pattern."},{"Start":"01:00.100 ","End":"01:03.705","Text":"Let\u0027s start differentiating."},{"Start":"01:03.705 ","End":"01:08.645","Text":"What we get is f prime of x."},{"Start":"01:08.645 ","End":"01:15.650","Text":"Natural log means it\u0027s 1 over 1 plus x times the internal derivative is 1,"},{"Start":"01:15.650 ","End":"01:19.465","Text":"so it\u0027s just 1 over 1 plus x."},{"Start":"01:19.465 ","End":"01:25.055","Text":"Allow me to write it as 1 plus x to the power of minus 1."},{"Start":"01:25.055 ","End":"01:29.045","Text":"Next derivative, f double prime of x."},{"Start":"01:29.045 ","End":"01:30.830","Text":"According to the exponents,"},{"Start":"01:30.830 ","End":"01:41.545","Text":"it\u0027s minus 1 times 1 plus x to the power of minus 2."},{"Start":"01:41.545 ","End":"01:45.010","Text":"Again, the internal derivative is 1."},{"Start":"01:45.010 ","End":"01:48.255","Text":"It\u0027s as if it was x almost."},{"Start":"01:48.255 ","End":"01:55.205","Text":"Next, f triple prime of x third derivative minus 1,"},{"Start":"01:55.205 ","End":"01:57.640","Text":"and then times minus 2,"},{"Start":"01:57.640 ","End":"02:00.210","Text":"and then 1 plus x,"},{"Start":"02:00.210 ","End":"02:04.300","Text":"reduce the power by 1 to the power of minus 3."},{"Start":"02:04.300 ","End":"02:09.230","Text":"I\u0027ll do one more. Let\u0027s start using the brackets notation for derivatives."},{"Start":"02:09.230 ","End":"02:15.260","Text":"Fourth derivative is minus 1 times minus 2 times the"},{"Start":"02:15.260 ","End":"02:22.325","Text":"exponent which is minus 3 times 1 plus x to the power of minus 4."},{"Start":"02:22.325 ","End":"02:25.010","Text":"I think the pattern is emerging."},{"Start":"02:25.010 ","End":"02:27.275","Text":"I\u0027m looking at this and I see,"},{"Start":"02:27.275 ","End":"02:29.630","Text":"I have a 4 here."},{"Start":"02:29.630 ","End":"02:31.910","Text":"I have 1 plus x to the minus 4."},{"Start":"02:31.910 ","End":"02:34.075","Text":"This 4 is this 4, 3,"},{"Start":"02:34.075 ","End":"02:37.055","Text":"3, 2, 2 with a minus."},{"Start":"02:37.055 ","End":"02:40.670","Text":"In front of it, I have 3 factorial,"},{"Start":"02:40.670 ","End":"02:44.780","Text":"1 times 2 times 3, but with minus signs with these."},{"Start":"02:44.780 ","End":"02:47.330","Text":"What I think I\u0027ll do is just write the"},{"Start":"02:47.330 ","End":"02:50.300","Text":"fifth derivative what I guess it to be according to the pattern."},{"Start":"02:50.300 ","End":"02:52.325","Text":"Then we\u0027ll check if that\u0027s really the case."},{"Start":"02:52.325 ","End":"02:55.130","Text":"According to the pattern and not by differentiating,"},{"Start":"02:55.130 ","End":"02:57.110","Text":"I\u0027ll get minus 1, minus 2,"},{"Start":"02:57.110 ","End":"03:02.475","Text":"minus 3, minus 4 here each time I increase by 1."},{"Start":"03:02.475 ","End":"03:05.255","Text":"The other way of looking at it is that the last number,"},{"Start":"03:05.255 ","End":"03:07.310","Text":"the 3 is 1 less than 4,"},{"Start":"03:07.310 ","End":"03:10.805","Text":"and 2 is 1 less than 3."},{"Start":"03:10.805 ","End":"03:12.685","Text":"If it\u0027s 5 here,"},{"Start":"03:12.685 ","End":"03:14.940","Text":"I want to go up to 4 to 1 less than."},{"Start":"03:14.940 ","End":"03:18.030","Text":"If it\u0027s 4 here, I go up to 3, and so on."},{"Start":"03:18.030 ","End":"03:22.110","Text":"Then I\u0027m going to copy the same 1 plus x that we said."},{"Start":"03:22.110 ","End":"03:25.730","Text":"We said we put a minus on the same number here as here."},{"Start":"03:25.730 ","End":"03:28.460","Text":"This is what it would be if I continue the pattern."},{"Start":"03:28.460 ","End":"03:29.990","Text":"Actually, this is also correct."},{"Start":"03:29.990 ","End":"03:32.390","Text":"It is the derivative because when we differentiate this,"},{"Start":"03:32.390 ","End":"03:35.915","Text":"we get the extra minus 4 and we also lower it by 1."},{"Start":"03:35.915 ","End":"03:37.790","Text":"It looks like we\u0027ve got it all set."},{"Start":"03:37.790 ","End":"03:40.120","Text":"Let\u0027s try and write the formula."},{"Start":"03:40.120 ","End":"03:42.860","Text":"For a moment, let\u0027s ignore the minuses here."},{"Start":"03:42.860 ","End":"03:45.785","Text":"Suppose it was 1 times 2 times 3 times 4,"},{"Start":"03:45.785 ","End":"03:48.230","Text":"then it would be 4 factorial."},{"Start":"03:48.230 ","End":"03:50.260","Text":"At the fourth entry,"},{"Start":"03:50.260 ","End":"03:52.480","Text":"we have 3 factorial."},{"Start":"03:52.480 ","End":"03:54.260","Text":"At the third derivative,"},{"Start":"03:54.260 ","End":"03:56.945","Text":"we only have 2 factorial."},{"Start":"03:56.945 ","End":"03:59.060","Text":"At the first derivative,"},{"Start":"03:59.060 ","End":"04:01.055","Text":"we have 1 factorial,"},{"Start":"04:01.055 ","End":"04:04.510","Text":"0 factorial because one doesn\u0027t hurt to put it anyway."},{"Start":"04:04.510 ","End":"04:09.680","Text":"Now, let\u0027s try and write a general formula."},{"Start":"04:09.680 ","End":"04:12.635","Text":"We\u0027ll take care of the minus also."},{"Start":"04:12.635 ","End":"04:15.785","Text":"The nth derivative of x will equal,"},{"Start":"04:15.785 ","End":"04:18.155","Text":"well, it\u0027s easiest to start at the last bit."},{"Start":"04:18.155 ","End":"04:22.640","Text":"We see that I have 1 plus x to the power of minus n,"},{"Start":"04:22.640 ","End":"04:24.020","Text":"because the 5 minus 5,"},{"Start":"04:24.020 ","End":"04:26.045","Text":"4 minus 4, and so on."},{"Start":"04:26.045 ","End":"04:29.390","Text":"The next thing we could say is that in front of it,"},{"Start":"04:29.390 ","End":"04:32.690","Text":"we have, this is 5 and this is 4 factorial."},{"Start":"04:32.690 ","End":"04:34.520","Text":"This is n, it\u0027s 1 less."},{"Start":"04:34.520 ","End":"04:37.470","Text":"It\u0027s n minus 1 factorial."},{"Start":"04:37.470 ","End":"04:40.435","Text":"Just like at 4, we had 3 factorial."},{"Start":"04:40.435 ","End":"04:44.165","Text":"The final thing to take care of is these minuses."},{"Start":"04:44.165 ","End":"04:48.230","Text":"Notice that if we compute the sign at each level,"},{"Start":"04:48.230 ","End":"04:50.030","Text":"here, I have minus, minus, minus,"},{"Start":"04:50.030 ","End":"04:52.185","Text":"minus, which is plus."},{"Start":"04:52.185 ","End":"04:53.550","Text":"Here, I have minus, minus,"},{"Start":"04:53.550 ","End":"04:56.085","Text":"minus which is minus."},{"Start":"04:56.085 ","End":"04:58.500","Text":"Here, I have minus, minus which is plus."},{"Start":"04:58.500 ","End":"05:00.095","Text":"Here, I have a minus."},{"Start":"05:00.095 ","End":"05:03.020","Text":"Here, I have a plus."},{"Start":"05:03.020 ","End":"05:04.400","Text":"This alternates."},{"Start":"05:04.400 ","End":"05:07.640","Text":"This formula doesn\u0027t quite work if n is 0."},{"Start":"05:07.640 ","End":"05:09.654","Text":"We\u0027ll do that separately."},{"Start":"05:09.654 ","End":"05:13.790","Text":"Anyway, back to this minus, plus business."},{"Start":"05:13.790 ","End":"05:20.450","Text":"The usual thing to do is to put minus 1 to the power of n. Each time it changes the sign,"},{"Start":"05:20.450 ","End":"05:24.000","Text":"minus 1 to the power of an odd number is minus 1,"},{"Start":"05:24.000 ","End":"05:26.495","Text":"and to an even number, it\u0027s plus 1."},{"Start":"05:26.495 ","End":"05:31.630","Text":"But here, I think we\u0027ve got it reversed because if I put here,"},{"Start":"05:31.630 ","End":"05:34.270","Text":"for example, minus 1 to the power of 3,"},{"Start":"05:34.270 ","End":"05:36.865","Text":"it comes out minus and we have plus."},{"Start":"05:36.865 ","End":"05:39.005","Text":"If I put minus 1 to the fourth,"},{"Start":"05:39.005 ","End":"05:41.260","Text":"it comes up plus and we have minus."},{"Start":"05:41.260 ","End":"05:45.070","Text":"When that happens, you just correct it by adding an extra minus,"},{"Start":"05:45.070 ","End":"05:48.490","Text":"which means another minus 1 just increases the power by 1."},{"Start":"05:48.490 ","End":"05:51.190","Text":"I think if you\u0027ll check now except for the first one,"},{"Start":"05:51.190 ","End":"05:54.360","Text":"let\u0027s try say, the fourth when n is 4."},{"Start":"05:54.360 ","End":"05:57.895","Text":"When n is 4, what we have here is minus, minus, minus,"},{"Start":"05:57.895 ","End":"06:03.900","Text":"which is minus and minus 1 to the power of 5,"},{"Start":"06:03.900 ","End":"06:07.530","Text":"because if n is 4, this is 5 is also minus,"},{"Start":"06:07.530 ","End":"06:08.850","Text":"so it works out."},{"Start":"06:08.850 ","End":"06:13.840","Text":"For example, this one is minus 3 factorial,"},{"Start":"06:13.840 ","End":"06:17.515","Text":"1 plus x to the minus 4."},{"Start":"06:17.515 ","End":"06:19.250","Text":"But if I take an odd number,"},{"Start":"06:19.250 ","End":"06:25.740","Text":"this 1 is plus 4 factorial 1 plus x to the power of minus 5."},{"Start":"06:25.740 ","End":"06:28.400","Text":"We took n equals 4 and we saw that this works."},{"Start":"06:28.400 ","End":"06:30.715","Text":"If I take n equals 5, it also works."},{"Start":"06:30.715 ","End":"06:36.020","Text":"What I\u0027ve highlighted here is the formula for the nth derivative of f of x,"},{"Start":"06:36.020 ","End":"06:39.410","Text":"which is this natural log of 1 plus x."},{"Start":"06:39.410 ","End":"06:43.850","Text":"Note that technically, this formula doesn\u0027t apply when n equals 0."},{"Start":"06:43.850 ","End":"06:48.310","Text":"The nth derivative of x is this for n bigger than 0."},{"Start":"06:48.310 ","End":"06:51.575","Text":"The 0th derivative is just the original function."},{"Start":"06:51.575 ","End":"06:54.740","Text":"But this formula holds from 1 onwards."},{"Start":"06:54.740 ","End":"07:00.815","Text":"But what I really want my formula is the value of these derivatives at 0."},{"Start":"07:00.815 ","End":"07:02.345","Text":"Let\u0027s write it over here."},{"Start":"07:02.345 ","End":"07:05.000","Text":"From here, I get that f is 0."},{"Start":"07:05.000 ","End":"07:09.415","Text":"If I substitute 0, natural log of 1 is 0."},{"Start":"07:09.415 ","End":"07:14.060","Text":"If I substitute here,1 plus 0 to the minus"},{"Start":"07:14.060 ","End":"07:19.195","Text":"1 is 1 to the minus 1 is 1 and 0 factorial is 1."},{"Start":"07:19.195 ","End":"07:24.000","Text":"F prime of 0 is 1, see, well,"},{"Start":"07:24.000 ","End":"07:26.810","Text":"all of these are going to be 1 because 1 plus x is 1,"},{"Start":"07:26.810 ","End":"07:28.760","Text":"and 1 to any power is 1."},{"Start":"07:28.760 ","End":"07:33.620","Text":"I\u0027m just going to get 1 factorial."},{"Start":"07:33.620 ","End":"07:35.480","Text":"This is the 0 factorial,"},{"Start":"07:35.480 ","End":"07:36.890","Text":"but from here onwards,"},{"Start":"07:36.890 ","End":"07:44.930","Text":"I\u0027m going to get f double prime of 0 is minus 1 factorial."},{"Start":"07:44.930 ","End":"07:48.830","Text":"Then I\u0027m going to get plus 2 factorial,"},{"Start":"07:48.830 ","End":"07:51.229","Text":"and I\u0027m going to get minus 3 factorial,"},{"Start":"07:51.229 ","End":"07:54.710","Text":"then I\u0027m going to get plus 4 factorial that I got ahead of myself."},{"Start":"07:54.710 ","End":"08:00.140","Text":"Let\u0027s write that as f triple prime of 0 is equal to that."},{"Start":"08:00.140 ","End":"08:03.725","Text":"F fourth derivative of 0 is that."},{"Start":"08:03.725 ","End":"08:08.540","Text":"F fifth derivative at 0 is equal to this."},{"Start":"08:08.540 ","End":"08:18.990","Text":"In general, the general term would be the nth derivative of 0 is going to be,"},{"Start":"08:18.990 ","End":"08:23.970","Text":"here, I\u0027m going to have n minus 1 factorial."},{"Start":"08:24.840 ","End":"08:28.360","Text":"Then we need the bit about the plus or minus."},{"Start":"08:28.360 ","End":"08:38.440","Text":"It\u0027s minus 1 to the power of n plus 1."},{"Start":"08:38.440 ","End":"08:41.120","Text":"Leave quite enough room there but you\u0027ll see."},{"Start":"08:43.400 ","End":"08:47.880","Text":"Let\u0027s call that the first part of finding the Maclaurin series"},{"Start":"08:47.880 ","End":"08:52.639","Text":"just to find the values of f to the power of n 0."},{"Start":"08:52.639 ","End":"09:02.420","Text":"Now we need to find the coefficients and we\u0027ll call that part 2 and scroll up a bit."},{"Start":"09:02.450 ","End":"09:08.130","Text":"Here we recall the Maclaurin series expansion of f of x."},{"Start":"09:08.130 ","End":"09:11.560","Text":"Let\u0027s write f of x or better still instead of f of"},{"Start":"09:11.560 ","End":"09:16.340","Text":"x I\u0027ll just write what it really is that\u0027s natural log of 1 plus x,"},{"Start":"09:16.680 ","End":"09:19.885","Text":"not using the sigma notation,"},{"Start":"09:19.885 ","End":"09:29.035","Text":"it\u0027s equal to f of 0 over 0 factorial plus f prime of 0"},{"Start":"09:29.035 ","End":"09:33.400","Text":"over 1 factorial times x to the"},{"Start":"09:33.400 ","End":"09:39.989","Text":"1 plus f double prime of 0 over 2 factorial."},{"Start":"09:39.989 ","End":"09:42.555","Text":"I will write these factorials from 2 onwards,"},{"Start":"09:42.555 ","End":"09:49.830","Text":"x squared plus f triple prime of 0 over 3 factorial times"},{"Start":"09:49.830 ","End":"10:02.155","Text":"x cubed plus the fourth derivative of f of 0 over 4 factorial times x^4,"},{"Start":"10:02.155 ","End":"10:07.810","Text":"and here we have room for more plus the fifth"},{"Start":"10:07.810 ","End":"10:15.985","Text":"derivative at 0 over 5 factorial x^5, and so on."},{"Start":"10:15.985 ","End":"10:20.275","Text":"At this point we can start substituting."},{"Start":"10:20.275 ","End":"10:23.440","Text":"We have these derivatives,"},{"Start":"10:23.440 ","End":"10:26.725","Text":"f of 0 is 0 so that means that this term goes."},{"Start":"10:26.725 ","End":"10:30.400","Text":"Now, f prime of 0 we said is 1,"},{"Start":"10:30.400 ","End":"10:38.470","Text":"so that\u0027s just x. f double prime of 0 is minus 1 factorial,"},{"Start":"10:38.470 ","End":"10:41.150","Text":"which is minus 1."},{"Start":"10:42.810 ","End":"10:48.880","Text":"I\u0027ll write it as factorial over 2 factorial x squared"},{"Start":"10:48.880 ","End":"10:56.305","Text":"plus 2 factorial over 3 factorial,"},{"Start":"10:56.305 ","End":"11:03.625","Text":"x cubed minus 3 factorial over 4 factorial x^4."},{"Start":"11:03.625 ","End":"11:06.909","Text":"All this is from the formula for the Laurent series,"},{"Start":"11:06.909 ","End":"11:08.710","Text":"which is pretty much like this,"},{"Start":"11:08.710 ","End":"11:13.280","Text":"but with the x to the power of n and added."},{"Start":"11:13.290 ","End":"11:20.360","Text":"Then we get plus 4 factorial over 5 factorial x^5."},{"Start":"11:20.360 ","End":"11:23.350","Text":"I\u0027ll put another one in their."},{"Start":"11:23.350 ","End":"11:30.940","Text":"5 factorial, we can see the pattern already over 6 factorial x^6,"},{"Start":"11:30.940 ","End":"11:33.140","Text":"and then and so on."},{"Start":"11:33.150 ","End":"11:37.885","Text":"Now, there is something that cancels here."},{"Start":"11:37.885 ","End":"11:40.255","Text":"This could be left as an answer."},{"Start":"11:40.255 ","End":"11:47.050","Text":"But for example, let\u0027s say at this 4 factorial over 5 factorial."},{"Start":"11:47.050 ","End":"11:51.355","Text":"If I say what 4 factorial over 5 factorial is,"},{"Start":"11:51.355 ","End":"11:59.065","Text":"it\u0027s equal to 4 times 3 times 2 times 1 over 5 times 4 times 3 times 2 times 1,"},{"Start":"11:59.065 ","End":"12:03.415","Text":"everything cancels we\u0027re just left with 1/5."},{"Start":"12:03.415 ","End":"12:10.630","Text":"In general, if we take the coefficient which is minus"},{"Start":"12:10.630 ","End":"12:20.065","Text":"1 to the n plus 1 times n minus 1 factorial."},{"Start":"12:20.065 ","End":"12:27.490","Text":"The term that it goes with is it\u0027s times x to the n over n factorial."},{"Start":"12:27.490 ","End":"12:32.335","Text":"That\u0027s the nth term in the power series."},{"Start":"12:32.335 ","End":"12:36.295","Text":"What we\u0027re left with is just the plus or minus,"},{"Start":"12:36.295 ","End":"12:40.765","Text":"which is minus 1 to the power of n plus 1."},{"Start":"12:40.765 ","End":"12:46.670","Text":"But then the factorials cancel just like 4 factorial over 5 factorial is 1/5."},{"Start":"12:46.670 ","End":"12:51.990","Text":"Here I get n minus 1 factorial over n factorial is just 1 over n,"},{"Start":"12:51.990 ","End":"12:55.200","Text":"because this is the product of all the numbers from 1 to n minus 1,"},{"Start":"12:55.200 ","End":"12:57.360","Text":"and here there\u0027s an extra one, then n at the end,"},{"Start":"12:57.360 ","End":"12:59.925","Text":"just like here we had the extra one at the end."},{"Start":"12:59.925 ","End":"13:02.260","Text":"We have 1 to n minus 1,"},{"Start":"13:02.260 ","End":"13:03.520","Text":"and here we have 1 to n,"},{"Start":"13:03.520 ","End":"13:09.430","Text":"it\u0027s just 1 over n. It\u0027s over n times x to"},{"Start":"13:09.430 ","End":"13:17.124","Text":"the n. This is the nth term of the Laurent series."},{"Start":"13:17.124 ","End":"13:19.675","Text":"If we just wrote it out like this,"},{"Start":"13:19.675 ","End":"13:29.440","Text":"we could continue over here and say that natural log of 1 plus x is,"},{"Start":"13:29.440 ","End":"13:31.045","Text":"putting all these things here,"},{"Start":"13:31.045 ","End":"13:40.884","Text":"is x minus x squared over 2 plus x cubed over 3,"},{"Start":"13:40.884 ","End":"13:43.510","Text":"minus x^4 over 4,"},{"Start":"13:43.510 ","End":"13:48.895","Text":"plus x^5 over 5, and so on."},{"Start":"13:48.895 ","End":"13:54.565","Text":"This is in fact the Maclaurin series for natural log of 1 plus x."},{"Start":"13:54.565 ","End":"13:59.920","Text":"If we wanted it in Sigma notation there\u0027s no problem giving you that."},{"Start":"13:59.920 ","End":"14:05.965","Text":"We could also say that the natural log of 1 plus x is the sum,"},{"Start":"14:05.965 ","End":"14:08.440","Text":"and since the 0 term is 0,"},{"Start":"14:08.440 ","End":"14:18.070","Text":"we can start the sum from 1 to infinity of minus 1 to the power of n plus"},{"Start":"14:18.070 ","End":"14:27.790","Text":"1 over n times x to the power of n. Let\u0027s check it."},{"Start":"14:27.790 ","End":"14:31.930","Text":"For example, if I let n equals 4 here, if n is 4,"},{"Start":"14:31.930 ","End":"14:37.045","Text":"I get x^4, here 4 and minus 1^5 is minus."},{"Start":"14:37.045 ","End":"14:39.610","Text":"Yes, indeed we get the right way around."},{"Start":"14:39.610 ","End":"14:41.230","Text":"I\u0027ll highlight this to."},{"Start":"14:41.230 ","End":"14:47.365","Text":"I want to say that this is definitely not the question that you would expect on an exam."},{"Start":"14:47.365 ","End":"14:52.465","Text":"More likely what you would get in an exam is to build on this,"},{"Start":"14:52.465 ","End":"14:55.060","Text":"and I\u0027ll show you what I mean."},{"Start":"14:55.060 ","End":"14:57.175","Text":"I\u0027d like to give you a typical,"},{"Start":"14:57.175 ","End":"15:00.650","Text":"let\u0027s call it an exam question."},{"Start":"15:01.710 ","End":"15:05.890","Text":"That\u0027s a possible question they would ask."},{"Start":"15:05.890 ","End":"15:08.485","Text":"They will say something like,"},{"Start":"15:08.485 ","End":"15:12.235","Text":"this whole thing that we computed would be a given,"},{"Start":"15:12.235 ","End":"15:14.830","Text":"that we have the series of this."},{"Start":"15:14.830 ","End":"15:23.320","Text":"Compute the series or let\u0027s say the Maclaurin series of something similar to this."},{"Start":"15:23.320 ","End":"15:30.820","Text":"They would say of x squared times natural log of 1 plus 4x."},{"Start":"15:30.820 ","End":"15:34.810","Text":"Given that this is the case,"},{"Start":"15:34.810 ","End":"15:40.580","Text":"we have to now compute a variation of this, which is this."},{"Start":"15:41.250 ","End":"15:44.320","Text":"Why don\u0027t I just work on the solution."},{"Start":"15:44.320 ","End":"15:46.075","Text":"I\u0027ll show you the solution now."},{"Start":"15:46.075 ","End":"15:49.885","Text":"It\u0027s a good question that they would ask."},{"Start":"15:49.885 ","End":"15:56.530","Text":"Solution, this is the question and this is the solution."},{"Start":"15:56.530 ","End":"16:03.415","Text":"What we have to do is start with this and then build up to this in stages."},{"Start":"16:03.415 ","End":"16:05.650","Text":"Start with this, I\u0027m not going to copy it again."},{"Start":"16:05.650 ","End":"16:09.070","Text":"What would happen if I replace x in this formula,"},{"Start":"16:09.070 ","End":"16:11.200","Text":"instead of x I put 4x,"},{"Start":"16:11.200 ","End":"16:15.880","Text":"then I would get that the natural logarithm of 1 plus 4x,"},{"Start":"16:15.880 ","End":"16:25.780","Text":"which is what we\u0027re looking for is equal to the sum from 1 to infinity of minus 1 to"},{"Start":"16:25.780 ","End":"16:31.990","Text":"the n plus 1 over n times 4x to the power"},{"Start":"16:31.990 ","End":"16:38.800","Text":"of n. Next thing I do is multiply by the x squared."},{"Start":"16:38.800 ","End":"16:46.780","Text":"I get that x squared times natural log of 1 plus 4x is this thing times x squared."},{"Start":"16:46.780 ","End":"16:51.430","Text":"Now, x squared is the constant as far as n is concerned, n is changing."},{"Start":"16:51.430 ","End":"16:55.315","Text":"At every n I can multiply by x squared."},{"Start":"16:55.315 ","End":"16:57.354","Text":"I get the sum,"},{"Start":"16:57.354 ","End":"17:03.895","Text":"n goes from 1 to infinity minus 1 to the n plus 1"},{"Start":"17:03.895 ","End":"17:11.535","Text":"over n times 4x to the n x squared."},{"Start":"17:11.535 ","End":"17:16.275","Text":"I\u0027ll just show you what is 4x to the n times x squared"},{"Start":"17:16.275 ","End":"17:21.634","Text":"is equal to 4 to the n, x to the n x squared,"},{"Start":"17:21.634 ","End":"17:28.500","Text":"which finally equals 4 to the n x to the n plus 2."},{"Start":"17:28.500 ","End":"17:31.535","Text":"This is equal, if I just simplify it a bit,"},{"Start":"17:31.535 ","End":"17:41.189","Text":"as the sum n goes from 1 to infinity minus 1"},{"Start":"17:41.380 ","End":"17:48.020","Text":"to the n plus 1 times 4 to"},{"Start":"17:48.020 ","End":"17:57.670","Text":"the n over n times x to the power of n plus 2,"},{"Start":"17:57.670 ","End":"18:01.325","Text":"and that I can write as the answer."},{"Start":"18:01.325 ","End":"18:04.980","Text":"There\u0027s one more thing we haven\u0027t talked about."},{"Start":"18:04.980 ","End":"18:06.980","Text":"I think I\u0027ll leave it for the next time,"},{"Start":"18:06.980 ","End":"18:11.935","Text":"but I\u0027ll just let you know what the topic is the radius of convergence."},{"Start":"18:11.935 ","End":"18:16.820","Text":"The power series has a radius of convergence and we\u0027ll talk about that in the next clip,"},{"Start":"18:16.820 ","End":"18:19.320","Text":"meanwhile here we\u0027re done."}],"ID":10341},{"Watched":false,"Name":"Taylor Series and Interval of Convergence","Duration":"6m 35s","ChapterTopicVideoID":10120,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10120.jpeg","UploadDate":"2020-01-16T13:05:00.6770000","DurationForVideoObject":"PT6M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.260 ","End":"00:04.530","Text":"There\u0027s something about Taylor series I didn\u0027t talk about,"},{"Start":"00:04.530 ","End":"00:07.560","Text":"I didn\u0027t want to overburden you with everything at once,"},{"Start":"00:07.560 ","End":"00:11.925","Text":"but we have to mention its interval of convergence."},{"Start":"00:11.925 ","End":"00:18.840","Text":"We discussed the example of the natural log of 1 plus x and found"},{"Start":"00:18.840 ","End":"00:25.845","Text":"its Taylor series to be the sum from 1 to infinity of minus 1 to the n,"},{"Start":"00:25.845 ","End":"00:30.365","Text":"x to the n over n. Now,"},{"Start":"00:30.365 ","End":"00:33.770","Text":"this Taylor series is a power series,"},{"Start":"00:33.770 ","End":"00:38.105","Text":"and when you studied power series you must have"},{"Start":"00:38.105 ","End":"00:43.250","Text":"talked about radius of convergence and interval of convergence."},{"Start":"00:43.250 ","End":"00:45.330","Text":"The question is, well,"},{"Start":"00:45.330 ","End":"00:49.275","Text":"there are 2 questions I\u0027d like to ask here and in general."},{"Start":"00:49.275 ","End":"00:54.125","Text":"The first is, what is this series radius of convergence?"},{"Start":"00:54.125 ","End":"00:57.780","Text":"Secondly, this is an interesting question,"},{"Start":"00:57.780 ","End":"01:04.715","Text":"does the function equal its power series within its interval of convergence?"},{"Start":"01:04.715 ","End":"01:08.855","Text":"Now, the radius of convergence,"},{"Start":"01:08.855 ","End":"01:11.795","Text":"you can compute using your usual techniques."},{"Start":"01:11.795 ","End":"01:15.945","Text":"When you get these series from formula sheet,"},{"Start":"01:15.945 ","End":"01:17.250","Text":"or from the lecture,"},{"Start":"01:17.250 ","End":"01:20.480","Text":"you\u0027ll already get it with its radius of convergence."},{"Start":"01:20.480 ","End":"01:25.940","Text":"In this case, it happens to be that minus 1 is strictly less than x,"},{"Start":"01:25.940 ","End":"01:28.195","Text":"less than or equal to 1."},{"Start":"01:28.195 ","End":"01:31.815","Text":"That\u0027s as far as the integral convergence,"},{"Start":"01:31.815 ","End":"01:35.584","Text":"and that\u0027s for the question about whether the series equals the function."},{"Start":"01:35.584 ","End":"01:37.235","Text":"In this case, yes."},{"Start":"01:37.235 ","End":"01:41.705","Text":"In fact, in pretty much all cases you\u0027ll encounter the answer is yes."},{"Start":"01:41.705 ","End":"01:45.890","Text":"But theoretically, the answer is no in general."},{"Start":"01:45.890 ","End":"01:49.115","Text":"There are examples, a bit bizarre, maybe rare,"},{"Start":"01:49.115 ","End":"01:53.825","Text":"but where the function is not at all equal to the Taylor series,"},{"Start":"01:53.825 ","End":"01:56.555","Text":"but you won\u0027t encounter this most likely."},{"Start":"01:56.555 ","End":"01:58.190","Text":"Generally they are equal,"},{"Start":"01:58.190 ","End":"02:01.865","Text":"at least within the interval of convergence."},{"Start":"02:01.865 ","End":"02:04.580","Text":"Because we have an interval of convergence, we can\u0027t,"},{"Start":"02:04.580 ","End":"02:06.950","Text":"for example answer a question like,"},{"Start":"02:06.950 ","End":"02:09.290","Text":"what is the natural log of 3?"},{"Start":"02:09.290 ","End":"02:12.365","Text":"We can answer it but not using this technique."},{"Start":"02:12.365 ","End":"02:17.315","Text":"The most we can get is when x is 1 and we could get natural log of 2."},{"Start":"02:17.315 ","End":"02:19.985","Text":"That\u0027s interesting in itself."},{"Start":"02:19.985 ","End":"02:23.570","Text":"Natural log of 2, and this is"},{"Start":"02:23.570 ","End":"02:27.065","Text":"just on the border of the interval of convergence, would be equal."},{"Start":"02:27.065 ","End":"02:29.330","Text":"If I substitute x equals 1 here,"},{"Start":"02:29.330 ","End":"02:32.080","Text":"all the x to the ends come out 1,"},{"Start":"02:32.080 ","End":"02:41.390","Text":"and this comes out to be sum from 1 to infinity of minus 1^n over n. If I write it out,"},{"Start":"02:41.390 ","End":"02:42.905","Text":"what this series is,"},{"Start":"02:42.905 ","End":"02:47.940","Text":"this is equal to, sorry, n plus 1."},{"Start":"02:47.940 ","End":"02:50.040","Text":"Forgot the plus 1."},{"Start":"02:50.040 ","End":"02:52.670","Text":"Yeah. This is equal to when n is 1,"},{"Start":"02:52.670 ","End":"02:54.080","Text":"we get minus 1 to the 2,"},{"Start":"02:54.080 ","End":"02:56.420","Text":"so it\u0027s plus 1 over 1."},{"Start":"02:56.420 ","End":"03:01.225","Text":"In general, you can see that it comes out to 1 minus 1/2 plus 1/3,"},{"Start":"03:01.225 ","End":"03:05.345","Text":"minus 1/4 plus 1/5, and so on."},{"Start":"03:05.345 ","End":"03:06.875","Text":"The more terms you take,"},{"Start":"03:06.875 ","End":"03:10.340","Text":"the closer it gets to the actual answer."},{"Start":"03:10.340 ","End":"03:14.380","Text":"Just by the way, if I put x equals minus 1 here,"},{"Start":"03:14.380 ","End":"03:23.730","Text":"what I get is simply 1 plus a 1/2 plus 1/3 plus a 1/4 plus 1/5 plus 1/6."},{"Start":"03:23.730 ","End":"03:28.100","Text":"Actually, it\u0027s a famous series called the harmonic series and it doesn\u0027t converge."},{"Start":"03:28.100 ","End":"03:30.510","Text":"It goes to infinity even though it\u0027s very slowly."},{"Start":"03:30.510 ","End":"03:32.360","Text":"Of course, on the other side of the equation,"},{"Start":"03:32.360 ","End":"03:37.255","Text":"we can\u0027t expect to put x equals minus 1 because then we get natural log of 0."},{"Start":"03:37.255 ","End":"03:39.255","Text":"That\u0027s just a by the way."},{"Start":"03:39.255 ","End":"03:42.600","Text":"Okay. Now, about exam questions."},{"Start":"03:42.600 ","End":"03:45.440","Text":"As we said, there are certain standard series that are"},{"Start":"03:45.440 ","End":"03:48.680","Text":"well-known and not only are the series given,"},{"Start":"03:48.680 ","End":"03:51.950","Text":"but it\u0027s given together with the interval of convergence."},{"Start":"03:51.950 ","End":"03:56.720","Text":"What you\u0027re often asked to do is to take a variation of this."},{"Start":"03:56.720 ","End":"03:59.075","Text":"They will say to you, okay,"},{"Start":"03:59.075 ","End":"04:05.915","Text":"given that natural log of 1 plus x is this with such and such interval of convergence,"},{"Start":"04:05.915 ","End":"04:12.635","Text":"now find the series and the interval of convergence for, let\u0027s say,"},{"Start":"04:12.635 ","End":"04:16.100","Text":"for f of x is equal to,"},{"Start":"04:16.100 ","End":"04:23.070","Text":"let\u0027s say x cubed times the natural log of 1 plus 2x."},{"Start":"04:23.150 ","End":"04:26.810","Text":"What we would say, would be, first of all,"},{"Start":"04:26.810 ","End":"04:35.990","Text":"forget about the x cubed and we\u0027d say that natural log of 1 plus 2x is equal to the sum"},{"Start":"04:35.990 ","End":"04:41.105","Text":"minus 1 to the n plus 1 of"},{"Start":"04:41.105 ","End":"04:49.740","Text":"2x to the n over n. N goes from 1 to infinity."},{"Start":"04:49.900 ","End":"04:53.975","Text":"Just highlight that."},{"Start":"04:53.975 ","End":"04:58.305","Text":"I\u0027ve got 2x instead of x,"},{"Start":"04:58.305 ","End":"05:03.485","Text":"and so also in the radius of convergence,"},{"Start":"05:03.485 ","End":"05:06.890","Text":"I would say minus 1 is less than,"},{"Start":"05:06.890 ","End":"05:10.055","Text":"2x is less than or equal to 1,"},{"Start":"05:10.055 ","End":"05:11.810","Text":"and there\u0027s my 2x."},{"Start":"05:11.810 ","End":"05:17.900","Text":"Of course, I would write this in the end as simply, instead of this,"},{"Start":"05:17.900 ","End":"05:22.175","Text":"I would just write minus a half less than x,"},{"Start":"05:22.175 ","End":"05:24.990","Text":"less than or equal to a half,"},{"Start":"05:24.990 ","End":"05:29.090","Text":"and this would be the interval of convergence."},{"Start":"05:29.090 ","End":"05:31.120","Text":"This is its power series."},{"Start":"05:31.120 ","End":"05:34.905","Text":"Then when we multiply by x cubed."},{"Start":"05:34.905 ","End":"05:41.910","Text":"X cubed, just multiplying by a function doesn\u0027t change its radius of convergence."},{"Start":"05:41.910 ","End":"05:43.520","Text":"By multiplying by x cubed,"},{"Start":"05:43.520 ","End":"05:44.750","Text":"it\u0027s still going to converge."},{"Start":"05:44.750 ","End":"05:47.540","Text":"I\u0027m just going to multiply it by x cubed,"},{"Start":"05:47.540 ","End":"05:53.630","Text":"I would say that x cubed times natural log of 1 plus 2x is equal to"},{"Start":"05:53.630 ","End":"06:02.640","Text":"the same thing times 2x to the n over n. I just have to multiply it by x cubed."},{"Start":"06:03.460 ","End":"06:06.990","Text":"Sorry, this will be 2^nx^n."},{"Start":"06:09.140 ","End":"06:11.420","Text":"When I multiply by x cubed,"},{"Start":"06:11.420 ","End":"06:15.365","Text":"I could just add 3 here because I multiply each term by x cubed,"},{"Start":"06:15.365 ","End":"06:17.620","Text":"which means raising the power by 3,"},{"Start":"06:17.620 ","End":"06:21.230","Text":"and I would say that still the radius of convergence is the"},{"Start":"06:21.230 ","End":"06:25.325","Text":"same as this minus 1/2 less than x,"},{"Start":"06:25.325 ","End":"06:28.800","Text":"less than or equal 2, plus 1/2."},{"Start":"06:28.800 ","End":"06:32.565","Text":"That\u0027s the kind of question you might be asked."},{"Start":"06:32.565 ","End":"06:36.160","Text":"We\u0027re done with this clip."}],"ID":10342},{"Watched":false,"Name":"Computations with Taylor Series and a given precision","Duration":"9m 41s","ChapterTopicVideoID":10121,"CourseChapterTopicPlaylistID":84684,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10121.jpeg","UploadDate":"2020-01-16T13:04:22.3270000","DurationForVideoObject":"PT9M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In the exercises following this clip,"},{"Start":"00:02.490 ","End":"00:07.140","Text":"you\u0027ll be asked to do all kinds of computations involving Taylor series."},{"Start":"00:07.140 ","End":"00:12.660","Text":"Typical question would be something like compute the value of e to within"},{"Start":"00:12.660 ","End":"00:21.390","Text":"3 decimal places or compute 1/e to an accuracy of 0.002 or something."},{"Start":"00:21.390 ","End":"00:24.270","Text":"That\u0027s the kind of exercise we\u0027ll be doing."},{"Start":"00:24.270 ","End":"00:27.700","Text":"It\u0027s making a computation using Taylor series,"},{"Start":"00:27.700 ","End":"00:31.020","Text":"and when you want to compute a value to a given precision."},{"Start":"00:31.020 ","End":"00:33.630","Text":"The main theorem we\u0027re going to be using is"},{"Start":"00:33.630 ","End":"00:38.250","Text":"the following Leibnig alternating series test."},{"Start":"00:38.250 ","End":"00:41.000","Text":"I just want to say that a Leibnig series and"},{"Start":"00:41.000 ","End":"00:44.340","Text":"an alternating series generally the same thing."},{"Start":"00:44.560 ","End":"00:49.830","Text":"What we have is we\u0027re given a Leibnig series."},{"Start":"00:50.090 ","End":"00:52.700","Text":"Given the Leibnig series,"},{"Start":"00:52.700 ","End":"00:54.574","Text":"which is an alternating series,"},{"Start":"00:54.574 ","End":"01:02.335","Text":"the following sum to infinity of minus 1 to the n makes the alternation and a_n."},{"Start":"01:02.335 ","End":"01:05.600","Text":"Well, of course, a_n has to be bigger than 0 as part of"},{"Start":"01:05.600 ","End":"01:08.795","Text":"the definition of an alternating series for it\u0027s redundant."},{"Start":"01:08.795 ","End":"01:10.460","Text":"By the way don\u0027t have to start from 1."},{"Start":"01:10.460 ","End":"01:13.295","Text":"We can start from anywhere as long as we go to infinity."},{"Start":"01:13.295 ","End":"01:17.555","Text":"Because with a series the beginning is never important, only the end."},{"Start":"01:17.555 ","End":"01:23.180","Text":"It\u0027s non-negative and want it to be decreasing,"},{"Start":"01:23.180 ","End":"01:34.130","Text":"a_n is a decreasing series and also want a_n to tend to 0 as n goes to infinity."},{"Start":"01:36.500 ","End":"01:39.870","Text":"Given all those conditions,"},{"Start":"01:39.870 ","End":"01:46.785","Text":"the error in summing the first n terms to estimate s. I\u0027ll stop here in a minute."},{"Start":"01:46.785 ","End":"01:52.795","Text":"Usually, what we do here is we can\u0027t sum it to infinity."},{"Start":"01:52.795 ","End":"01:56.500","Text":"We estimate the sum by adding the first, I don\u0027t know,"},{"Start":"01:56.500 ","End":"02:01.170","Text":"a 100 terms or 5 terms, whatever."},{"Start":"02:01.170 ","End":"02:06.440","Text":"Then we use that partial sum of the first terms to estimate the total sum."},{"Start":"02:06.440 ","End":"02:09.635","Text":"But we need to know how much the error is in doing that."},{"Start":"02:09.635 ","End":"02:12.695","Text":"If we use n terms to estimate,"},{"Start":"02:12.695 ","End":"02:16.100","Text":"then the error between the estimated value and"},{"Start":"02:16.100 ","End":"02:21.815","Text":"the actual value is less than a_n plus 1 in absolute value."},{"Start":"02:21.815 ","End":"02:24.290","Text":"In other words, if I take it, make it positive,"},{"Start":"02:24.290 ","End":"02:26.120","Text":"the error is negative and make it positive."},{"Start":"02:26.120 ","End":"02:28.910","Text":"It\u0027s still less than a_n plus 1."},{"Start":"02:28.910 ","End":"02:33.290","Text":"I\u0027m going to explain this more because it\u0027s important to understand this theorem."},{"Start":"02:33.290 ","End":"02:36.950","Text":"It\u0027s the key to solving the exercises."},{"Start":"02:36.950 ","End":"02:39.155","Text":"Let me explain further."},{"Start":"02:39.155 ","End":"02:46.540","Text":"Suppose I want to estimate as part of the exercise, 1/e."},{"Start":"02:46.540 ","End":"02:53.585","Text":"Later we\u0027ll introduce the matter of precision and 1/e is e to the minus 1."},{"Start":"02:53.585 ","End":"02:58.510","Text":"Now I want to get a series to converge to 1/e,"},{"Start":"02:58.510 ","End":"03:03.560","Text":"and that series is going to be the Taylor series evaluated at some point."},{"Start":"03:03.560 ","End":"03:08.600","Text":"Now, e^x in general is"},{"Start":"03:08.600 ","End":"03:13.790","Text":"1 plus x plus x^2/2 factorial"},{"Start":"03:13.790 ","End":"03:21.480","Text":"plus x^3/3 factorial plus x^4/4 factorial, et cetera."},{"Start":"03:21.480 ","End":"03:25.905","Text":"If we substitute x equals minus 1,"},{"Start":"03:25.905 ","End":"03:30.845","Text":"then we get that e^-1 is equal."},{"Start":"03:30.845 ","End":"03:35.515","Text":"We get the infinite series 1 minus x."},{"Start":"03:35.515 ","End":"03:40.490","Text":"What\u0027s going to happen is this is going to become alternating because all the odd powers,"},{"Start":"03:40.490 ","End":"03:44.599","Text":"if you put x is minus 1 will be negative and the even powers will be positive."},{"Start":"03:44.599 ","End":"03:49.340","Text":"So plus x^2/2 factorial minus"},{"Start":"03:49.340 ","End":"03:55.745","Text":"x^3/3 factorial plus x^4/4 factorial,"},{"Start":"03:55.745 ","End":"03:58.775","Text":"et cetera, et cetera, et cetera."},{"Start":"03:58.775 ","End":"04:04.580","Text":"Now what we have is a series that meets all these conditions for this test."},{"Start":"04:04.580 ","End":"04:10.899","Text":"Because well, we know that this series converges."},{"Start":"04:10.899 ","End":"04:14.500","Text":"If it converges, then of course the general term goes to 0."},{"Start":"04:14.500 ","End":"04:16.345","Text":"That\u0027s a necessary condition."},{"Start":"04:16.345 ","End":"04:19.010","Text":"It is decreasing."},{"Start":"04:20.100 ","End":"04:23.600","Text":"Now, just start this again."},{"Start":"04:28.720 ","End":"04:34.715","Text":"If we let x equals minus 1 to give us 1/e,"},{"Start":"04:34.715 ","End":"04:37.520","Text":"Then we get the series for 1/e,"},{"Start":"04:37.520 ","End":"04:46.620","Text":"which is 1 minus 1 plus 1/2 factorial,"},{"Start":"04:46.840 ","End":"04:54.020","Text":"minus 1/3 factorial plus 1/4 factorial,"},{"Start":"04:54.020 ","End":"04:56.880","Text":"et cetera, et cetera."},{"Start":"04:57.220 ","End":"05:06.935","Text":"I could write this as the sum and goes from 0 to infinity of plus or minus 1,"},{"Start":"05:06.935 ","End":"05:15.740","Text":"sorry, minus 1 to the n. That\u0027s what gives me my plus or minus times a_n."},{"Start":"05:15.740 ","End":"05:19.685","Text":"Where a_n is just the factorial,"},{"Start":"05:19.685 ","End":"05:22.640","Text":"a_n is 1 over n factorial."},{"Start":"05:22.640 ","End":"05:28.610","Text":"When n is 3, I get minus 1 to the 3 is minus, minus 1/3 factorial."},{"Start":"05:28.610 ","End":"05:30.875","Text":"This is a general term."},{"Start":"05:30.875 ","End":"05:35.630","Text":"I wrote it in this form because I wanted to check if the a_n"},{"Start":"05:35.630 ","End":"05:40.245","Text":"satisfies the alternating series test and they do."},{"Start":"05:40.245 ","End":"05:44.340","Text":"First of all, a_n\u0027s are all non-negative."},{"Start":"05:44.340 ","End":"05:48.355","Text":"1 factorial is positive or non-negative."},{"Start":"05:48.355 ","End":"05:51.210","Text":"They decrease n gets bigger,"},{"Start":"05:51.210 ","End":"05:52.530","Text":"n factorial gets bigger,"},{"Start":"05:52.530 ","End":"05:54.255","Text":"and so a_n gets smaller,"},{"Start":"05:54.255 ","End":"05:57.740","Text":"and it does tend to 0 because n goes to infinity and"},{"Start":"05:57.740 ","End":"06:02.315","Text":"factorial goes even faster to infinity and 1 over infinity is 0."},{"Start":"06:02.315 ","End":"06:10.360","Text":"It meets all the conditions in the Leibnig alternating series test,"},{"Start":"06:10.360 ","End":"06:13.055","Text":"I can also take the conclusions,"},{"Start":"06:13.055 ","End":"06:16.760","Text":"which is that if I take only n terms,"},{"Start":"06:16.760 ","End":"06:18.970","Text":"the sum of the terms up to"},{"Start":"06:18.970 ","End":"06:23.220","Text":"a_n to estimate then the error is going to be less than a_n plus 1."},{"Start":"06:23.220 ","End":"06:24.750","Text":"Let me show what this means."},{"Start":"06:24.750 ","End":"06:26.720","Text":"I\u0027ll give a couple of examples."},{"Start":"06:26.720 ","End":"06:33.299","Text":"Suppose I want to take s_3,"},{"Start":"06:33.299 ","End":"06:35.490","Text":"the sum of up to 3."},{"Start":"06:35.490 ","End":"06:36.960","Text":"I\u0027ll start the counting at 0."},{"Start":"06:36.960 ","End":"06:38.190","Text":"This is 0, 1, 2,"},{"Start":"06:38.190 ","End":"06:46.215","Text":"3. s_3 would be 1 minus 1 plus 1/2 minus 6,"},{"Start":"06:46.215 ","End":"06:52.090","Text":"and that is equal to 1 third."},{"Start":"06:52.760 ","End":"06:56.945","Text":"When I use a 1 third to estimate 1/e,"},{"Start":"06:56.945 ","End":"07:01.160","Text":"then the error, according to the theorem,"},{"Start":"07:01.160 ","End":"07:04.830","Text":"is less than a_n plus 1."},{"Start":"07:05.780 ","End":"07:11.340","Text":"If the error would be less than n plus 1 in this case is 4,"},{"Start":"07:11.340 ","End":"07:15.570","Text":"is less than 4 which is 1/24."},{"Start":"07:15.570 ","End":"07:18.540","Text":"Yeah, there\u0027s 24. 4 factorial is 24,"},{"Start":"07:18.540 ","End":"07:21.935","Text":"5 factorial is a 120, and so on."},{"Start":"07:21.935 ","End":"07:26.585","Text":"But if I took 1/24 is good but not great."},{"Start":"07:26.585 ","End":"07:28.190","Text":"If I take another term,"},{"Start":"07:28.190 ","End":"07:35.250","Text":"as 4 it\u0027s 1 minus 1 plus 1/2 minus 1/6 plus 1/24."},{"Start":"07:35.250 ","End":"07:38.415","Text":"This equals, this."},{"Start":"07:38.415 ","End":"07:44.940","Text":"Plus 124 comes out to be 17/24,"},{"Start":"07:44.940 ","End":"07:48.005","Text":"if I estimate 1/e at this and compute it,"},{"Start":"07:48.005 ","End":"07:56.130","Text":"then the error is guaranteed to be less than 1 over the next term,"},{"Start":"07:56.130 ","End":"07:59.730","Text":"which is 5 factorial is 120,"},{"Start":"07:59.730 ","End":"08:01.860","Text":"so it\u0027s less than 1/120."},{"Start":"08:01.860 ","End":"08:05.270","Text":"For example, it\u0027s already good within 2 decimal places because it\u0027s less"},{"Start":"08:05.270 ","End":"08:08.805","Text":"than a 100th and so on."},{"Start":"08:08.805 ","End":"08:10.640","Text":"The more terms you take,"},{"Start":"08:10.640 ","End":"08:13.940","Text":"the smaller the error is going to be because the terms are tending to"},{"Start":"08:13.940 ","End":"08:17.715","Text":"0 and they\u0027re decreasing to 0."},{"Start":"08:17.715 ","End":"08:26.360","Text":"That\u0027s basically how we use Leibnig test to do estimations and to estimate the error."},{"Start":"08:26.360 ","End":"08:29.480","Text":"When I say error, I mean absolute value of the error,"},{"Start":"08:29.480 ","End":"08:32.870","Text":"which means 1/e minus 1/24."},{"Start":"08:32.870 ","End":"08:36.200","Text":"For example, you just compare the 2 and take"},{"Start":"08:36.200 ","End":"08:40.200","Text":"the absolute value of the difference or 1/e compared to,"},{"Start":"08:40.200 ","End":"08:42.750","Text":"I meant 17 over."},{"Start":"08:42.750 ","End":"08:46.680","Text":"No, sorry. I meant 1 over."},{"Start":"08:46.680 ","End":"08:50.920","Text":"Sorry. If I compare,"},{"Start":"08:50.920 ","End":"08:55.570","Text":"let me just write down the estimates."},{"Start":"08:55.570 ","End":"08:59.355","Text":"This is 1/e is what we\u0027re trying to estimate,"},{"Start":"08:59.355 ","End":"09:08.470","Text":"and we ought 2 estimates in 1 is the estimate is 1/3,"},{"Start":"09:08.470 ","End":"09:14.045","Text":"and the other estimate we have is 17/24."},{"Start":"09:14.045 ","End":"09:17.015","Text":"Now if I take the difference between this and this,"},{"Start":"09:17.015 ","End":"09:20.210","Text":"it\u0027s going to be less than and probably a lot less than 1 over 24,"},{"Start":"09:20.210 ","End":"09:23.375","Text":"and if I take the difference between this and this an absolute value,"},{"Start":"09:23.375 ","End":"09:25.175","Text":"It\u0027s going to be less than a 100,"},{"Start":"09:25.175 ","End":"09:29.300","Text":"less than a 120, quite probably a lot less."},{"Start":"09:30.730 ","End":"09:35.045","Text":"I\u0027m done here and remember to do"},{"Start":"09:35.045 ","End":"09:40.890","Text":"the exercises following this as many as you can. That\u0027s all."}],"ID":10343}],"Thumbnail":null,"ID":84684},{"Name":"Basic Exercises with Maclaurin Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"3m 5s","ChapterTopicVideoID":6077,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6077.jpeg","UploadDate":"2016-05-08T11:08:29.8070000","DurationForVideoObject":"PT3M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have to find the Taylor series of"},{"Start":"00:03.570 ","End":"00:07.550","Text":"the function f of x equals sine of 2x."},{"Start":"00:07.550 ","End":"00:09.825","Text":"But we\u0027re not going to do it from scratch."},{"Start":"00:09.825 ","End":"00:14.430","Text":"Remember Taylor around x equals 0 is a Maclaurin series."},{"Start":"00:14.430 ","End":"00:15.975","Text":"So we go to the appendix,"},{"Start":"00:15.975 ","End":"00:20.025","Text":"find the closest 1 to this that we have,"},{"Start":"00:20.025 ","End":"00:26.380","Text":"and we find this 1 here which I copy pasted."},{"Start":"00:26.780 ","End":"00:32.005","Text":"What we have to do is just replace x with 2x."},{"Start":"00:32.005 ","End":"00:38.175","Text":"So we see that here\u0027s x and here\u0027s x and I\u0027ll use the Sigma form."},{"Start":"00:38.175 ","End":"00:40.710","Text":"Never mind about this part."},{"Start":"00:40.710 ","End":"00:46.100","Text":"So all we have to do is take this and instead of x, put 2x."},{"Start":"00:46.100 ","End":"00:52.730","Text":"So I get the sine of 2x is equal to,"},{"Start":"00:52.730 ","End":"00:59.855","Text":"I\u0027m just copying this at first and equals 0 to infinity minus 1 to the n."},{"Start":"00:59.855 ","End":"01:04.415","Text":"But instead of x, I\u0027m now going to put 2x."},{"Start":"01:04.415 ","End":"01:06.755","Text":"I need to put brackets here though,"},{"Start":"01:06.755 ","End":"01:16.195","Text":"to the power of 2_n plus 1 over 2_n plus 1 factorial."},{"Start":"01:16.195 ","End":"01:18.660","Text":"So here I had x,"},{"Start":"01:18.660 ","End":"01:21.915","Text":"now I have 2_x instead."},{"Start":"01:21.915 ","End":"01:25.120","Text":"Let\u0027s just tidy up a bit."},{"Start":"01:25.360 ","End":"01:28.880","Text":"I\u0027ll write it as the sum,"},{"Start":"01:28.880 ","End":"01:33.110","Text":"as n goes from 0 to infinity minus 1 to the n."},{"Start":"01:33.110 ","End":"01:40.189","Text":"I\u0027ll take the constant 2 and its exponent outside the brackets or the division."},{"Start":"01:40.189 ","End":"01:47.690","Text":"I\u0027ll put here 2 to the power of 2_n plus 1."},{"Start":"01:47.690 ","End":"01:55.650","Text":"Then x to the power of 2_n plus 1 over 2_n plus 1 factorial"},{"Start":"01:55.650 ","End":"01:58.175","Text":"and we\u0027re almost done."},{"Start":"01:58.175 ","End":"02:00.470","Text":"I mean, technically we are done,"},{"Start":"02:00.470 ","End":"02:04.100","Text":"but we have to take care of 1 other technical matter,"},{"Start":"02:04.100 ","End":"02:06.200","Text":"the radius of convergence."},{"Start":"02:06.200 ","End":"02:09.064","Text":"It doesn\u0027t say so specifically in the question,"},{"Start":"02:09.064 ","End":"02:10.850","Text":"but it\u0027s a good idea to relate to it."},{"Start":"02:10.850 ","End":"02:15.660","Text":"Now, this series holds for all x and"},{"Start":"02:15.660 ","End":"02:21.845","Text":"so this 1 also would hold for all x."},{"Start":"02:21.845 ","End":"02:25.279","Text":"Now if you want to get more technical,"},{"Start":"02:25.279 ","End":"02:31.500","Text":"we sometimes write it as x between minus infinity and infinity."},{"Start":"02:31.970 ","End":"02:39.125","Text":"What I would normally do if it wasn\u0027t infinity would also be to replace x by 2_x here."},{"Start":"02:39.125 ","End":"02:43.210","Text":"So I have minus infinity less than"},{"Start":"02:43.210 ","End":"02:47.240","Text":"2_x less than infinity for the radius of convergence of this 1."},{"Start":"02:47.240 ","End":"02:48.905","Text":"Then you divide by 2."},{"Start":"02:48.905 ","End":"02:53.225","Text":"Since infinity over 2 is infinity,"},{"Start":"02:53.225 ","End":"02:55.525","Text":"so I also get,"},{"Start":"02:55.525 ","End":"02:57.355","Text":"I\u0027ve just highlighted that,"},{"Start":"02:57.355 ","End":"03:01.190","Text":"we also get that this is true for all x if you want to put it,"},{"Start":"03:01.190 ","End":"03:06.150","Text":"in other words, and that\u0027s it. We\u0027re done."}],"ID":6085},{"Watched":false,"Name":"Exercise 2","Duration":"3m 31s","ChapterTopicVideoID":6078,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6078.jpeg","UploadDate":"2016-05-08T11:08:56.6970000","DurationForVideoObject":"PT3M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this exercise, we have to find the Taylor series of the function"},{"Start":"00:04.335 ","End":"00:09.375","Text":"x squared e to the minus 4x around x equals 0."},{"Start":"00:09.375 ","End":"00:13.260","Text":"Remember that Taylor about 0 is the same as Maclaurin."},{"Start":"00:13.260 ","End":"00:15.810","Text":"We don\u0027t do these exercises from scratch."},{"Start":"00:15.810 ","End":"00:18.495","Text":"We go to the appendix and find something similar."},{"Start":"00:18.495 ","End":"00:21.405","Text":"In this case, we find e to the x,"},{"Start":"00:21.405 ","End":"00:24.390","Text":"which I copy pasted here."},{"Start":"00:24.390 ","End":"00:27.570","Text":"We have both the Sigma form and the expanded form."},{"Start":"00:27.570 ","End":"00:29.775","Text":"I\u0027ll use the Sigma form."},{"Start":"00:29.775 ","End":"00:35.895","Text":"What we have to do is use this to work our way up to this."},{"Start":"00:35.895 ","End":"00:37.770","Text":"We\u0027re going to do it in 2 stages."},{"Start":"00:37.770 ","End":"00:40.515","Text":"The first thing is to take care of the minus 4 here."},{"Start":"00:40.515 ","End":"00:43.310","Text":"Wherever we see x which is here and here,"},{"Start":"00:43.310 ","End":"00:46.945","Text":"I can replace that by minus 4x."},{"Start":"00:46.945 ","End":"00:54.869","Text":"That will give me that e to the power minus 4x is equal to the sum"},{"Start":"00:54.869 ","End":"00:59.330","Text":"n goes from 0 to infinity of minus 4x"},{"Start":"00:59.330 ","End":"01:06.510","Text":"and it brackets to the power of n over n factorial."},{"Start":"01:07.060 ","End":"01:11.615","Text":"Now, there\u0027s also the technical matter of radius of convergence,"},{"Start":"01:11.615 ","End":"01:15.485","Text":"the e to the x, and it says so in the table is for all x."},{"Start":"01:15.485 ","End":"01:20.735","Text":"In other words, we write that sometimes as x between minus infinity and infinity,"},{"Start":"01:20.735 ","End":"01:23.075","Text":"same thing as all x."},{"Start":"01:23.075 ","End":"01:27.170","Text":"That\u0027s the radius of convergence for x."},{"Start":"01:27.170 ","End":"01:30.080","Text":"Now, if it wasn\u0027t for minus infinity to infinity,"},{"Start":"01:30.080 ","End":"01:32.240","Text":"it would be important to write"},{"Start":"01:32.240 ","End":"01:37.145","Text":"or to compute the same thing now that we have minus 4x,"},{"Start":"01:37.145 ","End":"01:41.970","Text":"so minus 4x is between minus infinity and infinity."},{"Start":"01:42.200 ","End":"01:45.950","Text":"You can see that if you divide everything by minus 4,"},{"Start":"01:45.950 ","End":"01:49.310","Text":"but then you also have to change directions,"},{"Start":"01:49.310 ","End":"01:54.055","Text":"we just end up with same thing."},{"Start":"01:54.055 ","End":"01:56.580","Text":"If I divide this by minus 4,"},{"Start":"01:56.580 ","End":"01:57.780","Text":"I get minus infinity."},{"Start":"01:57.780 ","End":"01:59.600","Text":"Here I get infinity because it\u0027s a minus,"},{"Start":"01:59.600 ","End":"02:00.920","Text":"so I also have to change."},{"Start":"02:00.920 ","End":"02:05.364","Text":"I\u0027ll just end up with this or it\u0027s still true for all x."},{"Start":"02:05.364 ","End":"02:09.895","Text":"Let\u0027s just simplify it by taking the minus 4 out."},{"Start":"02:09.895 ","End":"02:16.525","Text":"I have the sum also from 0 to infinity."},{"Start":"02:16.525 ","End":"02:18.130","Text":"I\u0027ll take the minus 4 out."},{"Start":"02:18.130 ","End":"02:21.535","Text":"In fact, I\u0027ll even split the minus 4 as minus 1 times 4."},{"Start":"02:21.535 ","End":"02:25.840","Text":"I have minus 1 to the n, and then 4 to the n,"},{"Start":"02:25.840 ","End":"02:30.790","Text":"and then x to the n over n factorial."},{"Start":"02:30.790 ","End":"02:32.830","Text":"That\u0027s just this bit."},{"Start":"02:32.830 ","End":"02:37.510","Text":"But our function f of x has also an x squared in it."},{"Start":"02:37.510 ","End":"02:40.565","Text":"If I just multiply this by x squared,"},{"Start":"02:40.565 ","End":"02:43.635","Text":"I\u0027ll do x squared and I\u0027ll just copy paste it."},{"Start":"02:43.635 ","End":"02:49.800","Text":"There we are, just to make it a bit further, and that\u0027s better."},{"Start":"02:49.800 ","End":"02:52.255","Text":"I\u0027ll just separate it off."},{"Start":"02:52.255 ","End":"02:57.250","Text":"Now what we have to do is take the x squared and put it inside the Sigma,"},{"Start":"02:57.250 ","End":"03:01.335","Text":"which means I can just raise every n by 2."},{"Start":"03:01.335 ","End":"03:08.070","Text":"We end up with getting the sum n goes from 0 to infinity"},{"Start":"03:08.070 ","End":"03:09.990","Text":"minus 1 to the n, 4 to the n,"},{"Start":"03:09.990 ","End":"03:12.955","Text":"and the x squared I\u0027m putting in here,"},{"Start":"03:12.955 ","End":"03:19.940","Text":"x squared times x to the n is x to the n plus 2 over n factorial."},{"Start":"03:20.150 ","End":"03:23.160","Text":"This is our answer,"},{"Start":"03:23.160 ","End":"03:30.010","Text":"and it\u0027s true for all x because the x squared is defined for all x."}],"ID":6086},{"Watched":false,"Name":"Exercise 3","Duration":"6m 59s","ChapterTopicVideoID":6079,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6079.jpeg","UploadDate":"2016-05-08T11:09:51.8700000","DurationForVideoObject":"PT6M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, you have to find the Taylor series of"},{"Start":"00:03.210 ","End":"00:08.040","Text":"the hyperbolic sine function around x equals 0."},{"Start":"00:08.040 ","End":"00:15.810","Text":"But if we look up the function f of x equals sine hyperbolic of x in our little appendix,"},{"Start":"00:15.810 ","End":"00:17.295","Text":"we don\u0027t find it,"},{"Start":"00:17.295 ","End":"00:19.185","Text":"and so what to do?"},{"Start":"00:19.185 ","End":"00:25.125","Text":"But then luckily, we remember that the definition of sine hyperbolic of x,"},{"Start":"00:25.125 ","End":"00:27.780","Text":"this is equal to by definition,"},{"Start":"00:27.780 ","End":"00:35.805","Text":"e^x minus e^minus x over 2 or 1/2 times this."},{"Start":"00:35.805 ","End":"00:46.130","Text":"We do have the Maclaurin series for e^power of x and here it is both in Sigma form"},{"Start":"00:46.130 ","End":"00:50.090","Text":"and in the dot-dot-dot expanded form and this is true for"},{"Start":"00:50.090 ","End":"00:56.360","Text":"all x or sometimes between minus infinity and infinity."},{"Start":"00:56.360 ","End":"01:00.305","Text":"We have to use this to somehow get this."},{"Start":"01:00.305 ","End":"01:05.735","Text":"Shouldn\u0027t be too difficult because e^x we have expanded,"},{"Start":"01:05.735 ","End":"01:11.060","Text":"e^minus x will get by substituting minus x for x and we can add and"},{"Start":"01:11.060 ","End":"01:16.740","Text":"subtract series term wise and also multiply by a constant or divide by a constant."},{"Start":"01:16.740 ","End":"01:21.320","Text":"Let\u0027s see from this we\u0027re going to need e^minus x."},{"Start":"01:21.320 ","End":"01:27.970","Text":"For e^minus x, take this formula and where we see x instead of it,"},{"Start":"01:27.970 ","End":"01:30.620","Text":"we\u0027ll put minus x."},{"Start":"01:30.800 ","End":"01:36.685","Text":"We\u0027ll get the sum n goes from 0 to infinity"},{"Start":"01:36.685 ","End":"01:45.010","Text":"of minus x^n over n factorial."},{"Start":"01:45.010 ","End":"01:54.720","Text":"What I want do is take out the minus 1 outside the brackets so we get"},{"Start":"01:54.720 ","End":"02:00.510","Text":"the sum from 0 to infinity minus"},{"Start":"02:00.510 ","End":"02:07.750","Text":"1^n times x^n over n factorial."},{"Start":"02:08.930 ","End":"02:12.110","Text":"Now let\u0027s do the subtraction and at the end,"},{"Start":"02:12.110 ","End":"02:13.700","Text":"we\u0027ll divide by 2."},{"Start":"02:13.700 ","End":"02:19.510","Text":"What we have is that not the final answer,"},{"Start":"02:19.510 ","End":"02:27.130","Text":"but e^x minus e^minus x is going to"},{"Start":"02:27.130 ","End":"02:33.600","Text":"equal the sum from"},{"Start":"02:33.600 ","End":"02:39.485","Text":"0 to infinity of this here,"},{"Start":"02:39.485 ","End":"02:41.780","Text":"which I\u0027ll just write over here."},{"Start":"02:41.780 ","End":"02:47.075","Text":"X^n over n factorial minus"},{"Start":"02:47.075 ","End":"02:52.730","Text":"the sum from 0 to infinity of this one,"},{"Start":"02:52.730 ","End":"02:59.225","Text":"which is minus 1^n,"},{"Start":"02:59.225 ","End":"03:02.020","Text":"x^n over n factorial."},{"Start":"03:02.020 ","End":"03:07.695","Text":"Basically, if I combine these 2 into 1 series,"},{"Start":"03:07.695 ","End":"03:16.340","Text":"I\u0027ve got the sum from n goes from 0 to infinity of the x^n over n factorials in"},{"Start":"03:16.340 ","End":"03:26.880","Text":"common and what I have is 1 minus minus 1^n times x^n over n factorial."},{"Start":"03:27.300 ","End":"03:30.235","Text":"Let me just write this at the side."},{"Start":"03:30.235 ","End":"03:36.940","Text":"We have the 1 minus, minus 1^n is equal to."},{"Start":"03:36.940 ","End":"03:38.755","Text":"We said that if n is even,"},{"Start":"03:38.755 ","End":"03:40.945","Text":"we get 1 minus 1 is 0."},{"Start":"03:40.945 ","End":"03:45.759","Text":"That\u0027s n even and if n is odd,"},{"Start":"03:45.759 ","End":"03:48.940","Text":"then minus 1 to an odd power is minus 1."},{"Start":"03:48.940 ","End":"03:54.580","Text":"1 minus minus 1 is 2 if n is odd."},{"Start":"03:54.580 ","End":"03:59.185","Text":"Now I\u0027m going to write something that\u0027s not quite precise and then we\u0027ll fix it."},{"Start":"03:59.185 ","End":"04:08.215","Text":"We\u0027re going to say that this is equal to the sum from 0 to infinity, but n odd,"},{"Start":"04:08.215 ","End":"04:10.885","Text":"it\u0027s not conventional notation, I\u0027ll fix that,"},{"Start":"04:10.885 ","End":"04:17.845","Text":"of 2 times x^n over n factorial."},{"Start":"04:17.845 ","End":"04:22.285","Text":"But how do I make this more precise and say n is odd?"},{"Start":"04:22.285 ","End":"04:24.675","Text":"Well, the idea is this,"},{"Start":"04:24.675 ","End":"04:26.440","Text":"if n is odd,"},{"Start":"04:26.440 ","End":"04:31.765","Text":"and I\u0027ll do something at the side here and is odd means n is 1,"},{"Start":"04:31.765 ","End":"04:37.450","Text":"3 ,5 ,7, and so on."},{"Start":"04:37.450 ","End":"04:40.150","Text":"But in a Sigma we can\u0027t have jumps of 2."},{"Start":"04:40.150 ","End":"04:45.200","Text":"So the trick is to write n equals 2k plus 1,"},{"Start":"04:45.200 ","End":"04:48.115","Text":"where k goes regular,"},{"Start":"04:48.115 ","End":"04:52.150","Text":"0, 1, 2, 3, etc."},{"Start":"04:52.150 ","End":"04:54.780","Text":"Notice that this works if k is 0,"},{"Start":"04:54.780 ","End":"04:56.715","Text":"twice 0 plus 1 is this."},{"Start":"04:56.715 ","End":"04:58.380","Text":"Twice 1 plus 1 is this,"},{"Start":"04:58.380 ","End":"04:59.910","Text":"twice 2 plus 1 is this."},{"Start":"04:59.910 ","End":"05:05.395","Text":"Twice 3 plus 1 is 7 and then if we do this substitution,"},{"Start":"05:05.395 ","End":"05:08.000","Text":"then we can get this."},{"Start":"05:10.800 ","End":"05:21.470","Text":"This will equal the sum k equals 0 to"},{"Start":"05:21.470 ","End":"05:28.470","Text":"infinity and I can take the 2 out front twice and it\u0027s going to"},{"Start":"05:28.470 ","End":"05:37.940","Text":"be x^2k plus 1 over 2k plus 1 factorial."},{"Start":"05:37.940 ","End":"05:40.880","Text":"There\u0027s only 1 thing we haven\u0027t taken care of."},{"Start":"05:40.880 ","End":"05:46.025","Text":"We\u0027ve done this, our function also has divided by 2."},{"Start":"05:46.025 ","End":"05:55.775","Text":"From here we can get that sine hyperbolic of x is equal to just without the 2."},{"Start":"05:55.775 ","End":"06:01.170","Text":"The sum, k goes from 0 to infinity."},{"Start":"06:02.950 ","End":"06:08.640","Text":"X^2k plus 1 over 2k plus"},{"Start":"06:08.640 ","End":"06:14.810","Text":"1 factorial and that\u0027s the answer."},{"Start":"06:14.810 ","End":"06:19.730","Text":"But I\u0027d like to say that we have a letter K and we usually use n"},{"Start":"06:19.730 ","End":"06:24.650","Text":"so it wouldn\u0027t hurt if I just now go and change k back to n everywhere,"},{"Start":"06:24.650 ","End":"06:33.090","Text":"which is what I\u0027m going to do now and this is how you will see it in the books."},{"Start":"06:36.580 ","End":"06:39.740","Text":"Well, I started to say this is for all x."},{"Start":"06:39.740 ","End":"06:41.660","Text":"So obviously with minus x,"},{"Start":"06:41.660 ","End":"06:44.490","Text":"it\u0027s also going to be true for all x and"},{"Start":"06:44.490 ","End":"06:47.660","Text":"dividing by 2 is not going to affect the radius of convergence."},{"Start":"06:47.660 ","End":"06:51.260","Text":"Basically, this is true for all x"},{"Start":"06:51.260 ","End":"06:58.770","Text":"and I\u0027ll just highlight it before declaring that we are done."}],"ID":6087},{"Watched":false,"Name":"Exercise 4","Duration":"7m 17s","ChapterTopicVideoID":6071,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6071.jpeg","UploadDate":"2016-05-08T11:03:39.4470000","DurationForVideoObject":"PT7M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.700","Text":"We have to find the Taylor series of the function sine squared of x around x equals 0,"},{"Start":"00:08.700 ","End":"00:12.150","Text":"which means that it\u0027s effectively a Maclaurin series."},{"Start":"00:12.150 ","End":"00:14.520","Text":"Now in the appendix,"},{"Start":"00:14.520 ","End":"00:17.430","Text":"there is no sine squared x,"},{"Start":"00:17.430 ","End":"00:19.515","Text":"so we have to find something close."},{"Start":"00:19.515 ","End":"00:22.140","Text":"You might think sine x,"},{"Start":"00:22.140 ","End":"00:25.965","Text":"but that\u0027s going to be difficult to square a power series."},{"Start":"00:25.965 ","End":"00:28.170","Text":"What we do instead, it\u0027s a standard trick,"},{"Start":"00:28.170 ","End":"00:30.945","Text":"is to use trigonometrical identities."},{"Start":"00:30.945 ","End":"00:34.795","Text":"I\u0027m actually going to use the cosine, as you\u0027ll see."},{"Start":"00:34.795 ","End":"00:41.655","Text":"From trigonometry, there\u0027s a formula that sine squared of Alpha"},{"Start":"00:41.655 ","End":"00:51.420","Text":"is 1/2 of 1 minus cosine of 2 Alpha in general."},{"Start":"00:51.420 ","End":"00:54.970","Text":"In our case, we\u0027ll take Alpha as x,"},{"Start":"00:55.130 ","End":"01:04.210","Text":"and so this will equal 1/2 of 1 minus cosine of 2x,"},{"Start":"01:04.210 ","End":"01:08.390","Text":"and we\u0027ll start with cosine x,"},{"Start":"01:08.390 ","End":"01:12.485","Text":"for which we do have a Taylor series, and here it is."},{"Start":"01:12.485 ","End":"01:20.895","Text":"It gives cosine x as a Maclaurin series in the Sigma form,"},{"Start":"01:20.895 ","End":"01:24.210","Text":"and expanded with dot dot dot,"},{"Start":"01:24.210 ","End":"01:26.620","Text":"we use the Sigma form."},{"Start":"01:26.840 ","End":"01:32.925","Text":"We\u0027ll build up to this f of x in 2 or 3 stages."},{"Start":"01:32.925 ","End":"01:38.850","Text":"In the first stage I\u0027ll do cosine 2x and then we\u0027ll worry about the 1 minus,"},{"Start":"01:38.850 ","End":"01:43.125","Text":"and multiplying by a 1/2, that\u0027s fairly straightforward."},{"Start":"01:43.125 ","End":"01:51.610","Text":"If I replace x here and here with 2x,"},{"Start":"01:51.800 ","End":"01:57.600","Text":"then I get that cosine of"},{"Start":"01:57.600 ","End":"02:04.275","Text":"2x is equal to the sum as above,"},{"Start":"02:04.275 ","End":"02:06.585","Text":"except that instead of x,"},{"Start":"02:06.585 ","End":"02:15.675","Text":"I put 2x and leave that in brackets, over 2n factorial."},{"Start":"02:15.675 ","End":"02:19.770","Text":"That\u0027s the matter of the 2x,"},{"Start":"02:19.770 ","End":"02:21.980","Text":"and I\u0027ll just slightly rewrite it."},{"Start":"02:21.980 ","End":"02:28.040","Text":"I\u0027ll take this constant out of the exponent,"},{"Start":"02:28.040 ","End":"02:31.150","Text":"and we get the sum,"},{"Start":"02:31.150 ","End":"02:34.755","Text":"same limits, minus 1_n."},{"Start":"02:34.755 ","End":"02:39.465","Text":"From here, 2_ 2n and then"},{"Start":"02:39.465 ","End":"02:49.990","Text":"x_2n over 2n factorial."},{"Start":"02:50.600 ","End":"02:54.950","Text":"Let me note at this point that the interval of"},{"Start":"02:54.950 ","End":"02:59.375","Text":"convergence for this power series is all x,"},{"Start":"02:59.375 ","End":"03:03.925","Text":"it\u0027s written in the appendix also."},{"Start":"03:03.925 ","End":"03:06.320","Text":"If this is true for all x,"},{"Start":"03:06.320 ","End":"03:08.330","Text":"then when I replace x by 2x,"},{"Start":"03:08.330 ","End":"03:10.340","Text":"it\u0027s still for all x."},{"Start":"03:10.340 ","End":"03:15.680","Text":"Let\u0027s not say anymore about radius of convergence, it\u0027s all x."},{"Start":"03:15.680 ","End":"03:21.680","Text":"Let\u0027s worry now about the technical part of how to get to the result."},{"Start":"03:21.680 ","End":"03:25.460","Text":"We need to still do 1 minus the series,"},{"Start":"03:25.460 ","End":"03:27.455","Text":"then to multiply it by a 1/2."},{"Start":"03:27.455 ","End":"03:31.070","Text":"Now, how do I do 1 minus a series?"},{"Start":"03:31.070 ","End":"03:35.760","Text":"The idea, this is a standard trick,"},{"Start":"03:35.760 ","End":"03:39.350","Text":"is to split the series up into the first term,"},{"Start":"03:39.350 ","End":"03:41.570","Text":"and all the remainder."},{"Start":"03:41.570 ","End":"03:46.695","Text":"In general, if I have the sum from,"},{"Start":"03:46.695 ","End":"03:48.870","Text":"let\u0027s say, 0 to infinity,"},{"Start":"03:48.870 ","End":"03:51.915","Text":"if anything gets called at a_n,"},{"Start":"03:51.915 ","End":"03:58.770","Text":"it\u0027s always going to be true that I can take just the first term, which is a_0,"},{"Start":"03:58.790 ","End":"04:04.100","Text":"plus the remaining terms,"},{"Start":"04:04.100 ","End":"04:10.740","Text":"which means that we start the count from 1 to infinity of a_n."},{"Start":"04:11.420 ","End":"04:14.360","Text":"If you still can\u0027t see this,"},{"Start":"04:14.360 ","End":"04:15.995","Text":"maybe if we write it out,"},{"Start":"04:15.995 ","End":"04:20.555","Text":"a_0 plus a_1, plus a_2 plus a_3,"},{"Start":"04:20.555 ","End":"04:24.985","Text":"etc., is going to equal a_0"},{"Start":"04:24.985 ","End":"04:31.575","Text":"separately and then I can just look at a_1 plus a_2 and so on in brackets."},{"Start":"04:31.575 ","End":"04:36.650","Text":"This is this, and this is just the sum from 1 onwards."},{"Start":"04:36.650 ","End":"04:44.280","Text":"Now if I play this general idea of peeling off the first term to here,"},{"Start":"04:44.320 ","End":"04:49.205","Text":"I can write this as the first term with n equals 0."},{"Start":"04:49.205 ","End":"04:51.800","Text":"Let\u0027s see what that is. When n is 0,"},{"Start":"04:51.800 ","End":"04:55.440","Text":"anything to the 0 is 1,"},{"Start":"04:55.670 ","End":"04:58.715","Text":"and here if n is 0,"},{"Start":"04:58.715 ","End":"05:02.540","Text":"it also gives me x_0 is 1,"},{"Start":"05:02.540 ","End":"05:07.985","Text":"and 0 factorial is also 1,"},{"Start":"05:07.985 ","End":"05:16.005","Text":"so this is just equal to 1 plus the sum moved on 1,"},{"Start":"05:16.005 ","End":"05:19.780","Text":"only from 1 to infinity of the same thing."},{"Start":"05:22.220 ","End":"05:29.655","Text":"Now, I can write down what is 1 minus cosine of 2x,"},{"Start":"05:29.655 ","End":"05:33.180","Text":"and say that it\u0027s 1 minus this."},{"Start":"05:33.180 ","End":"05:36.600","Text":"The 1 minus 1 cancels to 0,"},{"Start":"05:36.600 ","End":"05:39.030","Text":"and all we get is minus,"},{"Start":"05:39.030 ","End":"05:41.925","Text":"and again I\u0027ll copy paste this."},{"Start":"05:41.925 ","End":"05:46.460","Text":"There we are. The 1 drops out and we get a minus."},{"Start":"05:46.460 ","End":"05:49.520","Text":"Now we\u0027re very, very close to what we need."},{"Start":"05:49.520 ","End":"05:52.995","Text":"We need just to multiply by a half,"},{"Start":"05:52.995 ","End":"05:57.425","Text":"so what I get is that sine squared x"},{"Start":"05:57.425 ","End":"06:03.780","Text":"is 1/2 of what we have here and so it\u0027s minus a 1/2."},{"Start":"06:03.780 ","End":"06:06.490","Text":"Again, I\u0027ll just copy this here."},{"Start":"06:07.070 ","End":"06:13.250","Text":"Here we are. Let\u0027s just do a bit of simplification."},{"Start":"06:13.250 ","End":"06:15.800","Text":"I have minus 1 to a power,"},{"Start":"06:15.800 ","End":"06:17.435","Text":"and 2 to a power,"},{"Start":"06:17.435 ","End":"06:20.870","Text":"and I also have here a minus 1 times a 1/2,"},{"Start":"06:20.870 ","End":"06:23.060","Text":"which is like dividing by 2."},{"Start":"06:23.060 ","End":"06:30.045","Text":"What I\u0027m saying is that the minus can be added to the minus 1_n."},{"Start":"06:30.045 ","End":"06:34.875","Text":"Let me just write the Sigma first, 0 to infinity."},{"Start":"06:34.875 ","End":"06:39.964","Text":"The minus just increases this n to n plus 1,"},{"Start":"06:39.964 ","End":"06:42.620","Text":"and the 1/2 is dividing by 2,"},{"Start":"06:42.620 ","End":"06:44.720","Text":"will decrease this exponent,"},{"Start":"06:44.720 ","End":"06:48.900","Text":"so we\u0027ll have 2^2n minus 1,"},{"Start":"06:48.900 ","End":"06:50.790","Text":"and this part the same,"},{"Start":"06:50.790 ","End":"06:56.865","Text":"x_2n over 2n factorial."},{"Start":"06:56.865 ","End":"07:04.760","Text":"This is the Taylor series for sine squared x about 0 of the Maclaurin series,"},{"Start":"07:04.760 ","End":"07:06.860","Text":"let me highlight this as well,"},{"Start":"07:06.860 ","End":"07:13.400","Text":"and we already said that the radius of convergence is all x."},{"Start":"07:13.400 ","End":"07:18.090","Text":"Maybe you highlight that also. We\u0027re done."}],"ID":6088},{"Watched":false,"Name":"Exercise 5","Duration":"5m 3s","ChapterTopicVideoID":6072,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6072.jpeg","UploadDate":"2016-05-08T11:04:18.5970000","DurationForVideoObject":"PT5M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this exercise, we are given a function f of x is"},{"Start":"00:05.010 ","End":"00:10.080","Text":"cosine squared x and we have to find its Taylor series around 0,"},{"Start":"00:10.080 ","End":"00:13.500","Text":"which is essentially the same as the Maclaurin series."},{"Start":"00:13.500 ","End":"00:15.150","Text":"We\u0027re not going to do it from scratch."},{"Start":"00:15.150 ","End":"00:18.675","Text":"We\u0027re going to use some functions from the appendix."},{"Start":"00:18.675 ","End":"00:21.150","Text":"But cosine squared x is not there,"},{"Start":"00:21.150 ","End":"00:23.710","Text":"we have to find something similar."},{"Start":"00:25.910 ","End":"00:29.700","Text":"What we can do is, first of all,"},{"Start":"00:29.700 ","End":"00:36.765","Text":"use some trigonometric identities because we do have cosine and sine in the appendix."},{"Start":"00:36.765 ","End":"00:42.115","Text":"We know that cosine squared can be written in terms of a double angle."},{"Start":"00:42.115 ","End":"00:48.920","Text":"In fact, the trigonometrical identity we need is that cosine squared"},{"Start":"00:48.920 ","End":"00:58.175","Text":"x is equal to 1/2 of 1 plus cosine of 2x."},{"Start":"00:58.175 ","End":"01:01.820","Text":"We actually had a similar exercise earlier."},{"Start":"01:01.820 ","End":"01:04.370","Text":"You may have seen it with the sine squared,"},{"Start":"01:04.370 ","End":"01:08.030","Text":"and that also uses the cosine of the double angle."},{"Start":"01:08.030 ","End":"01:11.780","Text":"Anyway, so we\u0027re going to use cosine x,"},{"Start":"01:11.780 ","End":"01:14.765","Text":"but not in the sense of that we\u0027re going to square it,"},{"Start":"01:14.765 ","End":"01:18.120","Text":"but we\u0027re going to use trigonometric identities."},{"Start":"01:18.770 ","End":"01:23.940","Text":"Here\u0027s what I found in the appendix, cosine x."},{"Start":"01:23.940 ","End":"01:26.630","Text":"We\u0027re going to use the Sigma notation."},{"Start":"01:26.630 ","End":"01:30.900","Text":"This is just how it would look if I started expanding it out."},{"Start":"01:31.970 ","End":"01:37.610","Text":"What we need is cosine of 2x in order to use this."},{"Start":"01:37.610 ","End":"01:40.740","Text":"What I\u0027m saying is that this is equal to"},{"Start":"01:41.630 ","End":"01:48.360","Text":"1 plus cosine 2x"},{"Start":"01:48.360 ","End":"01:55.335","Text":"and then 1/2 of that and 1/2 in front."},{"Start":"01:55.335 ","End":"01:59.930","Text":"Let\u0027s, first of all, take care of the cosine 2x,"},{"Start":"01:59.930 ","End":"02:02.830","Text":"and then we\u0027ll add 1, and then we\u0027ll divide by 2."},{"Start":"02:02.830 ","End":"02:05.840","Text":"I see the x here and here,"},{"Start":"02:05.840 ","End":"02:08.940","Text":"and I\u0027m going to replace it with 2x."},{"Start":"02:09.610 ","End":"02:16.130","Text":"We can get that cosine of 2x is equal to"},{"Start":"02:16.130 ","End":"02:23.075","Text":"the sum and goes from 0 to infinity minus 1^n and instead of x,"},{"Start":"02:23.075 ","End":"02:30.060","Text":"2x, but I need brackets to the 2n over 2n factorial."},{"Start":"02:30.060 ","End":"02:34.870","Text":"Same as this, but 2x instead of x."},{"Start":"02:34.870 ","End":"02:43.640","Text":"If I want 1 plus cosine of 2x,"},{"Start":"02:43.640 ","End":"02:46.370","Text":"and this is just equal to 1 plus,"},{"Start":"02:46.370 ","End":"02:48.880","Text":"I\u0027ll just copy-paste this here."},{"Start":"02:48.880 ","End":"02:54.225","Text":"Now, all that\u0027s missing to get to cosine squared x,"},{"Start":"02:54.225 ","End":"02:57.690","Text":"which is this, is just the 1/2."},{"Start":"02:57.690 ","End":"03:03.450","Text":"So this is equal to 1/2 plus"},{"Start":"03:03.450 ","End":"03:11.250","Text":"the sum from 0 to infinity of minus 1^n."},{"Start":"03:12.220 ","End":"03:19.635","Text":"Now, what I\u0027m going to do is put the 1/2 inside here."},{"Start":"03:19.635 ","End":"03:25.925","Text":"Basically, what I have is I can put the 1/2 in here."},{"Start":"03:25.925 ","End":"03:29.675","Text":"The other thing I can do is take 2 to the power of 2n."},{"Start":"03:29.675 ","End":"03:35.450","Text":"I mean, I could split this as a product to a power is the power of each of the factors."},{"Start":"03:35.450 ","End":"03:40.190","Text":"It\u0027s 2^2n and then"},{"Start":"03:40.190 ","End":"03:46.900","Text":"x to the 2n over 2n factorial."},{"Start":"03:47.210 ","End":"03:50.445","Text":"I forgot the parentheses here,"},{"Start":"03:50.445 ","End":"03:52.350","Text":"and it is important because otherwise,"},{"Start":"03:52.350 ","End":"03:56.560","Text":"you would just mean twice n factorial. Sorry about that."},{"Start":"03:57.890 ","End":"04:00.300","Text":"That\u0027s basically it."},{"Start":"04:00.300 ","End":"04:02.315","Text":"It\u0027s just a little bit of tidying up."},{"Start":"04:02.315 ","End":"04:08.000","Text":"Instead of 1/2 of 2^2n,"},{"Start":"04:08.000 ","End":"04:16.250","Text":"I would write this as 2 to the power of 2n minus 1,"},{"Start":"04:16.250 ","End":"04:18.305","Text":"because it\u0027s like dividing by 2."},{"Start":"04:18.305 ","End":"04:21.040","Text":"The rest of it, the same."},{"Start":"04:21.040 ","End":"04:26.315","Text":"This is the answer for the Taylor series for cosine squared"},{"Start":"04:26.315 ","End":"04:31.880","Text":"x. I should have said a word about radius of convergence."},{"Start":"04:31.880 ","End":"04:35.210","Text":"This applies to all x."},{"Start":"04:35.210 ","End":"04:38.240","Text":"It should say that in the table."},{"Start":"04:38.240 ","End":"04:42.180","Text":"Sometimes they write that as minus infinity, less than x."},{"Start":"04:42.180 ","End":"04:45.725","Text":"Less than infinity means any x really."},{"Start":"04:45.725 ","End":"04:49.790","Text":"Now replacing x by 2x doesn\u0027t change the fact that it\u0027s still"},{"Start":"04:49.790 ","End":"04:54.320","Text":"all x and adding the constant and dividing by 2."},{"Start":"04:54.320 ","End":"04:59.455","Text":"Basically, this is also going to apply for all x."},{"Start":"04:59.455 ","End":"05:02.490","Text":"We are done."}],"ID":6089},{"Watched":false,"Name":"Exercise 6","Duration":"3m 55s","ChapterTopicVideoID":6073,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6073.jpeg","UploadDate":"2016-05-08T11:04:48.0130000","DurationForVideoObject":"PT3M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we are given the function f of x equals"},{"Start":"00:04.500 ","End":"00:09.060","Text":"2 to the x and we have to find its Taylor series around x equal 0,"},{"Start":"00:09.060 ","End":"00:11.565","Text":"which means it\u0027s a Maclaurin series."},{"Start":"00:11.565 ","End":"00:12.970","Text":"We don\u0027t do it from scratch,"},{"Start":"00:12.970 ","End":"00:16.170","Text":"we go to the appendix and look for something close."},{"Start":"00:16.170 ","End":"00:18.105","Text":"We don\u0027t have 2^x,"},{"Start":"00:18.105 ","End":"00:20.620","Text":"but we do have e^x,"},{"Start":"00:20.620 ","End":"00:22.230","Text":"and this is going to be useful to us."},{"Start":"00:22.230 ","End":"00:24.195","Text":"Let me just copy it in."},{"Start":"00:24.195 ","End":"00:26.864","Text":"Here\u0027s the formula."},{"Start":"00:26.864 ","End":"00:28.890","Text":"We\u0027re going to use the Sigma form,"},{"Start":"00:28.890 ","End":"00:31.530","Text":"it gives you the first few terms also."},{"Start":"00:31.530 ","End":"00:36.465","Text":"As far as interval of convergence goes,"},{"Start":"00:36.465 ","End":"00:39.880","Text":"this is good for all x."},{"Start":"00:40.060 ","End":"00:48.930","Text":"Or sometimes it\u0027s written as x goes from minus infinity to infinity, same thing."},{"Start":"00:49.810 ","End":"00:58.490","Text":"The question is, how do we get from e ^x to 2^x is an algebraic trick."},{"Start":"00:58.490 ","End":"01:03.020","Text":"If you remember your logarithms and exponents."},{"Start":"01:03.020 ","End":"01:09.140","Text":"1of the rules is that a^b is equal"},{"Start":"01:09.140 ","End":"01:17.010","Text":"to e^b natural log of a."},{"Start":"01:17.480 ","End":"01:22.415","Text":"1 way to quickly verify it would be detected natural logarithm of both sides."},{"Start":"01:22.415 ","End":"01:27.590","Text":"Natural logarithm of e to the something is just this and natural logarithm,"},{"Start":"01:27.590 ","End":"01:33.835","Text":"the left-hand side, the exponent comes in front and we get b natural log of a, so anyway."},{"Start":"01:33.835 ","End":"01:40.460","Text":"In our case, I\u0027m going to get that 2^x is equal e to"},{"Start":"01:40.460 ","End":"01:48.005","Text":"the power of and thus would be x times natural log of 2,"},{"Start":"01:48.005 ","End":"01:51.440","Text":"which I could also write the other way around."},{"Start":"01:51.440 ","End":"01:57.110","Text":"I might decide to put the natural log of 2 before the x, might be easier."},{"Start":"01:57.110 ","End":"02:03.350","Text":"Now, to get what 2^x is,"},{"Start":"02:03.350 ","End":"02:13.195","Text":"all I have to do is to substitute the x here and here with x natural log of 2."},{"Start":"02:13.195 ","End":"02:16.365","Text":"Like I said, I think I\u0027ll use it the other way around."},{"Start":"02:16.365 ","End":"02:20.770","Text":"So we get the sum, and instead of x,"},{"Start":"02:20.770 ","End":"02:25.075","Text":"I\u0027ll put natural log of 2 times x,"},{"Start":"02:25.075 ","End":"02:30.015","Text":"all this to the power of n over n factorial,"},{"Start":"02:30.015 ","End":"02:33.820","Text":"n goes from 0 to infinity."},{"Start":"02:34.580 ","End":"02:41.220","Text":"Rewrite will give me that this is the sum from 0 to infinity,"},{"Start":"02:41.220 ","End":"02:47.520","Text":"I\u0027ll take the natural log of 2 separately to the power of n. In fact,"},{"Start":"02:47.520 ","End":"02:50.185","Text":"you can even write the n over here."},{"Start":"02:50.185 ","End":"02:53.895","Text":"Well no, it\u0027s going to look like this n,"},{"Start":"02:53.895 ","End":"02:56.485","Text":"I\u0027ll use the brackets form,"},{"Start":"02:56.485 ","End":"03:01.580","Text":"and then x to the n over n factorial."},{"Start":"03:01.580 ","End":"03:09.480","Text":"So it\u0027s similar to this except that we have this extra power of this constant log of 2."},{"Start":"03:11.230 ","End":"03:19.830","Text":"That\u0027s basically it except that I need to mention the radius,"},{"Start":"03:19.830 ","End":"03:21.665","Text":"so interval of convergence."},{"Start":"03:21.665 ","End":"03:24.260","Text":"Well, if x goes from minus infinity to infinity,"},{"Start":"03:24.260 ","End":"03:34.070","Text":"this 1 goes for x log 2 between infinity and minus infinity,"},{"Start":"03:34.070 ","End":"03:38.675","Text":"and I can just divide everything by the log 2,"},{"Start":"03:38.675 ","End":"03:44.825","Text":"so really it\u0027s all x. I\u0027ll just highlight"},{"Start":"03:44.825 ","End":"03:51.755","Text":"that this is equal to this and it\u0027s good for all x,"},{"Start":"03:51.755 ","End":"03:54.150","Text":"and that\u0027s the answer."}],"ID":6090},{"Watched":false,"Name":"Exercise 7","Duration":"3m 33s","ChapterTopicVideoID":6074,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6074.jpeg","UploadDate":"2016-05-08T11:05:15.9500000","DurationForVideoObject":"PT3M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"Here we have to find the Taylor series of the function"},{"Start":"00:04.515 ","End":"00:10.620","Text":"x times cosine of 4x squared around x equals 0,"},{"Start":"00:10.620 ","End":"00:13.050","Text":"which makes it a Maclaurin series,"},{"Start":"00:13.050 ","End":"00:17.054","Text":"and so we go to the appendix and look for something similar,"},{"Start":"00:17.054 ","End":"00:20.290","Text":"and we find we have the cosine."},{"Start":"00:21.530 ","End":"00:26.940","Text":"This expansion is good for all x. How do I know this?"},{"Start":"00:26.940 ","End":"00:31.695","Text":"Because in the table it says minus infinity,"},{"Start":"00:31.695 ","End":"00:33.765","Text":"less than x, less than infinity."},{"Start":"00:33.765 ","End":"00:36.240","Text":"Same thing as saying all x,"},{"Start":"00:36.240 ","End":"00:41.270","Text":"and the way we use this to get to here is first of all,"},{"Start":"00:41.270 ","End":"00:46.260","Text":"we\u0027re going to replace x with 4x squared and then we\u0027re going to multiply by x."},{"Start":"00:47.030 ","End":"00:54.410","Text":"First we\u0027ll do the substitution and figure out cosine of 4x squared."},{"Start":"00:54.410 ","End":"00:56.720","Text":"We\u0027re going to use the Sigma form."},{"Start":"00:56.720 ","End":"00:59.915","Text":"Now instead of x here and here,"},{"Start":"00:59.915 ","End":"01:02.375","Text":"I\u0027m going to put 4x squared."},{"Start":"01:02.375 ","End":"01:10.145","Text":"This will equal, this part is the same sum from 0 to infinity minus 1^n."},{"Start":"01:10.145 ","End":"01:12.214","Text":"But now instead of x,"},{"Start":"01:12.214 ","End":"01:21.300","Text":"I need to put 4x squared to the power of 2n over 2n factorial 3,"},{"Start":"01:21.300 ","End":"01:30.480","Text":"4x squared takes the place of x. I can just modify this a bit."},{"Start":"01:30.890 ","End":"01:37.760","Text":"From n equals 0 to infinity minus 1^n."},{"Start":"01:37.760 ","End":"01:40.010","Text":"Now, I can split this up."},{"Start":"01:40.010 ","End":"01:43.070","Text":"It\u0027s a product 4^2n,"},{"Start":"01:43.070 ","End":"01:48.440","Text":"which I\u0027ll put in front of this fraction,"},{"Start":"01:48.440 ","End":"01:54.860","Text":"and then we\u0027ll do the easy bit to 2n factorial."},{"Start":"01:54.860 ","End":"01:57.830","Text":"All I\u0027m left with now is x squared to the 2n."},{"Start":"01:57.830 ","End":"01:59.975","Text":"Using the rules of exponents,"},{"Start":"01:59.975 ","End":"02:03.760","Text":"this gives me x^4n."},{"Start":"02:03.760 ","End":"02:07.385","Text":"Now I\u0027ve got up to the cosine of 4x squared."},{"Start":"02:07.385 ","End":"02:11.175","Text":"Now all I have to do is multiply by x."},{"Start":"02:11.175 ","End":"02:14.285","Text":"What I get is, and I\u0027ll write it over here,"},{"Start":"02:14.285 ","End":"02:23.295","Text":"x times cosine of 4x squared it\u0027s going to be x times,"},{"Start":"02:23.295 ","End":"02:29.010","Text":"let us copy paste this and now I can throw the x inside."},{"Start":"02:29.010 ","End":"02:30.905","Text":"If I do that,"},{"Start":"02:30.905 ","End":"02:34.380","Text":"pretty much everything will be the same."},{"Start":"02:34.510 ","End":"02:38.900","Text":"Sorry, I said I\u0027m going to put x inside."},{"Start":"02:38.900 ","End":"02:45.815","Text":"Sum n goes from 0 to infinity minus 1^n, 4^2n,"},{"Start":"02:45.815 ","End":"02:52.650","Text":"2n factorial, I\u0027m multiplying by x,"},{"Start":"02:52.650 ","End":"02:57.540","Text":"I can multiply it term-wise we\u0027ll just raise the power by 1."},{"Start":"02:57.540 ","End":"03:04.380","Text":"It\u0027ll be x^4n plus 1."},{"Start":"03:04.380 ","End":"03:06.230","Text":"As for the radius of convergence,"},{"Start":"03:06.230 ","End":"03:11.480","Text":"it\u0027s also for all x. I can put any x here I want,"},{"Start":"03:11.480 ","End":"03:13.520","Text":"I can also put 4x squared."},{"Start":"03:13.520 ","End":"03:20.475","Text":"This is also going to be for all x. I\u0027ll just highlight the answer."},{"Start":"03:20.475 ","End":"03:22.095","Text":"This is the answer,"},{"Start":"03:22.095 ","End":"03:28.760","Text":"it\u0027s true for all x and that\u0027s what the cosine 4x squared is equal to."},{"Start":"03:28.760 ","End":"03:32.760","Text":"This is the Taylor and Maclaurin series. We\u0027re done."}],"ID":6091},{"Watched":false,"Name":"Exercise 8","Duration":"11m 59s","ChapterTopicVideoID":6075,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6075.jpeg","UploadDate":"2016-05-08T11:06:50.5030000","DurationForVideoObject":"PT11M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.765","Text":"Here we\u0027re given a function f of x equals natural log of 2 minus 3x plus x squared,"},{"Start":"00:09.765 ","End":"00:12.510","Text":"and we have to find the Taylor series around 0,"},{"Start":"00:12.510 ","End":"00:14.940","Text":"which means Maclaurin series."},{"Start":"00:14.940 ","End":"00:19.274","Text":"We don\u0027t do it from scratch we go to look at the table in the appendix,"},{"Start":"00:19.274 ","End":"00:21.060","Text":"look for something similar."},{"Start":"00:21.060 ","End":"00:23.220","Text":"Nothing really similar."},{"Start":"00:23.220 ","End":"00:30.730","Text":"The closest we have is one with natural log in it and it\u0027s this one."},{"Start":"00:30.800 ","End":"00:37.245","Text":"This, which is given in Sigma form and I\u0027ve expanded a few terms."},{"Start":"00:37.245 ","End":"00:44.120","Text":"It also says there that it\u0027s true the interval of convergence is minus 1,"},{"Start":"00:44.120 ","End":"00:46.550","Text":"less than x, less than,"},{"Start":"00:46.550 ","End":"00:48.200","Text":"or equal to 1."},{"Start":"00:48.200 ","End":"00:49.880","Text":"It could be equal to 1,"},{"Start":"00:49.880 ","End":"00:52.505","Text":"but it can\u0027t be minus 1."},{"Start":"00:52.505 ","End":"00:58.335","Text":"We somehow have to get from here to this."},{"Start":"00:58.335 ","End":"01:00.000","Text":"Well, here\u0027s the idea."},{"Start":"01:00.000 ","End":"01:02.470","Text":"We\u0027re going to factorize this,"},{"Start":"01:02.470 ","End":"01:06.770","Text":"the fact that the log of a product is the sum of the logs."},{"Start":"01:06.770 ","End":"01:10.230","Text":"First of all, let\u0027s do the factorization."},{"Start":"01:10.330 ","End":"01:13.444","Text":"To make it easier on the quadratic,"},{"Start":"01:13.444 ","End":"01:17.705","Text":"I\u0027ll just write it in the usual form in decreasing order,"},{"Start":"01:17.705 ","End":"01:22.245","Text":"x squared minus 3x plus 2."},{"Start":"01:22.245 ","End":"01:24.180","Text":"There are several ways to factorize."},{"Start":"01:24.180 ","End":"01:29.210","Text":"One way is to find the roots and that is equal to 0."},{"Start":"01:29.210 ","End":"01:33.845","Text":"I\u0027m not going to waste time with solving quadratic equations."},{"Start":"01:33.845 ","End":"01:40.070","Text":"X turns out to be equal to 1 or 2."},{"Start":"01:40.070 ","End":"01:44.300","Text":"You can plug them in and check that at least it works,1 minus 3 plus 2 is"},{"Start":"01:44.300 ","End":"01:50.285","Text":"0 and 4 minus 6 plus 2 is also 0, that\u0027s okay."},{"Start":"01:50.285 ","End":"01:57.185","Text":"Then we know that this thing factorizes as x minus 1 one of the roots,"},{"Start":"01:57.185 ","End":"02:02.185","Text":"and x minus the other root."},{"Start":"02:02.185 ","End":"02:07.985","Text":"Now, notice that I really prefer"},{"Start":"02:07.985 ","End":"02:13.360","Text":"to have the number before the x because I wanted to somehow look a bit more like this."},{"Start":"02:13.360 ","End":"02:16.104","Text":"If we reverse the order of a difference,"},{"Start":"02:16.104 ","End":"02:18.080","Text":"it\u0027s like throwing a minus in,"},{"Start":"02:18.080 ","End":"02:20.840","Text":"but if we do it for both of them, we\u0027ll be all right."},{"Start":"02:20.840 ","End":"02:24.994","Text":"If I reverse these and reverse these,"},{"Start":"02:24.994 ","End":"02:31.880","Text":"then I\u0027ve got this is equal to 1 minus x, 2 minus x."},{"Start":"02:31.880 ","End":"02:33.980","Text":"This is still not quite close enough,"},{"Start":"02:33.980 ","End":"02:35.920","Text":"I really do want a 1 here."},{"Start":"02:35.920 ","End":"02:38.070","Text":"How about I take 2 out of here,"},{"Start":"02:38.070 ","End":"02:42.020","Text":"so there I put 2, 1 minus x."},{"Start":"02:42.020 ","End":"02:43.625","Text":"If I take the 2 out,"},{"Start":"02:43.625 ","End":"02:51.670","Text":"then I have 1 minus x over 2."},{"Start":"02:51.670 ","End":"02:56.629","Text":"Remember that the rule of logarithms that if we have log of a product,"},{"Start":"02:56.629 ","End":"02:59.645","Text":"it could be even 3 things which we have here,"},{"Start":"02:59.645 ","End":"03:01.610","Text":"a times b times c,"},{"Start":"03:01.610 ","End":"03:08.690","Text":"that becomes natural log of a plus natural log of b plus natural log of c, for example."},{"Start":"03:08.690 ","End":"03:13.480","Text":"Here we get that this is equal to the product of 3 things."},{"Start":"03:13.480 ","End":"03:15.930","Text":"This is the product of 3 things."},{"Start":"03:15.930 ","End":"03:18.575","Text":"We get the sum of 3 things."},{"Start":"03:18.575 ","End":"03:28.380","Text":"We get the natural log of 2 plus the natural log of 1 minus x,"},{"Start":"03:28.460 ","End":"03:37.820","Text":"and then plus the natural log of 1 minus x over 2."},{"Start":"03:37.970 ","End":"03:40.810","Text":"This is a constant and here,"},{"Start":"03:40.810 ","End":"03:45.590","Text":"and here we\u0027ll be able to use this formula with the variation."},{"Start":"03:45.590 ","End":"03:49.745","Text":"Let me just get rid of this scratchwork here."},{"Start":"03:49.745 ","End":"03:55.765","Text":"Now I want the natural log of 1 minus x."},{"Start":"03:55.765 ","End":"04:01.210","Text":"1 minus x, I can just write it as 1 plus negative x."},{"Start":"04:01.210 ","End":"04:07.870","Text":"All I have to do is replace the x in this formula"},{"Start":"04:07.870 ","End":"04:15.555","Text":"by minus x so I get the sum from 0 to infinity,"},{"Start":"04:15.555 ","End":"04:19.340","Text":"minus 1 to the n, and instead of x,"},{"Start":"04:19.340 ","End":"04:28.020","Text":"I have minus x in brackets to the n plus 1 over n plus 1."},{"Start":"04:28.300 ","End":"04:34.235","Text":"Now, this actually simplifies"},{"Start":"04:34.235 ","End":"04:42.350","Text":"because if I take the minus 1 out here,"},{"Start":"04:42.350 ","End":"04:46.445","Text":"it\u0027s really minus 1 to the n plus 1,"},{"Start":"04:46.445 ","End":"04:50.850","Text":"and then multiply with minus 1^n,"},{"Start":"04:50.900 ","End":"04:59.570","Text":"what I get is minus 1 to the power of 2n plus 1."},{"Start":"04:59.570 ","End":"05:04.100","Text":"Now, this is definitely an odd number because 2n is always even,"},{"Start":"05:04.100 ","End":"05:07.320","Text":"so this is just equal to minus 1."},{"Start":"05:07.320 ","End":"05:10.770","Text":"A minus can be brought in front,"},{"Start":"05:10.770 ","End":"05:17.960","Text":"so really what I get is minus the sum from 0 to infinity of"},{"Start":"05:17.960 ","End":"05:25.760","Text":"just x to the n plus 1 over n plus 1."},{"Start":"05:25.760 ","End":"05:30.525","Text":"So that\u0027s this one here."},{"Start":"05:30.525 ","End":"05:33.650","Text":"Now, what about this bit here?"},{"Start":"05:33.650 ","End":"05:37.650","Text":"Natural log of 1,"},{"Start":"05:38.470 ","End":"05:43.510","Text":"I want to minus x over 2."},{"Start":"05:43.510 ","End":"05:47.260","Text":"What would this be equal to?"},{"Start":"05:47.570 ","End":"05:53.630","Text":"Now look here, I have minus x and minus x"},{"Start":"05:53.630 ","End":"06:00.530","Text":"and I got this by replacing x with minus x here."},{"Start":"06:00.530 ","End":"06:03.990","Text":"I could actually use this formula because this is even closer,"},{"Start":"06:03.990 ","End":"06:05.240","Text":"it already has a minus,"},{"Start":"06:05.240 ","End":"06:10.385","Text":"but instead of x, I\u0027ll replace x with x over 2."},{"Start":"06:10.385 ","End":"06:14.764","Text":"I\u0027m just going to take this formula and make a switch,"},{"Start":"06:14.764 ","End":"06:18.410","Text":"x is replaced by x over 2."},{"Start":"06:18.410 ","End":"06:27.575","Text":"What I get is just copying from here minus the sum and goes from 0 to infinity."},{"Start":"06:27.575 ","End":"06:38.040","Text":"Instead of x, x over 2 to the power of n plus 1 over n plus 1."},{"Start":"06:38.040 ","End":"06:45.570","Text":"We\u0027ll get to the interval of convergence later just continue developing here."},{"Start":"06:45.770 ","End":"06:53.105","Text":"This comes out to be minus the sum from 0 to infinity"},{"Start":"06:53.105 ","End":"07:03.155","Text":"of x^n plus 1 and the 2^n plus 1 I\u0027ll just throw it in the denominator,"},{"Start":"07:03.155 ","End":"07:05.810","Text":"2^n plus 1 times,"},{"Start":"07:05.810 ","End":"07:08.840","Text":"and now I need a bracket around the n plus 1."},{"Start":"07:08.840 ","End":"07:14.315","Text":"Maybe it\u0027s a good point to talk about interval of convergence."},{"Start":"07:14.315 ","End":"07:20.885","Text":"Starting with this one which was x between minus 1 and 1, including the 1."},{"Start":"07:20.885 ","End":"07:24.959","Text":"If I replace x by minus x,"},{"Start":"07:28.130 ","End":"07:35.265","Text":"we would get minus one less than minus x less than or equal to 1,"},{"Start":"07:35.265 ","End":"07:38.720","Text":"and if I divide everything by minus 1,"},{"Start":"07:38.720 ","End":"07:43.245","Text":"and I also have to reverse the directions of the inequalities,"},{"Start":"07:43.245 ","End":"07:50.530","Text":"what I\u0027m going to get is minus 1 less than or equal to x less than 1."},{"Start":"07:52.160 ","End":"07:55.195","Text":"As for this one,"},{"Start":"07:55.195 ","End":"08:00.890","Text":"we just have to replace x with x over 2 here so we"},{"Start":"08:00.890 ","End":"08:07.235","Text":"get minus 1 less than or equal to x over 2,"},{"Start":"08:07.235 ","End":"08:15.780","Text":"less than 1 and this comes out to be minus 2 less than or equal to x, less than 2."},{"Start":"08:15.780 ","End":"08:19.310","Text":"Now it\u0027s time to put the pieces together."},{"Start":"08:19.310 ","End":"08:21.274","Text":"This bit is just a constant."},{"Start":"08:21.274 ","End":"08:24.275","Text":"Now, this one here,"},{"Start":"08:24.275 ","End":"08:28.200","Text":"let me just underline it in green,"},{"Start":"08:28.790 ","End":"08:33.420","Text":"corresponds to this bit here,"},{"Start":"08:33.420 ","End":"08:40.715","Text":"I\u0027ll just circle that and the inequality is this one here."},{"Start":"08:40.715 ","End":"08:43.160","Text":"As for the other bit,"},{"Start":"08:43.160 ","End":"08:45.605","Text":"I\u0027ll do it in this color."},{"Start":"08:45.605 ","End":"08:50.045","Text":"This corresponds to this here."},{"Start":"08:50.045 ","End":"08:58.470","Text":"The inequality is this one here."},{"Start":"08:59.280 ","End":"09:01.525","Text":"When I combine them,"},{"Start":"09:01.525 ","End":"09:06.370","Text":"I need for all the inequalities to hold for this and this to hold."},{"Start":"09:06.370 ","End":"09:09.040","Text":"If I take the end of these 2,"},{"Start":"09:09.040 ","End":"09:10.780","Text":"sometimes called the intersection,"},{"Start":"09:10.780 ","End":"09:15.265","Text":"what we\u0027ll get is this one because this one\u0027s included in this one."},{"Start":"09:15.265 ","End":"09:24.160","Text":"The interval of convergence"},{"Start":"09:24.160 ","End":"09:29.065","Text":"is just minus 1 less than or equal to x,"},{"Start":"09:29.065 ","End":"09:34.110","Text":"less than 1 for the whole thing."},{"Start":"09:34.110 ","End":"09:37.200","Text":"This one I\u0027ll highlight in green."},{"Start":"09:37.200 ","End":"09:41.455","Text":"Let me just scroll a bit for a put the pieces together."},{"Start":"09:41.455 ","End":"09:45.425","Text":"What we get is that f of x,"},{"Start":"09:45.425 ","End":"09:46.909","Text":"or in other words,"},{"Start":"09:46.909 ","End":"09:55.925","Text":"the natural log of 2 minus 3x plus x squared is equal to."},{"Start":"09:55.925 ","End":"09:58.085","Text":"This is the constant bit,"},{"Start":"09:58.085 ","End":"10:03.035","Text":"natural log of 2 then I need this,"},{"Start":"10:03.035 ","End":"10:08.645","Text":"which is what is written here."},{"Start":"10:08.645 ","End":"10:10.040","Text":"It\u0027s not really a plus,"},{"Start":"10:10.040 ","End":"10:15.380","Text":"it\u0027s a minus and it\u0027s the sum of x^n plus"},{"Start":"10:15.380 ","End":"10:21.755","Text":"1 over n plus 1 from 0 to infinity."},{"Start":"10:21.755 ","End":"10:23.210","Text":"Then I need this bit,"},{"Start":"10:23.210 ","End":"10:30.065","Text":"so it\u0027s another minus the sum from 0 to infinity"},{"Start":"10:30.065 ","End":"10:40.235","Text":"of x^n plus 1 over 2^n plus 1 n plus 1."},{"Start":"10:40.235 ","End":"10:44.015","Text":"We\u0027re almost done. Just want to simplify a bit."},{"Start":"10:44.015 ","End":"10:45.910","Text":"These 2 are very similar,"},{"Start":"10:45.910 ","End":"10:50.145","Text":"the x^n plus 1 over n plus 1 is the same."},{"Start":"10:50.145 ","End":"10:53.900","Text":"I can say that this is natural log of 2"},{"Start":"10:53.900 ","End":"11:01.660","Text":"minus the sum from 0 to infinity."},{"Start":"11:03.080 ","End":"11:05.400","Text":"The common bit is that,"},{"Start":"11:05.400 ","End":"11:09.300","Text":"and here I\u0027m going to take out 1."},{"Start":"11:09.300 ","End":"11:11.120","Text":"It\u0027s just going to be this thing,"},{"Start":"11:11.120 ","End":"11:14.960","Text":"but here I got 1 over 2^n plus 1."},{"Start":"11:14.960 ","End":"11:17.180","Text":"It\u0027s a plus because the minus is in front,"},{"Start":"11:17.180 ","End":"11:22.550","Text":"1/2^n plus 1 times this common bit x^n"},{"Start":"11:22.550 ","End":"11:31.980","Text":"plus 1 over n plus 1."},{"Start":"11:31.980 ","End":"11:33.480","Text":"This is really the answer."},{"Start":"11:33.480 ","End":"11:37.870","Text":"I know there\u0027s a constant besides the series, it doesn\u0027t matter,"},{"Start":"11:37.870 ","End":"11:42.425","Text":"this is still considered a Taylor and Maclaurin series."},{"Start":"11:42.425 ","End":"11:48.160","Text":"I\u0027ll highlight the answer here."},{"Start":"11:49.040 ","End":"11:51.650","Text":"This is what it\u0027s equal to,"},{"Start":"11:51.650 ","End":"11:56.465","Text":"and this is the interval of convergence."},{"Start":"11:56.465 ","End":"11:59.340","Text":"We are done."}],"ID":6092},{"Watched":false,"Name":"Exercise 9","Duration":"8m 47s","ChapterTopicVideoID":6076,"CourseChapterTopicPlaylistID":84685,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6076.jpeg","UploadDate":"2016-05-08T11:08:06.0830000","DurationForVideoObject":"PT8M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, we have to find the Taylor series of"},{"Start":"00:03.990 ","End":"00:09.825","Text":"the function arcsine of x around x equals 0."},{"Start":"00:09.825 ","End":"00:16.740","Text":"Around x equals 0 means that it\u0027s a Maclaurin series and we have a table in the appendix,"},{"Start":"00:16.740 ","End":"00:22.230","Text":"and we look for something that looks like arcsine and we don\u0027t find anything."},{"Start":"00:22.230 ","End":"00:26.550","Text":"What to do, I\u0027m going to teach you a trick here."},{"Start":"00:26.550 ","End":"00:33.915","Text":"Sometimes, if you have a function and you don\u0027t know it\u0027s Maclaurin expansion or series,"},{"Start":"00:33.915 ","End":"00:36.630","Text":"you can differentiate or integrate."},{"Start":"00:36.630 ","End":"00:43.475","Text":"If that has a series then you can integrate or differentiate back,"},{"Start":"00:43.475 ","End":"00:47.555","Text":"do the opposite of what you did before and get the original function."},{"Start":"00:47.555 ","End":"00:52.485","Text":"In this case, I\u0027ll tell you that what we want to do is differentiate this."},{"Start":"00:52.485 ","End":"00:58.850","Text":"We get that f prime of x is equal to 1"},{"Start":"00:58.850 ","End":"01:06.150","Text":"over the square root of 1 minus x squared."},{"Start":"01:06.230 ","End":"01:11.050","Text":"This still doesn\u0027t look like very much in the appendix there,"},{"Start":"01:11.050 ","End":"01:18.570","Text":"but if I write it as 1 minus x squared ^ minus 1/2,"},{"Start":"01:18.570 ","End":"01:22.730","Text":"then there is something that looks a bit like this."},{"Start":"01:22.730 ","End":"01:25.820","Text":"That is, if I think of this minus 1/2 is m,"},{"Start":"01:25.820 ","End":"01:29.800","Text":"I do have 1 plus x^m, let me bring it in."},{"Start":"01:29.800 ","End":"01:32.415","Text":"Here\u0027s the 1 we\u0027re going to use,"},{"Start":"01:32.415 ","End":"01:37.675","Text":"we\u0027re going to use it with m equal minus 1/2."},{"Start":"01:37.675 ","End":"01:41.810","Text":"Now, the radius of convergence is also given depending on m,"},{"Start":"01:41.810 ","End":"01:43.400","Text":"and if you look at the different cases,"},{"Start":"01:43.400 ","End":"01:48.870","Text":"the 1 that we want is that the case between minus 1 and 0."},{"Start":"01:49.460 ","End":"01:57.905","Text":"In this range, the interval of convergence is from minus 1 to 1,"},{"Start":"01:57.905 ","End":"02:00.755","Text":"inclusive here and not inclusive here."},{"Start":"02:00.755 ","End":"02:09.565","Text":"What we\u0027re going to do is replace the x here and here by minus x squared."},{"Start":"02:09.565 ","End":"02:11.130","Text":"I\u0027ll just make a note of that,"},{"Start":"02:11.130 ","End":"02:15.120","Text":"replace x with minus x squared."},{"Start":"02:15.120 ","End":"02:21.180","Text":"Then we\u0027ll get that 1 minus x squared,"},{"Start":"02:21.180 ","End":"02:30.240","Text":"we also replace m to be minus 1/2 is going to equal 1 plus the sum,"},{"Start":"02:30.240 ","End":"02:34.575","Text":"n goes from 1 to infinity."},{"Start":"02:34.575 ","End":"02:36.735","Text":"Remember m is 1/2,"},{"Start":"02:36.735 ","End":"02:40.290","Text":"so we get 1/2,"},{"Start":"02:40.290 ","End":"02:43.755","Text":"1/2 minus 1 times,"},{"Start":"02:43.755 ","End":"02:46.590","Text":"and so on and so on and so on,"},{"Start":"02:46.590 ","End":"02:54.075","Text":"keep decreasing by 1 until we get to 1/2 minus n plus 1."},{"Start":"02:54.075 ","End":"02:58.335","Text":"This is over n factorial,"},{"Start":"02:58.335 ","End":"03:04.480","Text":"x is replaced by minus x squared^n."},{"Start":"03:06.790 ","End":"03:10.730","Text":"I replaced x by minus x squared here and here,"},{"Start":"03:10.730 ","End":"03:14.245","Text":"but I also have to do it in the interval of convergence."},{"Start":"03:14.245 ","End":"03:18.065","Text":"What we get is that we write this,"},{"Start":"03:18.065 ","End":"03:23.885","Text":"but also with minus x squared instead of x."},{"Start":"03:23.885 ","End":"03:28.020","Text":"From here, I\u0027ll multiply everything by minus 1,"},{"Start":"03:28.020 ","End":"03:35.385","Text":"but then I also have to change the direction of the inequality and then reverse it."},{"Start":"03:35.385 ","End":"03:44.675","Text":"Anyway, you\u0027ll see that you get minus 1 less than or equal to x squared, less than 1."},{"Start":"03:44.675 ","End":"03:47.960","Text":"Here is the less than or equal to not here any"},{"Start":"03:47.960 ","End":"03:51.740","Text":"more because we reversed it, because of the minus."},{"Start":"03:51.740 ","End":"03:56.405","Text":"Now x squared is automatically bigger or equal to minus 1, it\u0027s always non-negative."},{"Start":"03:56.405 ","End":"03:57.980","Text":"This bit is redundant,"},{"Start":"03:57.980 ","End":"04:00.050","Text":"so we\u0027re just left with this bit."},{"Start":"04:00.050 ","End":"04:01.655","Text":"If I spell it out,"},{"Start":"04:01.655 ","End":"04:06.165","Text":"it comes out to be minus 1 less than x,"},{"Start":"04:06.165 ","End":"04:09.270","Text":"less than 1, not quite the same as the original."},{"Start":"04:09.270 ","End":"04:12.790","Text":"Here we had less than or equal to, here less than."},{"Start":"04:15.020 ","End":"04:18.770","Text":"Let me just slightly revise this and I\u0027ll write it over here."},{"Start":"04:18.770 ","End":"04:22.805","Text":"We have that f prime of x is equal to,"},{"Start":"04:22.805 ","End":"04:31.340","Text":"we\u0027ve got the 1 plus the Sigma from 1 to infinity."},{"Start":"04:31.340 ","End":"04:34.655","Text":"This bit is the same,"},{"Start":"04:34.655 ","End":"04:37.240","Text":"but I\u0027ve got a minus,"},{"Start":"04:37.240 ","End":"04:39.860","Text":"I\u0027ll split this up into minus 1 times x squared."},{"Start":"04:39.860 ","End":"04:42.110","Text":"We\u0027ll have a minus 1^n,"},{"Start":"04:42.110 ","End":"04:47.730","Text":"then I have this bit which is a nuisance sum to write,"},{"Start":"04:47.730 ","End":"04:49.964","Text":"but not so bad."},{"Start":"04:49.964 ","End":"04:53.880","Text":"1/2, 1/2 minus 1 times dot, dot, dot,"},{"Start":"04:53.880 ","End":"04:58.905","Text":"times 1/2 minus n plus 1"},{"Start":"04:58.905 ","End":"05:07.230","Text":"and x squared^n is x^2n."},{"Start":"05:07.230 ","End":"05:10.160","Text":"This is where we are with f prime of x."},{"Start":"05:10.160 ","End":"05:12.710","Text":"Now, to get to f of x,"},{"Start":"05:12.710 ","End":"05:14.405","Text":"we do an integration,"},{"Start":"05:14.405 ","End":"05:16.805","Text":"so f of x will equal."},{"Start":"05:16.805 ","End":"05:20.300","Text":"Now we can do termwise integration,"},{"Start":"05:20.300 ","End":"05:24.005","Text":"and later we\u0027ll talk about the interval of convergence."},{"Start":"05:24.005 ","End":"05:29.795","Text":"We get x plus Sigma,"},{"Start":"05:29.795 ","End":"05:33.595","Text":"and from 1 to infinity."},{"Start":"05:33.595 ","End":"05:40.050","Text":"This part\u0027s the same minus 1 to the n, 1/2,"},{"Start":"05:40.050 ","End":"05:51.310","Text":"1/2 minus 1, 1/2 minus n plus 1 over n factorial."},{"Start":"05:51.410 ","End":"05:55.744","Text":"Then we also have to take the integral of x to the 2n,"},{"Start":"05:55.744 ","End":"06:03.550","Text":"which is x to the 2n plus 1 over 2n plus 1."},{"Start":"06:03.550 ","End":"06:09.660","Text":"But we also have to add a constant of integration."},{"Start":"06:09.800 ","End":"06:12.210","Text":"To get the constant,"},{"Start":"06:12.210 ","End":"06:15.680","Text":"the easiest thing to do is to substitute a convenient value,"},{"Start":"06:15.680 ","End":"06:19.390","Text":"and I think 0 will be very convenient because all these are powers of x,"},{"Start":"06:19.390 ","End":"06:21.740","Text":"and if x is 0 everything will be 0."},{"Start":"06:21.740 ","End":"06:23.960","Text":"We just need to know what f of 0 is,"},{"Start":"06:23.960 ","End":"06:29.520","Text":"in other words we need that what is arcsine 0 and I\u0027ll just refresh your memory."},{"Start":"06:29.520 ","End":"06:31.790","Text":"Arcsine of 0 is 0,"},{"Start":"06:31.790 ","End":"06:34.560","Text":"or do it on your calculator."},{"Start":"06:35.300 ","End":"06:38.250","Text":"All this comes out to be 0,"},{"Start":"06:38.250 ","End":"06:46.365","Text":"so when we let x equals 0 we\u0027re going to get that 0 is equal to,"},{"Start":"06:46.365 ","End":"06:51.305","Text":"all these terms are going to be 0 plus C,"},{"Start":"06:51.305 ","End":"06:55.945","Text":"and so that gives us that C is equal to 0."},{"Start":"06:55.945 ","End":"07:01.385","Text":"Finally, what I could do would be to say,"},{"Start":"07:01.385 ","End":"07:05.120","Text":"well, I could copy it again or I could just say if C is 0,"},{"Start":"07:05.120 ","End":"07:07.760","Text":"I\u0027ll just put a box around this bit,"},{"Start":"07:07.760 ","End":"07:14.470","Text":"perhaps I\u0027ll just highlight the bits that we want which is up to here."},{"Start":"07:14.470 ","End":"07:19.770","Text":"Because like we said C is 0 so it\u0027s like crossing this out,"},{"Start":"07:19.770 ","End":"07:22.540","Text":"so this is the answer."},{"Start":"07:22.970 ","End":"07:28.225","Text":"Really I should replace f of x by arcsine x,"},{"Start":"07:28.225 ","End":"07:37.515","Text":"so I\u0027ll just copy it again and now I\u0027ll replace the f of x by arc sine x."},{"Start":"07:37.515 ","End":"07:44.020","Text":"All that\u0027s missing now is to say what is the interval of convergence."},{"Start":"07:44.020 ","End":"07:48.520","Text":"Now there\u0027s a theorem that when you do derivatives or integrals of power series,"},{"Start":"07:48.520 ","End":"07:52.910","Text":"you get the same radius of convergence."},{"Start":"07:53.160 ","End":"07:58.570","Text":"The radius of convergence here is 1 because around 0,"},{"Start":"07:58.570 ","End":"08:01.640","Text":"this is 0 plus or minus 1."},{"Start":"08:01.640 ","End":"08:04.740","Text":"Here also we\u0027re going to have a radius of 1,"},{"Start":"08:04.740 ","End":"08:08.335","Text":"but the radius of 1 doesn\u0027t exactly tell us the interval."},{"Start":"08:08.335 ","End":"08:14.080","Text":"It tells us that for sure we also have x between minus 1 and 1,"},{"Start":"08:14.080 ","End":"08:17.270","Text":"but we\u0027re doubtful about whether it could"},{"Start":"08:17.270 ","End":"08:21.710","Text":"possibly also be equal to and it will still have a radius of 1."},{"Start":"08:21.710 ","End":"08:24.900","Text":"So we have to actually manually check,"},{"Start":"08:24.900 ","End":"08:30.160","Text":"and it\u0027s really beyond the scope of this exercise."},{"Start":"08:30.160 ","End":"08:34.265","Text":"Anyway, I\u0027ll just quote the result because this is more than you\u0027d be expected to do."},{"Start":"08:34.265 ","End":"08:38.860","Text":"It actually does converge also at minus 1 to 1."},{"Start":"08:38.860 ","End":"08:46.860","Text":"This is power series for arcsine from minus 1 to 1 inclusive. We are done."}],"ID":6093}],"Thumbnail":null,"ID":84685},{"Name":"Expansions about General Point","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Taylor Series Expansion about general point","Duration":"4m 38s","ChapterTopicVideoID":10123,"CourseChapterTopicPlaylistID":84686,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10123.jpeg","UploadDate":"2020-01-16T13:03:24.9330000","DurationForVideoObject":"PT4M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.870","Text":"In the exercises following this clip,"},{"Start":"00:03.870 ","End":"00:07.350","Text":"you\u0027ll be typically given a function and asked to expand"},{"Start":"00:07.350 ","End":"00:11.910","Text":"it into a Taylor series about x equals 4,"},{"Start":"00:11.910 ","End":"00:14.190","Text":"x equals minus 3,"},{"Start":"00:14.190 ","End":"00:16.905","Text":"x equals 0, x equals 10."},{"Start":"00:16.905 ","End":"00:20.980","Text":"In general, you\u0027ll be given some number called x_0,"},{"Start":"00:20.980 ","End":"00:25.470","Text":"and you\u0027ll be asked to expand the function into a Taylor series about that point."},{"Start":"00:25.470 ","End":"00:28.095","Text":"This is a new concept."},{"Start":"00:28.095 ","End":"00:31.890","Text":"Up till now, when we\u0027re given a function f of x,"},{"Start":"00:31.890 ","End":"00:35.975","Text":"we expanded it into a Taylor or Maclaurin series."},{"Start":"00:35.975 ","End":"00:37.880","Text":"It will look something like this."},{"Start":"00:37.880 ","End":"00:43.730","Text":"The sum from 0 to infinity of some coefficient times a power of x."},{"Start":"00:43.730 ","End":"00:46.020","Text":"It\u0027s a power series."},{"Start":"00:46.550 ","End":"00:49.610","Text":"This is what we call the Taylor series."},{"Start":"00:49.610 ","End":"00:52.190","Text":"But I said it\u0027s also called the Maclaurin series,"},{"Start":"00:52.190 ","End":"00:55.505","Text":"because this is a Taylor series about x equals 0."},{"Start":"00:55.505 ","End":"00:57.050","Text":"Now, what do I mean?"},{"Start":"00:57.050 ","End":"00:58.440","Text":"What is this x equals 0,"},{"Start":"00:58.440 ","End":"01:00.005","Text":"and what is this x_0?"},{"Start":"01:00.005 ","End":"01:05.089","Text":"Well, it turns out there\u0027s a more general expansion when you\u0027re given a function,"},{"Start":"01:05.089 ","End":"01:09.740","Text":"and that is to expand it about some other point, x equals x_0."},{"Start":"01:09.740 ","End":"01:12.020","Text":"As I said, there could be x equals 3,"},{"Start":"01:12.020 ","End":"01:14.460","Text":"x equals 200, whatever."},{"Start":"01:14.460 ","End":"01:17.980","Text":"This expansion looks a bit different."},{"Start":"01:17.980 ","End":"01:19.800","Text":"It looks like this,"},{"Start":"01:19.800 ","End":"01:23.445","Text":"n goes from 0 to infinity a_n,"},{"Start":"01:23.445 ","End":"01:25.355","Text":"but instead of x_n,"},{"Start":"01:25.355 ","End":"01:28.000","Text":"we have x minus x_0_n."},{"Start":"01:28.000 ","End":"01:33.310","Text":"This could be x minus 4_n depending on what x_0 is."},{"Start":"01:33.440 ","End":"01:39.595","Text":"This is called expansion about x equals x_0,"},{"Start":"01:39.595 ","End":"01:43.469","Text":"which if x_0 happens to be 0,"},{"Start":"01:43.469 ","End":"01:45.200","Text":"it\u0027s just this expansion."},{"Start":"01:45.200 ","End":"01:47.450","Text":"Now this is in general is called"},{"Start":"01:47.450 ","End":"01:52.880","Text":"a Taylor series when it\u0027s not about 0 or even when it is about 0."},{"Start":"01:52.880 ","End":"01:54.740","Text":"This is also called Taylor,"},{"Start":"01:54.740 ","End":"01:57.050","Text":"but specifically it\u0027s called"},{"Start":"01:57.050 ","End":"02:03.125","Text":"a Maclaurin series because I\u0027m not exactly sure of the history of it."},{"Start":"02:03.125 ","End":"02:06.140","Text":"Anyway, both of these this could be also Taylor."},{"Start":"02:06.140 ","End":"02:08.615","Text":"Taylor\u0027s more generally, you can call everything Taylor."},{"Start":"02:08.615 ","End":"02:09.770","Text":"But when you say Maclaurin,"},{"Start":"02:09.770 ","End":"02:14.360","Text":"you mean specifically about 0 and you don\u0027t have to say about 0."},{"Start":"02:14.360 ","End":"02:18.440","Text":"That\u0027s the brief introduction"},{"Start":"02:18.440 ","End":"02:22.600","Text":"before you get to the exercises and don\u0027t know what they mean."},{"Start":"02:22.600 ","End":"02:25.490","Text":"In question 1, following,"},{"Start":"02:25.490 ","End":"02:32.855","Text":"you asked to expand the function f of x equals natural log of x,"},{"Start":"02:32.855 ","End":"02:37.235","Text":"about x equals 1."},{"Start":"02:37.235 ","End":"02:42.390","Text":"They wouldn\u0027t ask you about 0 because natural log is not defined at 0."},{"Start":"02:42.400 ","End":"02:53.280","Text":"This means that they want you to find f of x as a sum from 0 to infinity of a_n,"},{"Start":"02:53.280 ","End":"02:58.440","Text":"x minus 1 to the power of n. You have to find the coefficients a_n."},{"Start":"02:58.440 ","End":"02:59.970","Text":"If you have a formula for the coefficients,"},{"Start":"02:59.970 ","End":"03:01.530","Text":"then you\u0027ve got the function."},{"Start":"03:01.530 ","End":"03:05.150","Text":"That\u0027s 1 example that you will encounter."},{"Start":"03:05.150 ","End":"03:06.980","Text":"In the second example,"},{"Start":"03:06.980 ","End":"03:12.845","Text":"you will be given the task of expanding the function 1 over x,"},{"Start":"03:12.845 ","End":"03:17.005","Text":"about x equals 3."},{"Start":"03:17.005 ","End":"03:22.775","Text":"Again, you couldn\u0027t be asked to do it about 0 because it\u0027s not defined at 0."},{"Start":"03:22.775 ","End":"03:28.160","Text":"This means that they want you to write f of x as"},{"Start":"03:28.160 ","End":"03:35.415","Text":"the sum of a_n x minus 3_n."},{"Start":"03:35.415 ","End":"03:37.150","Text":"Here, x_0 is 3."},{"Start":"03:37.150 ","End":"03:39.680","Text":"Actually, I think it was x equals 2."},{"Start":"03:39.680 ","End":"03:42.049","Text":"The third one, if I recall,"},{"Start":"03:42.049 ","End":"03:46.505","Text":"was to expand the sine x"},{"Start":"03:46.505 ","End":"03:52.620","Text":"around x_0 is Pi over 2."},{"Start":"03:52.620 ","End":"03:57.500","Text":"Then that means they want you to write actually here f of x,"},{"Start":"03:57.500 ","End":"03:59.240","Text":"I should really write what it is,"},{"Start":"03:59.240 ","End":"04:03.860","Text":"and here it means that they want us to write f of x or sine"},{"Start":"04:03.860 ","End":"04:09.810","Text":"x as a power series centered on Pi over 2,"},{"Start":"04:09.810 ","End":"04:15.890","Text":"which means the sum of a_n x minus Pi over 2 to"},{"Start":"04:15.890 ","End":"04:22.880","Text":"the power of n. In some of the exercises they might give you to expand about x equals 0,"},{"Start":"04:22.880 ","End":"04:29.225","Text":"in which case, it\u0027s really a Maclaurin series and we write it like this."},{"Start":"04:29.225 ","End":"04:35.735","Text":"But this is just a special case of this when x_0 is 0."},{"Start":"04:35.735 ","End":"04:39.300","Text":"Let\u0027s go and do those 3 exercises."}],"ID":10345},{"Watched":false,"Name":"Exercise 1","Duration":"4m 50s","ChapterTopicVideoID":6080,"CourseChapterTopicPlaylistID":84686,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6080.jpeg","UploadDate":"2016-05-08T11:11:31.8030000","DurationForVideoObject":"PT4M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.960","Text":"In this exercise, we have to expand f(x) equals natural log of x,"},{"Start":"00:06.960 ","End":"00:10.019","Text":"but not about x equals 0,"},{"Start":"00:10.019 ","End":"00:13.110","Text":"but about x equals 1."},{"Start":"00:13.110 ","End":"00:16.560","Text":"It\u0027s a Taylor series, not a Maclaurin."},{"Start":"00:16.560 ","End":"00:22.280","Text":"But we have tables"},{"Start":"00:22.280 ","End":"00:28.265","Text":"of expansions of Maclaurin series around x equals 0."},{"Start":"00:28.265 ","End":"00:30.755","Text":"What we\u0027re going to do is a trick,"},{"Start":"00:30.755 ","End":"00:32.980","Text":"and it\u0027s a standard trick."},{"Start":"00:32.980 ","End":"00:37.885","Text":"It\u0027s a typical trick we\u0027ll be using for Taylor series when it\u0027s not around 0,"},{"Start":"00:37.885 ","End":"00:40.130","Text":"that is a substitution."},{"Start":"00:40.130 ","End":"00:42.440","Text":"The substitution is to use another letter,"},{"Start":"00:42.440 ","End":"00:49.260","Text":"say y, to be equal to x minus 1."},{"Start":"00:49.260 ","End":"00:52.650","Text":"The 1, because it\u0027s 1 here and in any particular case,"},{"Start":"00:52.650 ","End":"00:57.790","Text":"it\u0027s x minus whatever the point is we\u0027re expanding about."},{"Start":"00:57.790 ","End":"01:06.605","Text":"The reverse substitution, if we extract x in terms of y,"},{"Start":"01:06.605 ","End":"01:15.560","Text":"this is the same thing as x is equal to y plus 1."},{"Start":"01:15.560 ","End":"01:19.510","Text":"Notice that when x is 1,"},{"Start":"01:19.510 ","End":"01:27.330","Text":"y is 0, x is 1 if and only if y equals 0."},{"Start":"01:27.330 ","End":"01:29.450","Text":"In terms of y,"},{"Start":"01:29.450 ","End":"01:33.230","Text":"we\u0027re going to be doing an expansion around y equals"},{"Start":"01:33.230 ","End":"01:38.135","Text":"0 and be able to use the Maclaurin table."},{"Start":"01:38.135 ","End":"01:45.284","Text":"f(x) is actually equal to, in terms of y,"},{"Start":"01:45.284 ","End":"01:53.495","Text":"the natural log, and x is y plus 1 but allow me to write it as 1 plus y."},{"Start":"01:53.495 ","End":"02:02.930","Text":"The reason I\u0027m doing that is that we have this already in the table of Maclaurin,"},{"Start":"02:02.930 ","End":"02:13.040","Text":"but we have to use it in terms of y. I\u0027ll just copy it,"},{"Start":"02:13.040 ","End":"02:16.630","Text":"but substitute y instead of x and we\u0027ve got the sum."},{"Start":"02:16.630 ","End":"02:26.510","Text":"I copied in the formula for the expansion of natural log of 1 plus x but here,"},{"Start":"02:26.510 ","End":"02:29.180","Text":"we\u0027re going to replace x by y, as I said."},{"Start":"02:29.180 ","End":"02:36.600","Text":"We get n goes from 0 to infinity of minus 1^n,"},{"Start":"02:36.600 ","End":"02:44.750","Text":"y to the n plus 1 over n plus 1."},{"Start":"02:45.340 ","End":"02:52.685","Text":"That\u0027s not all because we usually required to give the interval of convergence."},{"Start":"02:52.685 ","End":"02:57.950","Text":"The interval of convergence here is stated as x is"},{"Start":"02:57.950 ","End":"03:05.680","Text":"between minus 1 and 1 and actually including the 1."},{"Start":"03:06.050 ","End":"03:12.975","Text":"Here, what I\u0027ll have to say is that"},{"Start":"03:12.975 ","End":"03:21.930","Text":"y is between 1 and minus 1,"},{"Start":"03:21.930 ","End":"03:25.500","Text":"vice versa, including this."},{"Start":"03:25.500 ","End":"03:32.185","Text":"If I put y equals x minus 1,"},{"Start":"03:32.185 ","End":"03:34.920","Text":"and then just add 1,"},{"Start":"03:34.920 ","End":"03:44.475","Text":"you\u0027ll see that we get that x is between 0 and 2."},{"Start":"03:44.475 ","End":"03:53.610","Text":"Just put x minus 1 here and then add 1 to the double inequality and we get this."},{"Start":"03:55.160 ","End":"04:05.340","Text":"The remaining thing to do is just to now substitute from y back to x here."},{"Start":"04:05.340 ","End":"04:08.240","Text":"We\u0027ve got f of x,"},{"Start":"04:08.240 ","End":"04:11.250","Text":"which is natural log of x,"},{"Start":"04:11.840 ","End":"04:20.910","Text":"is equal to the sum from 0 to infinity minus 1^n,"},{"Start":"04:20.910 ","End":"04:26.250","Text":"instead of y, x minus 1 to"},{"Start":"04:26.250 ","End":"04:32.820","Text":"the n plus 1 over n plus 1."},{"Start":"04:32.820 ","End":"04:36.345","Text":"I\u0027ll highlight this answer."},{"Start":"04:36.345 ","End":"04:39.420","Text":"I guess this is part of the answer 2,"},{"Start":"04:39.420 ","End":"04:44.390","Text":"the interval of convergence and we\u0027re back to"},{"Start":"04:44.390 ","End":"04:50.700","Text":"all in terms of x. This is it."}],"ID":6094},{"Watched":false,"Name":"Exercise 2","Duration":"6m 44s","ChapterTopicVideoID":6468,"CourseChapterTopicPlaylistID":84686,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6468.jpeg","UploadDate":"2016-06-30T15:57:46.9900000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.740","Text":"In this exercise, we\u0027re given the function f of x equals 1 over x,"},{"Start":"00:07.740 ","End":"00:14.265","Text":"and we want to expand it around x equals 2 to a Taylor Series."},{"Start":"00:14.265 ","End":"00:20.465","Text":"The general trick is to try and make it into a Maclaurin Series,"},{"Start":"00:20.465 ","End":"00:27.795","Text":"which means Taylor around x equals 0 because then we can use tables of known functions."},{"Start":"00:27.795 ","End":"00:30.345","Text":"That helps us, we don\u0027t have to do it from scratch."},{"Start":"00:30.345 ","End":"00:35.985","Text":"The standard trick is to make a substitution,"},{"Start":"00:35.985 ","End":"00:41.150","Text":"and we let y equal x minus 2,"},{"Start":"00:41.150 ","End":"00:43.565","Text":"it\u0027s x minus whatever\u0027s written here,"},{"Start":"00:43.565 ","End":"00:48.800","Text":"so that when x equals 2, y equals 0."},{"Start":"00:48.800 ","End":"00:55.190","Text":"What we get is that f of x is 1 over x,"},{"Start":"00:55.190 ","End":"00:56.930","Text":"and we need the reverse substitution,"},{"Start":"00:56.930 ","End":"01:00.320","Text":"which is x equals y plus 2."},{"Start":"01:00.320 ","End":"01:07.214","Text":"Just bring the 2 over to the other side so we have 1 over y plus 2,"},{"Start":"01:07.214 ","End":"01:12.490","Text":"but this time around y equals 0,"},{"Start":"01:13.630 ","End":"01:17.240","Text":"which is what I said, a Maclaurin series."},{"Start":"01:17.240 ","End":"01:24.790","Text":"Now we look amongst the table of functions that we have,"},{"Start":"01:24.790 ","End":"01:26.795","Text":"to see if there\u0027s anything similar to this."},{"Start":"01:26.795 ","End":"01:29.610","Text":"Well, the closest we can get,"},{"Start":"01:29.690 ","End":"01:32.790","Text":"this is the closest thing in the table,"},{"Start":"01:32.790 ","End":"01:36.620","Text":"it\u0027s 1 over 1 minus whatever the variable is,"},{"Start":"01:36.620 ","End":"01:39.295","Text":"it could be x, but it could be something else."},{"Start":"01:39.295 ","End":"01:41.510","Text":"We\u0027re going to use the Sigma form,"},{"Start":"01:41.510 ","End":"01:44.150","Text":"although it\u0027s also given in expanded form."},{"Start":"01:44.150 ","End":"01:51.595","Text":"The interval of convergence is that x is between minus 1 and 1."},{"Start":"01:51.595 ","End":"01:57.335","Text":"What we\u0027re going to do is modify this so it looks more like this."},{"Start":"01:57.335 ","End":"02:04.355","Text":"Well, the first thing I can do is write it as the number before the letter 2 plus y,"},{"Start":"02:04.355 ","End":"02:06.500","Text":"and I really like to have a 1 here,"},{"Start":"02:06.500 ","End":"02:11.780","Text":"so what I\u0027m going to do is take out a half, in other words,"},{"Start":"02:11.780 ","End":"02:14.105","Text":"take 2 out of the denominator,"},{"Start":"02:14.105 ","End":"02:20.140","Text":"and I\u0027ll be left with 1 plus y over 2."},{"Start":"02:20.140 ","End":"02:25.335","Text":"Now, this part here is just 1"},{"Start":"02:25.335 ","End":"02:33.345","Text":"over 1 minus minus y over 2,"},{"Start":"02:33.345 ","End":"02:35.235","Text":"just this bit here."},{"Start":"02:35.235 ","End":"02:37.490","Text":"That really looks like this,"},{"Start":"02:37.490 ","End":"02:43.880","Text":"if instead of x, which I have here and here,"},{"Start":"02:43.880 ","End":"02:49.025","Text":"I\u0027m going to replace it by minus y over 2,"},{"Start":"02:49.025 ","End":"02:59.080","Text":"and so I can write 1 over 1 plus y over 2."},{"Start":"02:59.290 ","End":"03:01.730","Text":"But I\u0027m reading from here,"},{"Start":"03:01.730 ","End":"03:05.690","Text":"the minus y over 2 is equal to the sum,"},{"Start":"03:05.690 ","End":"03:10.750","Text":"n goes from 0 to infinity of"},{"Start":"03:10.750 ","End":"03:18.995","Text":"minus y over 2 to the power of n. Instead of the x,"},{"Start":"03:18.995 ","End":"03:21.050","Text":"I have minus y over 2,"},{"Start":"03:21.050 ","End":"03:24.330","Text":"so it\u0027s minus y over 2 here."},{"Start":"03:25.850 ","End":"03:34.460","Text":"What I get is that 1 over 2 plus y is equal to the half here,"},{"Start":"03:34.460 ","End":"03:39.800","Text":"and this is equal to this, so it\u0027s Sigma,"},{"Start":"03:39.800 ","End":"03:49.625","Text":"n goes from 0 to infinity of minus y over 2 to the power of n."},{"Start":"03:49.625 ","End":"03:54.265","Text":"Now I want to substitute back from"},{"Start":"03:54.265 ","End":"04:01.015","Text":"y to x using this y equals x minus 2."},{"Start":"04:01.015 ","End":"04:04.730","Text":"This thing is just the original function f of x,"},{"Start":"04:04.730 ","End":"04:06.320","Text":"which is 1 over x,"},{"Start":"04:06.320 ","End":"04:10.920","Text":"so I get that 1 over x is equal to"},{"Start":"04:10.920 ","End":"04:17.620","Text":"1/2 the sum from 0 to infinity."},{"Start":"04:17.620 ","End":"04:24.125","Text":"Now, minus y over 2 would be,"},{"Start":"04:24.125 ","End":"04:26.315","Text":"well, I can just split it up,"},{"Start":"04:26.315 ","End":"04:30.890","Text":"minus y over 2 is minus a half times y."},{"Start":"04:30.890 ","End":"04:39.750","Text":"Now I can write this as minus 1/2 to the n,"},{"Start":"04:40.090 ","End":"04:43.490","Text":"and then y to the n,"},{"Start":"04:43.490 ","End":"04:46.890","Text":"but y is x minus 2."},{"Start":"04:47.840 ","End":"04:52.930","Text":"Now, just a little bit of simplifying,"},{"Start":"04:54.350 ","End":"05:04.205","Text":"what I can say is that this is equal to the sum n equals 0 to infinity."},{"Start":"05:04.205 ","End":"05:10.425","Text":"I can put minus 1 to the n here,"},{"Start":"05:10.425 ","End":"05:12.450","Text":"now over 2 to the n,"},{"Start":"05:12.450 ","End":"05:17.339","Text":"but I can also throw the 2 in so I can put 2 to the n plus 1,"},{"Start":"05:17.339 ","End":"05:23.510","Text":"and x minus 2 to the n. This is"},{"Start":"05:23.510 ","End":"05:29.880","Text":"the expansion of 1 over x around x equals 2."},{"Start":"05:29.880 ","End":"05:33.480","Text":"But we mustn\u0027t forget the radius of convergence,"},{"Start":"05:33.480 ","End":"05:36.340","Text":"I mean the interval of convergence."},{"Start":"05:37.880 ","End":"05:43.190","Text":"We look here, and we see that we replaced x by minus y over 2,"},{"Start":"05:43.190 ","End":"05:52.920","Text":"so this inequality becomes the inequality minus 1 less than minus y over 2 less than 1."},{"Start":"05:52.920 ","End":"05:58.910","Text":"If I multiply everything by minus 2,"},{"Start":"05:58.910 ","End":"06:03.575","Text":"I also have to reverse the direction."},{"Start":"06:03.575 ","End":"06:13.810","Text":"Basically, what we get in the end is that minus 2 is less than y is less than 2,"},{"Start":"06:13.810 ","End":"06:17.389","Text":"and now from y back to x,"},{"Start":"06:17.389 ","End":"06:21.040","Text":"y is x minus 2,"},{"Start":"06:21.040 ","End":"06:24.870","Text":"so I just add 2 to the double inequality,"},{"Start":"06:24.870 ","End":"06:32.205","Text":"and we get 0 less than x, less than 4."},{"Start":"06:32.205 ","End":"06:39.300","Text":"I\u0027ll just highlight the solution that 1 over x is expanded thus,"},{"Start":"06:39.300 ","End":"06:45.070","Text":"and the interval of convergence is this, and we\u0027re done."}],"ID":6495},{"Watched":false,"Name":"Exercise 3","Duration":"6m 35s","ChapterTopicVideoID":6081,"CourseChapterTopicPlaylistID":84686,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6081.jpeg","UploadDate":"2016-05-08T11:12:21.3830000","DurationForVideoObject":"PT6M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we\u0027re given the function f of x is equal to"},{"Start":"00:06.030 ","End":"00:15.700","Text":"sine x. I want to expand it as a Taylor series around x equals Pi over 2."},{"Start":"00:16.280 ","End":"00:20.640","Text":"In this exercise, we\u0027re given the function f of x,"},{"Start":"00:20.640 ","End":"00:22.530","Text":"which is sine x."},{"Start":"00:22.530 ","End":"00:27.510","Text":"We have to expand it as a Taylor series around x equals Pi over 2."},{"Start":"00:27.510 ","End":"00:29.729","Text":"What does that even mean?"},{"Start":"00:29.729 ","End":"00:30.960","Text":"Just to remind you,"},{"Start":"00:30.960 ","End":"00:39.915","Text":"it means that we have a sum from 0 to infinity of some coefficients, call them a_n,"},{"Start":"00:39.915 ","End":"00:45.100","Text":"and it will be x minus Pi over"},{"Start":"00:45.100 ","End":"00:52.310","Text":"2 to the power of n. We don\u0027t want to do this from scratch."},{"Start":"00:52.310 ","End":"00:57.610","Text":"We have tables of Maclaurin series and if it wasn\u0027t Pi over 2,"},{"Start":"00:57.610 ","End":"00:59.785","Text":"if it was 0, we\u0027d be okay."},{"Start":"00:59.785 ","End":"01:03.790","Text":"We use the standard trick of a substitution to let this x minus"},{"Start":"01:03.790 ","End":"01:08.665","Text":"Pi over 2 to be y [inaudible] could be t,"},{"Start":"01:08.665 ","End":"01:13.565","Text":"y equals x minus Pi over 2."},{"Start":"01:13.565 ","End":"01:17.400","Text":"Then when x is Pi over 2, y is 0."},{"Start":"01:17.400 ","End":"01:23.320","Text":"We can look for a Maclaurin series. Let me just write that."},{"Start":"01:23.320 ","End":"01:28.070","Text":"The reverse is that x is equal to"},{"Start":"01:28.070 ","End":"01:35.370","Text":"y plus Pi over 2 when we substitute back or substitute."},{"Start":"01:36.080 ","End":"01:40.305","Text":"What we have now is that f of x,"},{"Start":"01:40.305 ","End":"01:46.560","Text":"in terms of y, is equal to sine of x,"},{"Start":"01:46.560 ","End":"01:53.540","Text":"which is y plus Pi over 2 but this time we want it expanded around y equals 0,"},{"Start":"01:53.540 ","End":"01:59.900","Text":"which is the Maclaurin series."},{"Start":"01:59.900 ","End":"02:01.580","Text":"But if I look in the table,"},{"Start":"02:01.580 ","End":"02:05.750","Text":"I don\u0027t have something for y plus Pi over 2."},{"Start":"02:05.750 ","End":"02:10.730","Text":"I have sine and cosine of just a single variable."},{"Start":"02:10.730 ","End":"02:18.960","Text":"I need to use some trigonometric identities here to get this into just sine or cosine."},{"Start":"02:19.550 ","End":"02:23.555","Text":"Let\u0027s do a bit of trigonometry here."},{"Start":"02:23.555 ","End":"02:33.855","Text":"In general, we have that sine of Alpha is equal"},{"Start":"02:33.855 ","End":"02:40.860","Text":"to cosine of 90 degrees or"},{"Start":"02:40.860 ","End":"02:45.705","Text":"Pi over 2 minus Alpha."},{"Start":"02:45.705 ","End":"02:51.685","Text":"In our case, if I let Alpha be y plus Pi over 2,"},{"Start":"02:51.685 ","End":"02:58.150","Text":"then I\u0027ll have the sine of y plus Pi over 2 is"},{"Start":"02:58.150 ","End":"03:05.145","Text":"equal to cosine of Pi over 2 minus."},{"Start":"03:05.145 ","End":"03:07.755","Text":"You know what? I\u0027m going to do an extra step."},{"Start":"03:07.755 ","End":"03:15.600","Text":"Pi over 2 minus y plus Pi over 2."},{"Start":"03:15.600 ","End":"03:19.080","Text":"This is just cosine,"},{"Start":"03:19.080 ","End":"03:25.075","Text":"the Pi over 2 minus Pi over 2 cancels and I\u0027m just left with minus y."},{"Start":"03:25.075 ","End":"03:27.670","Text":"But remember, cosine is an even function."},{"Start":"03:27.670 ","End":"03:33.030","Text":"Cosine of minus y is cosine of y."},{"Start":"03:33.030 ","End":"03:42.280","Text":"This is very good because now f of x is equal to cosine of y."},{"Start":"03:42.280 ","End":"03:45.910","Text":"We want the Maclaurin expansion and still around y equals 0."},{"Start":"03:45.910 ","End":"03:51.080","Text":"We do have that in the table in the appendix."},{"Start":"03:51.110 ","End":"03:55.300","Text":"Here it is. We don\u0027t need this part."},{"Start":"03:55.300 ","End":"03:57.440","Text":"I\u0027m going to use the Sigma form."},{"Start":"03:57.440 ","End":"04:03.590","Text":"Anyway, the convergences for all x,"},{"Start":"04:03.590 ","End":"04:10.220","Text":"sometimes it\u0027s written as x between minus infinity infinity, same thing."},{"Start":"04:10.220 ","End":"04:12.110","Text":"Only here we have y,"},{"Start":"04:12.110 ","End":"04:15.470","Text":"so this is just equal to,"},{"Start":"04:15.470 ","End":"04:25.370","Text":"and I\u0027ll rewrite f of x is sine x. Cosine y from here would be the sum from n"},{"Start":"04:25.370 ","End":"04:31.579","Text":"goes from 0 to infinity of minus"},{"Start":"04:31.579 ","End":"04:38.000","Text":"1 to the n pair of x,"},{"Start":"04:38.000 ","End":"04:45.345","Text":"y, remember, y to the 2n over 2 n factorial."},{"Start":"04:45.345 ","End":"04:52.990","Text":"Now all that remains to do is to go from y back to x."},{"Start":"04:52.990 ","End":"04:57.429","Text":"We get that sine x equals the sum,"},{"Start":"04:57.429 ","End":"05:00.670","Text":"still from 0 to infinity minus 1 to the n,"},{"Start":"05:00.670 ","End":"05:04.900","Text":"but instead of y, we put x minus Pi over 2."},{"Start":"05:04.900 ","End":"05:09.290","Text":"X minus Pi over 2"},{"Start":"05:10.330 ","End":"05:19.830","Text":"over 2n factorial to the power of 2n."},{"Start":"05:20.800 ","End":"05:27.050","Text":"Then it\u0027s just a matter of interval of convergence."},{"Start":"05:27.050 ","End":"05:32.405","Text":"But this thing, when I"},{"Start":"05:32.405 ","End":"05:38.700","Text":"put it with y to the 2n over 2n factorial, is all y."},{"Start":"05:39.410 ","End":"05:46.325","Text":"If it\u0027s all y and I replace y by x minus Pi over 2,"},{"Start":"05:46.325 ","End":"05:49.910","Text":"it\u0027s like adding or subtracting Pi over 2 from infinity."},{"Start":"05:49.910 ","End":"05:51.715","Text":"It\u0027s still all x."},{"Start":"05:51.715 ","End":"05:54.570","Text":"If it\u0027s for all y then any x."},{"Start":"05:54.570 ","End":"05:59.115","Text":"Anyway, it\u0027s just all x is the radius of convergence."},{"Start":"05:59.115 ","End":"06:01.620","Text":"This does indeed look like this."},{"Start":"06:01.620 ","End":"06:08.550","Text":"If I rewrote it so that the a_n was minus 1 to the n. If you want,"},{"Start":"06:08.550 ","End":"06:14.915","Text":"you could write this as minus 1 to the n over 2n factorial."},{"Start":"06:14.915 ","End":"06:17.030","Text":"Don\u0027t have to do this. I\u0027m just saying,"},{"Start":"06:17.030 ","End":"06:21.875","Text":"if you want to hold me to my word that this is what we\u0027re looking for,"},{"Start":"06:21.875 ","End":"06:23.870","Text":"then this looks like this."},{"Start":"06:23.870 ","End":"06:26.975","Text":"If we let a_n be this."},{"Start":"06:26.975 ","End":"06:30.640","Text":"Anyway, this is the Taylor expansion."},{"Start":"06:30.640 ","End":"06:35.820","Text":"This is our answer and I\u0027ll highlight it and we are done."}],"ID":6095}],"Thumbnail":null,"ID":84686},{"Name":"Finding Nonzero Terms in Expansions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"7m 51s","ChapterTopicVideoID":6083,"CourseChapterTopicPlaylistID":84687,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6083.jpeg","UploadDate":"2016-05-08T11:15:07.4770000","DurationForVideoObject":"PT7M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.125","Text":"In this exercise, we\u0027re given the function f of x equals e to the minus x squared"},{"Start":"00:07.125 ","End":"00:14.085","Text":"cosine x and we have to find the first 4 non-zero terms of the Maclaurin series,"},{"Start":"00:14.085 ","End":"00:18.360","Text":"which means like the Taylor expansion around x equals 0."},{"Start":"00:18.360 ","End":"00:21.285","Text":"Now we\u0027re not going to do it from scratch,"},{"Start":"00:21.285 ","End":"00:28.065","Text":"which we could do by differentiating 4 times or more."},{"Start":"00:28.065 ","End":"00:32.325","Text":"We\u0027re going to use the table in the appendix,"},{"Start":"00:32.325 ","End":"00:34.860","Text":"we don\u0027t have it exactly for this function,"},{"Start":"00:34.860 ","End":"00:38.340","Text":"but we do have for e to the x and for cosine x."},{"Start":"00:38.340 ","End":"00:43.470","Text":"So let me bring in both those formulas and here they"},{"Start":"00:43.470 ","End":"00:49.765","Text":"are and they both defined for all x."},{"Start":"00:49.765 ","End":"00:57.630","Text":"Or if you like, the interval of convergence is minus infinity to infinity."},{"Start":"00:57.630 ","End":"01:02.600","Text":"Now 1 of them we have immediately is the cosine x,"},{"Start":"01:02.600 ","End":"01:04.610","Text":"the e to the minus x squared."},{"Start":"01:04.610 ","End":"01:08.945","Text":"We\u0027re going to have to replace x with x squared."},{"Start":"01:08.945 ","End":"01:10.590","Text":"Just make a note of that,"},{"Start":"01:10.590 ","End":"01:15.530","Text":"in this formula I\u0027m going to replace x by minus x squared."},{"Start":"01:15.530 ","End":"01:22.160","Text":"What we get is that e to the minus x squared equals."},{"Start":"01:22.160 ","End":"01:25.130","Text":"Now sometimes I use the Sigma form,"},{"Start":"01:25.130 ","End":"01:30.320","Text":"but here we\u0027re talking specifically about first terms."},{"Start":"01:30.320 ","End":"01:36.515","Text":"I\u0027d rather take the expanded and if I need more I\u0027ll add more terms."},{"Start":"01:36.515 ","End":"01:43.985","Text":"What we have is I\u0027m taking this and replacing wherever I see x,"},{"Start":"01:43.985 ","End":"01:45.200","Text":"I\u0027m not going to use the Sigma form,"},{"Start":"01:45.200 ","End":"01:46.520","Text":"I\u0027m going to use this form."},{"Start":"01:46.520 ","End":"01:50.615","Text":"Wherever I see x, I\u0027ll put minus x squared in it,"},{"Start":"01:50.615 ","End":"02:00.450","Text":"so we get 1 and then minus x squared over 1 factorial"},{"Start":"02:00.450 ","End":"02:07.290","Text":"plus minus x squared over"},{"Start":"02:07.290 ","End":"02:18.065","Text":"2 factorial plus minus x squared cubed over 3 factorial, and so on."},{"Start":"02:18.065 ","End":"02:22.370","Text":"Let\u0027s see what this slightly simplifies to."},{"Start":"02:22.370 ","End":"02:24.140","Text":"This is 1."},{"Start":"02:24.140 ","End":"02:26.105","Text":"Now 1 factorial is 1."},{"Start":"02:26.105 ","End":"02:29.490","Text":"This just gives us minus x squared."},{"Start":"02:29.740 ","End":"02:35.195","Text":"This thing squared gives us x to the fourth and 2 factorial is 2,"},{"Start":"02:35.195 ","End":"02:38.990","Text":"so plus x to the fourth over 2."},{"Start":"02:38.990 ","End":"02:43.175","Text":"Here we have minus cubed,"},{"Start":"02:43.175 ","End":"02:45.160","Text":"which is going to be minus."},{"Start":"02:45.160 ","End":"02:50.810","Text":"It\u0027s minus x to the sixth and 3 factorial is 6."},{"Start":"02:50.810 ","End":"02:56.450","Text":"I have a feeling that this will be enough terms."},{"Start":"02:56.450 ","End":"02:58.190","Text":"It goes minus, plus, minus,"},{"Start":"02:58.190 ","End":"03:00.964","Text":"plus and so on. That\u0027s this 1."},{"Start":"03:00.964 ","End":"03:04.100","Text":"The cosine x, all I need to do is copy."},{"Start":"03:04.100 ","End":"03:11.760","Text":"Cosine x is 1 minus x squared over"},{"Start":"03:11.760 ","End":"03:22.425","Text":"2 plus x to the fourth over 4 factorial is 24,"},{"Start":"03:22.425 ","End":"03:28.755","Text":"minus x to the 6 and 6 factorial is 720."},{"Start":"03:28.755 ","End":"03:33.850","Text":"You can check and etc."},{"Start":"03:34.160 ","End":"03:39.375","Text":"What I\u0027m going to do is multiply these 2 polynomials together"},{"Start":"03:39.375 ","End":"03:44.500","Text":"and just take first 4 non-zeros."},{"Start":"03:44.500 ","End":"03:47.735","Text":"I\u0027ll collect ones, x squared,"},{"Start":"03:47.735 ","End":"03:50.135","Text":"x to the fourths and x to the sixth."},{"Start":"03:50.135 ","End":"03:55.095","Text":"Anything beyond will give me beyond 4 terms."},{"Start":"03:55.095 ","End":"03:58.185","Text":"Let\u0027s see, if I multiply them together,"},{"Start":"03:58.185 ","End":"04:04.415","Text":"so e to the minus x squared cosine x equals."},{"Start":"04:04.415 ","End":"04:08.580","Text":"Now I\u0027m going to organize it properly."},{"Start":"04:09.130 ","End":"04:11.935","Text":"I\u0027ll just highlight them."},{"Start":"04:11.935 ","End":"04:17.975","Text":"This polynomial though really to series,"},{"Start":"04:17.975 ","End":"04:19.595","Text":"but I\u0027ll just take these,"},{"Start":"04:19.595 ","End":"04:25.810","Text":"multiply with these and let\u0027s see what we get."},{"Start":"04:25.810 ","End":"04:30.230","Text":"Let me take the 1 here and multiply by each of these."},{"Start":"04:30.230 ","End":"04:38.645","Text":"The first thing I\u0027ll get will be 1 minus x squared over 2 plus x to the fourth over 24,"},{"Start":"04:38.645 ","End":"04:42.485","Text":"minus x to the sixth over 720."},{"Start":"04:42.485 ","End":"04:47.120","Text":"The rest of it will be beyond x to the sixth."},{"Start":"04:47.120 ","End":"04:48.880","Text":"I won\u0027t take anymore."},{"Start":"04:48.880 ","End":"04:53.800","Text":"The next thing I\u0027ll do is take the minus x squared and multiply it with all of"},{"Start":"04:53.800 ","End":"04:59.565","Text":"these and I found this trick of putting them in rows,"},{"Start":"04:59.565 ","End":"05:03.720","Text":"layered, indented, like minus x squared"},{"Start":"05:03.720 ","End":"05:09.440","Text":"times 1 is minus x squared and it goes under the x squared."},{"Start":"05:09.750 ","End":"05:16.260","Text":"Minus x squared times minus x squared over 2 gives me x to the fourth over 2."},{"Start":"05:16.260 ","End":"05:21.220","Text":"It\u0027s plus x to the fourth over 2 and the x to the fourths are aligned."},{"Start":"05:21.220 ","End":"05:27.940","Text":"Then x squared with this 1 becomes minus x to the"},{"Start":"05:27.940 ","End":"05:33.020","Text":"sixth over 24 and the rest of"},{"Start":"05:33.020 ","End":"05:37.100","Text":"it doesn\u0027t matter because I only need 4 non-zero terms of just taking 1 x squared,"},{"Start":"05:37.100 ","End":"05:38.285","Text":"x fourth, x to the sixth."},{"Start":"05:38.285 ","End":"05:41.195","Text":"But here it goes on, here it goes up. I don\u0027t care."},{"Start":"05:41.195 ","End":"05:44.480","Text":"The next thing I\u0027ll do will be to take the x to the"},{"Start":"05:44.480 ","End":"05:48.230","Text":"fourth over 2 and multiply by however many I need."},{"Start":"05:48.230 ","End":"05:56.680","Text":"X to the fourth over 2 times 1 will be x to the fourth over 2."},{"Start":"05:56.680 ","End":"06:03.320","Text":"I will get that 1 again and x to the fourth over 2 times minus x"},{"Start":"06:03.320 ","End":"06:06.440","Text":"squared over 2 will give me minus x to the sixth over 2"},{"Start":"06:06.440 ","End":"06:10.355","Text":"times 2 minus x to the sixth over 4,"},{"Start":"06:10.355 ","End":"06:13.270","Text":"and the rest of it doesn\u0027t matter."},{"Start":"06:13.270 ","End":"06:18.290","Text":"Next is this 1 taken with however many of these I need."},{"Start":"06:18.290 ","End":"06:21.470","Text":"I only need 1 of them because minus x to the sixth over 6 times"},{"Start":"06:21.470 ","End":"06:26.240","Text":"1 is minus x to the sixth over 6,"},{"Start":"06:26.240 ","End":"06:29.235","Text":"and the rest of it is higher powers and whatever I do"},{"Start":"06:29.235 ","End":"06:32.510","Text":"now is just total them according to columns."},{"Start":"06:32.510 ","End":"06:36.485","Text":"I\u0027ll put a line here and say, okay, first column,"},{"Start":"06:36.485 ","End":"06:40.310","Text":"I have 1, next column for the x squared,"},{"Start":"06:40.310 ","End":"06:44.000","Text":"I have minus 1/2 and minus 1."},{"Start":"06:44.000 ","End":"06:51.565","Text":"That\u0027s minus 1.5 or minus 3 over 2 however you want it x squared."},{"Start":"06:51.565 ","End":"06:54.510","Text":"Next is the x to the fourth."},{"Start":"06:54.510 ","End":"06:56.330","Text":"I have first of all from here and here,"},{"Start":"06:56.330 ","End":"06:58.055","Text":"a half and a half is 1."},{"Start":"06:58.055 ","End":"07:03.770","Text":"1 over 1 over 24 as an improper fraction will be"},{"Start":"07:03.770 ","End":"07:10.340","Text":"25 over 24 x to the fourth and the last 1,"},{"Start":"07:10.340 ","End":"07:12.230","Text":"it\u0027s going to be minus."},{"Start":"07:12.230 ","End":"07:16.985","Text":"These are all minus is going to be something x to the sixth and what I have to do"},{"Start":"07:16.985 ","End":"07:22.745","Text":"is 1 over 720 plus 1 over 24 plus 1/4 plus 1/6."},{"Start":"07:22.745 ","End":"07:24.739","Text":"I did it at the side on the calculator."},{"Start":"07:24.739 ","End":"07:26.300","Text":"I\u0027ll just quote the result here."},{"Start":"07:26.300 ","End":"07:30.960","Text":"It comes out 331 over"},{"Start":"07:30.960 ","End":"07:36.890","Text":"720 and we are done."},{"Start":"07:36.890 ","End":"07:42.365","Text":"I just want to highlight the result and here we are."},{"Start":"07:42.365 ","End":"07:50.950","Text":"This is an approximation to this with 4 terms. We are done."}],"ID":6096},{"Watched":false,"Name":"Exercise 2","Duration":"9m 20s","ChapterTopicVideoID":6084,"CourseChapterTopicPlaylistID":84687,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6084.jpeg","UploadDate":"2016-05-08T11:16:19.3270000","DurationForVideoObject":"PT9M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.720","Text":"In this exercise, we\u0027re given the function f of x equals the tangent of x"},{"Start":"00:06.720 ","End":"00:13.825","Text":"and what we want to do is find the first 4 non-zero terms of the Maclaurin series."},{"Start":"00:13.825 ","End":"00:21.395","Text":"Now, we could do it straight off the basic definition or formula,"},{"Start":"00:21.395 ","End":"00:27.620","Text":"which is to keep differentiating and substituting 0 and so on,"},{"Start":"00:27.620 ","End":"00:29.390","Text":"but there was a hint in"},{"Start":"00:29.390 ","End":"00:35.420","Text":"the exercise book to use multiplication or division of polynomials,"},{"Start":"00:35.420 ","End":"00:37.295","Text":"and that\u0027s what we\u0027re going to do."},{"Start":"00:37.295 ","End":"00:41.585","Text":"We look in the table of"},{"Start":"00:41.585 ","End":"00:46.310","Text":"Maclaurin series for certain well-known series and we don\u0027t find tangent there,"},{"Start":"00:46.310 ","End":"00:48.665","Text":"but we do find sine and cosine."},{"Start":"00:48.665 ","End":"00:51.140","Text":"We remember from trigonometry,"},{"Start":"00:51.140 ","End":"00:57.030","Text":"that tangent is just the sine over the cosine."},{"Start":"01:00.260 ","End":"01:07.195","Text":"Both of these do appear in the table of Maclaurin series and I\u0027ll just copy them in."},{"Start":"01:07.195 ","End":"01:10.920","Text":"Here they are, sine and cosine."},{"Start":"01:10.920 ","End":"01:13.425","Text":"We don\u0027t need the Sigma form,"},{"Start":"01:13.425 ","End":"01:16.505","Text":"we\u0027re going to use the expanded form."},{"Start":"01:16.505 ","End":"01:21.730","Text":"In actual fact, we have 4 terms already."},{"Start":"01:21.730 ","End":"01:24.900","Text":"It turns out we just need these 4."},{"Start":"01:24.900 ","End":"01:28.920","Text":"If we need, we could always add a next 1,"},{"Start":"01:28.920 ","End":"01:31.130","Text":"x^9 over 9 factorial and so on."},{"Start":"01:31.130 ","End":"01:33.665","Text":"But let\u0027s see if we can get by with these 4."},{"Start":"01:33.665 ","End":"01:36.830","Text":"Let\u0027s do the long division problem."},{"Start":"01:36.830 ","End":"01:39.350","Text":"What we need to do,"},{"Start":"01:39.350 ","End":"01:41.420","Text":"is a long division."},{"Start":"01:41.420 ","End":"01:42.990","Text":"Let me choose a different color."},{"Start":"01:42.990 ","End":"01:47.050","Text":"First of all, let me write the division sign."},{"Start":"01:47.050 ","End":"01:53.520","Text":"Here I\u0027ll put the first few terms of the cosine. Let\u0027s see."},{"Start":"01:53.520 ","End":"02:03.390","Text":"1 minus x squared over 2 plus x^4 over 24."},{"Start":"02:03.390 ","End":"02:06.825","Text":"I need more room."},{"Start":"02:06.825 ","End":"02:11.850","Text":"6 factorial is 720,"},{"Start":"02:11.850 ","End":"02:18.700","Text":"so it\u0027s minus x^6 over 720."},{"Start":"02:18.700 ","End":"02:22.860","Text":"I\u0027ll just put in the dot-dot-dot to remind us that there are actually more."},{"Start":"02:22.860 ","End":"02:24.960","Text":"If we need more, we\u0027ll add more,"},{"Start":"02:24.960 ","End":"02:30.320","Text":"but from experience, we can get by with just these 4 here and here."},{"Start":"02:30.320 ","End":"02:33.700","Text":"Under here, I\u0027ll put the sine x."},{"Start":"02:33.700 ","End":"02:40.225","Text":"So that\u0027s x minus x cubed over 6,"},{"Start":"02:40.225 ","End":"02:45.220","Text":"assuming you know the factorials or you use the calculator,"},{"Start":"02:45.220 ","End":"02:57.020","Text":"plus x^5 over 120 minus x^7."},{"Start":"02:57.020 ","End":"02:58.200","Text":"This 1 I did look up."},{"Start":"02:58.200 ","End":"02:59.940","Text":"I know them up to 6 factorial."},{"Start":"02:59.940 ","End":"03:03.490","Text":"This came out to 5,040."},{"Start":"03:04.460 ","End":"03:09.640","Text":"Also, I\u0027ll put the dot-dot-dot to indicate that there are more,"},{"Start":"03:09.640 ","End":"03:12.580","Text":"but I don\u0027t think we\u0027ll need them."},{"Start":"03:12.580 ","End":"03:19.760","Text":"Here we go, 1 into x goes x times."},{"Start":"03:21.660 ","End":"03:25.345","Text":"Then we multiply x by this,"},{"Start":"03:25.345 ","End":"03:31.585","Text":"so we get x minus x cubed"},{"Start":"03:31.585 ","End":"03:41.565","Text":"over 2 plus x^5 over 24."},{"Start":"03:41.565 ","End":"03:43.080","Text":"It\u0027s the same denominators,"},{"Start":"03:43.080 ","End":"03:46.740","Text":"we\u0027re just raising the power of x by 1."},{"Start":"03:46.740 ","End":"03:57.915","Text":"Then minus x^7 over 720 plus dot-dot-dot."},{"Start":"03:57.915 ","End":"04:05.175","Text":"Now we subtract and we get x minus x is nothing."},{"Start":"04:05.175 ","End":"04:09.710","Text":"You always get nothing in the beginning if you\u0027ve done this correctly."},{"Start":"04:09.710 ","End":"04:16.585","Text":"Here we have minus 1/6 minus minus 1/2."},{"Start":"04:16.585 ","End":"04:19.305","Text":"It comes out to be plus 1/3,"},{"Start":"04:19.305 ","End":"04:21.120","Text":"the x cubed, of course."},{"Start":"04:21.120 ","End":"04:25.120","Text":"So it\u0027s 1/3x cubed."},{"Start":"04:26.060 ","End":"04:32.690","Text":"This minus this, I\u0027ll leave it to you to do fractions."},{"Start":"04:32.690 ","End":"04:41.135","Text":"1/120 minus 1/24 is minus 1/30."},{"Start":"04:41.135 ","End":"04:46.190","Text":"This is x^5."},{"Start":"04:46.190 ","End":"04:54.170","Text":"Minus 1/5,040 plus 1/720 comes out to"},{"Start":"04:54.170 ","End":"05:03.390","Text":"be plus 1/840 x^7 plus,"},{"Start":"05:03.390 ","End":"05:07.125","Text":"I don\u0027t think we\u0027ll need these, but dot-dot-dot."},{"Start":"05:07.125 ","End":"05:11.405","Text":"Now, we ask how many times does 1 go into this?"},{"Start":"05:11.405 ","End":"05:13.370","Text":"It\u0027s very convenient having a 1 here,"},{"Start":"05:13.370 ","End":"05:14.735","Text":"it makes it so much easier."},{"Start":"05:14.735 ","End":"05:25.125","Text":"1 into 1/3x cubed is just 1/3x cubed."},{"Start":"05:25.125 ","End":"05:28.620","Text":"Then we multiply 1/3x cubed by this,"},{"Start":"05:28.620 ","End":"05:31.784","Text":"so we get 1/3x cubed."},{"Start":"05:31.784 ","End":"05:35.685","Text":"All is well if the first 1 comes out the same."},{"Start":"05:35.685 ","End":"05:46.590","Text":"1/3 times this comes out to be the minus 1/6."},{"Start":"05:46.590 ","End":"05:52.845","Text":"1/3 times 1/2, yeah. x^5, and then 1/3"},{"Start":"05:52.845 ","End":"06:05.250","Text":"times 1/24 is 1/72x^7."},{"Start":"06:05.250 ","End":"06:08.055","Text":"I\u0027m not going to need the next 1."},{"Start":"06:08.055 ","End":"06:09.790","Text":"As I said, if it turns out we do,"},{"Start":"06:09.790 ","End":"06:11.425","Text":"we can always go back,"},{"Start":"06:11.425 ","End":"06:14.750","Text":"and then we do another subtraction."},{"Start":"06:14.940 ","End":"06:18.940","Text":"This comes out 2/15x^5."},{"Start":"06:18.940 ","End":"06:21.955","Text":"I\u0027ll leave you to do all the fractions."},{"Start":"06:21.955 ","End":"06:24.505","Text":"I don\u0027t want to waste time with that."},{"Start":"06:24.505 ","End":"06:37.780","Text":"Here we get minus 4/315x^7."},{"Start":"06:37.780 ","End":"06:43.370","Text":"Now, 1 is just this 2/15x^5."},{"Start":"06:45.800 ","End":"06:52.210","Text":"Multiply this by this and we get 2/15x^5."},{"Start":"06:52.210 ","End":"06:59.400","Text":"2/15 times 1/2 is just 1/15,"},{"Start":"06:59.400 ","End":"07:05.250","Text":"so it\u0027s minus 1/15."},{"Start":"07:05.250 ","End":"07:09.840","Text":"x^5 times x squared is x^7 so on."},{"Start":"07:09.840 ","End":"07:12.730","Text":"Another subtraction."},{"Start":"07:13.070 ","End":"07:15.840","Text":"This minus this cancels."},{"Start":"07:15.840 ","End":"07:18.510","Text":"This minus this turns out to"},{"Start":"07:18.510 ","End":"07:28.075","Text":"be 17/315x^7 plus higher-order terms."},{"Start":"07:28.075 ","End":"07:40.315","Text":"This into this goes 17/315x^7 times."},{"Start":"07:40.315 ","End":"07:49.440","Text":"We get 17/315x^7."},{"Start":"07:49.440 ","End":"07:52.050","Text":"I don\u0027t care about the others."},{"Start":"07:52.050 ","End":"07:58.800","Text":"Finally, when we subtract this we don\u0027t get 0,"},{"Start":"07:58.800 ","End":"08:06.285","Text":"but we just get the dot-dot-dot of the higher powers."},{"Start":"08:06.285 ","End":"08:09.660","Text":"We already have 4 non-zero terms here,"},{"Start":"08:09.660 ","End":"08:14.190","Text":"so I\u0027ll also indicate that there is a continuation here."},{"Start":"08:14.190 ","End":"08:18.800","Text":"But because the question did say first 4 non-zero terms,"},{"Start":"08:18.800 ","End":"08:22.425","Text":"that\u0027s it. Here we are."},{"Start":"08:22.425 ","End":"08:27.470","Text":"Let me just say a word about the interval of convergence."},{"Start":"08:27.470 ","End":"08:31.745","Text":"Sine and cosine, the Maclaurin series,"},{"Start":"08:31.745 ","End":"08:36.545","Text":"is for all x from minus infinity to infinity."},{"Start":"08:36.545 ","End":"08:40.055","Text":"The thing is, when we take sine over cosine,"},{"Start":"08:40.055 ","End":"08:43.110","Text":"the denominator can\u0027t be 0."},{"Start":"08:43.110 ","End":"08:54.315","Text":"Cosine is 0 at 90 degrees Pi over 2 and also at minus 90 degrees."},{"Start":"08:54.315 ","End":"08:56.400","Text":"We have trouble there."},{"Start":"08:56.400 ","End":"09:02.900","Text":"Actually, the interval of convergence will be where x is"},{"Start":"09:02.900 ","End":"09:11.045","Text":"between 90 degrees Pi over 2 in radians and minus 90 degrees minus Pi over 2 in radians."},{"Start":"09:11.045 ","End":"09:15.379","Text":"That will really complete the question."},{"Start":"09:15.379 ","End":"09:17.570","Text":"I\u0027ll highlight this too."},{"Start":"09:17.570 ","End":"09:21.040","Text":"Done."}],"ID":6097},{"Watched":false,"Name":"Exercise 3","Duration":"8m 58s","ChapterTopicVideoID":6082,"CourseChapterTopicPlaylistID":84687,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6082.jpeg","UploadDate":"2016-05-08T11:14:06.8430000","DurationForVideoObject":"PT8M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.090","Text":"In this exercise, we\u0027re given the function f of x equals sine x over e to the x,"},{"Start":"00:09.090 ","End":"00:13.980","Text":"and we have to find the first 4 non-zero terms of the Maclaurin series,"},{"Start":"00:13.980 ","End":"00:18.239","Text":"which is the Taylor series about x equal 0."},{"Start":"00:18.239 ","End":"00:22.695","Text":"We\u0027re going to do it using the tables in the appendix."},{"Start":"00:22.695 ","End":"00:25.590","Text":"Of course, we don\u0027t have exactly sine x over e to the x."},{"Start":"00:25.590 ","End":"00:28.320","Text":"We\u0027re going to have to do some work ourselves."},{"Start":"00:28.320 ","End":"00:31.875","Text":"We do have sine x there I know and we do have e to the x."},{"Start":"00:31.875 ","End":"00:35.590","Text":"Let me just copy those."},{"Start":"00:36.020 ","End":"00:39.730","Text":"Here we are, sine x and e to the x."},{"Start":"00:39.730 ","End":"00:42.440","Text":"We\u0027re not going to use the Sigma form,"},{"Start":"00:42.440 ","End":"00:44.420","Text":"we\u0027re going to use the expanded form,"},{"Start":"00:44.420 ","End":"00:48.450","Text":"and they\u0027ve even been nice enough to give us 4 terms."},{"Start":"00:49.040 ","End":"00:57.075","Text":"Now, the obvious thing to do would be division and I think we had one of those before."},{"Start":"00:57.075 ","End":"00:58.880","Text":"Let\u0027s go and do that."},{"Start":"00:58.880 ","End":"01:02.815","Text":"But I\u0027ll just mention that there is a shortcut you could take"},{"Start":"01:02.815 ","End":"01:08.065","Text":"and that is to write this as sine x,"},{"Start":"01:08.065 ","End":"01:12.725","Text":"e to the minus x and make it into a multiplication."},{"Start":"01:12.725 ","End":"01:15.635","Text":"But I\u0027m deliberately doing it with division"},{"Start":"01:15.635 ","End":"01:19.100","Text":"because that\u0027s the harder one and you need more practice with that."},{"Start":"01:19.100 ","End":"01:21.649","Text":"But bear in mind that there is a shortcut,"},{"Start":"01:21.649 ","End":"01:23.570","Text":"you could do it e to the minus x,"},{"Start":"01:23.570 ","End":"01:25.385","Text":"just replace x with minus x,"},{"Start":"01:25.385 ","End":"01:27.955","Text":"and then make it a multiplication."},{"Start":"01:27.955 ","End":"01:30.740","Text":"As far as the interval of convergence,"},{"Start":"01:30.740 ","End":"01:35.930","Text":"they both converge for all x and e to the x is never 0,"},{"Start":"01:35.930 ","End":"01:38.120","Text":"so there\u0027s no problem with the division."},{"Start":"01:38.120 ","End":"01:41.940","Text":"Let\u0027s go ahead and set up a division problem."},{"Start":"01:42.920 ","End":"01:46.600","Text":"Let\u0027s write a division sign,"},{"Start":"01:47.540 ","End":"01:51.800","Text":"and the numerator, the sine x goes in here,"},{"Start":"01:51.800 ","End":"01:53.900","Text":"so we\u0027re copying it from here,"},{"Start":"01:53.900 ","End":"02:03.360","Text":"x minus 1/6, 3 factorial is 6,"},{"Start":"02:03.360 ","End":"02:08.610","Text":"so it\u0027s minus 1/6x cubed."},{"Start":"02:08.610 ","End":"02:13.680","Text":"5 factorial is 120,"},{"Start":"02:13.680 ","End":"02:17.175","Text":"so here\u0027s the x to the 5th term,"},{"Start":"02:17.175 ","End":"02:21.390","Text":"and 7 factorial is 5,040,"},{"Start":"02:21.390 ","End":"02:31.365","Text":"so we have 1 over 5,040x to the 7th."},{"Start":"02:31.365 ","End":"02:35.230","Text":"I\u0027ll just indicate that it does continue."},{"Start":"02:35.810 ","End":"02:39.585","Text":"If we need more terms we can always get them,"},{"Start":"02:39.585 ","End":"02:42.350","Text":"x to the 9th over 9 factorial and so on."},{"Start":"02:42.350 ","End":"02:46.889","Text":"Here I\u0027ll put the first part of e to the x,"},{"Start":"02:46.889 ","End":"02:52.625","Text":"so we have 1 plus x to the 1 over 1 factorial is x."},{"Start":"02:52.625 ","End":"02:55.750","Text":"This is x squared over 2."},{"Start":"02:55.750 ","End":"03:00.700","Text":"This is x cubed over 6."},{"Start":"03:02.870 ","End":"03:10.260","Text":"I\u0027ll just write dot-dot-dot to show that we acknowledge that there is more,"},{"Start":"03:10.260 ","End":"03:12.700","Text":"we just want 4 terms."},{"Start":"03:12.770 ","End":"03:14.970","Text":"I start the division,"},{"Start":"03:14.970 ","End":"03:18.690","Text":"1 into x goes x times,"},{"Start":"03:18.690 ","End":"03:23.340","Text":"multiply x by this and we have x plus x"},{"Start":"03:23.340 ","End":"03:30.240","Text":"squared plus, and then you know what?"},{"Start":"03:30.240 ","End":"03:31.890","Text":"At this point I realize that we have"},{"Start":"03:31.890 ","End":"03:34.580","Text":"an alignment problem because there\u0027s missing bits here,"},{"Start":"03:34.580 ","End":"03:39.115","Text":"so why don\u0027t I expand this and leave blanks for the 0?"},{"Start":"03:39.115 ","End":"03:44.099","Text":"Then I stretched it a bit and then I left blanks for the odd powers,"},{"Start":"03:44.099 ","End":"03:45.330","Text":"x squared, x to the 4th,"},{"Start":"03:45.330 ","End":"03:46.800","Text":"and x to the 6th are missing."},{"Start":"03:46.800 ","End":"03:53.660","Text":"Back to here, x times this is 1 plus x squared plus x cubed over"},{"Start":"03:53.660 ","End":"04:01.400","Text":"2 and then plus"},{"Start":"04:01.400 ","End":"04:05.710","Text":"x to the 4th over 6."},{"Start":"04:05.710 ","End":"04:13.475","Text":"At this point I see I really should have taken another term here."},{"Start":"04:13.475 ","End":"04:17.320","Text":"Let me add 1 more here and I\u0027ll make some space."},{"Start":"04:17.320 ","End":"04:23.640","Text":"The next one in this series is x to the 4th over 4 factorial,"},{"Start":"04:23.640 ","End":"04:28.110","Text":"x to the 4th over 24."},{"Start":"04:28.110 ","End":"04:38.130","Text":"Now I can take the x with the x to the 4th over 24 and get x to the 5th over 24."},{"Start":"04:38.130 ","End":"04:42.230","Text":"I can see already I\u0027m not going to be needing this last one."},{"Start":"04:42.230 ","End":"04:44.730","Text":"Yeah, I just erase that."},{"Start":"04:44.730 ","End":"04:47.950","Text":"Now let\u0027s do the subtraction."},{"Start":"04:48.830 ","End":"04:53.940","Text":"Let\u0027s see, x minus x is nothing."},{"Start":"04:53.940 ","End":"04:57.030","Text":"Then we have minus x squared,"},{"Start":"04:57.030 ","End":"05:01.840","Text":"and then we have 1/2 minus a 1/6."},{"Start":"05:01.840 ","End":"05:07.050","Text":"No, we have minus a 1/6 minus a 1/2,"},{"Start":"05:07.050 ","End":"05:13.820","Text":"which is minus 2/3x cubed,"},{"Start":"05:13.820 ","End":"05:17.430","Text":"and then we have minus"},{"Start":"05:19.180 ","End":"05:24.935","Text":"1/6x to the 4th."},{"Start":"05:24.935 ","End":"05:35.250","Text":"Then if I do this subtraction I get minus 1/30 of x to the 5th."},{"Start":"05:37.220 ","End":"05:44.380","Text":"Next, we want to see how many times 1 goes into minus x squared."},{"Start":"05:44.380 ","End":"05:49.700","Text":"It\u0027s just minus x squared times multiply by this,"},{"Start":"05:49.700 ","End":"06:00.780","Text":"minus x squared minus x cubed minus 1/2x to the 4th."},{"Start":"06:00.860 ","End":"06:10.785","Text":"Next one would be minus a 1/6x to the 5th, and so on."},{"Start":"06:10.785 ","End":"06:17.680","Text":"Then a subtraction, and we get this of course is nothing."},{"Start":"06:17.680 ","End":"06:23.170","Text":"We have 1 minus 2/3 is 1/3x cubed."},{"Start":"06:23.170 ","End":"06:28.705","Text":"A 1/2 minus 1/6 is a 1/3x to the 4th."},{"Start":"06:28.705 ","End":"06:34.760","Text":"Here we have 1/6 minus 1/30,"},{"Start":"06:34.920 ","End":"06:40.330","Text":"which is 5/30 minus 1/30 is 4/30,"},{"Start":"06:40.330 ","End":"06:48.430","Text":"that\u0027s 2/15x to the 5th."},{"Start":"06:48.430 ","End":"06:58.045","Text":"Now, 1 into 1/3x cubed is 1/3x cubed."},{"Start":"06:58.045 ","End":"07:06.250","Text":"This times this 1/3x cubed plus 1/3x to"},{"Start":"07:06.250 ","End":"07:12.895","Text":"the 4th and the next one"},{"Start":"07:12.895 ","End":"07:18.890","Text":"is 1/6x to the 5th."},{"Start":"07:20.840 ","End":"07:24.675","Text":"The rest of it doesn\u0027t matter."},{"Start":"07:24.675 ","End":"07:27.730","Text":"Subtract."},{"Start":"07:27.800 ","End":"07:38.430","Text":"Both of these cancel which means that I\u0027m going to get a 0x to the 4th you\u0027ll see."},{"Start":"07:38.430 ","End":"07:42.525","Text":"Both these cancel, so all I have is the 2/15 minus the"},{"Start":"07:42.525 ","End":"07:50.650","Text":"1/6 and this difference gives me minus 1/30x to the 5th."},{"Start":"07:51.530 ","End":"07:57.990","Text":"Then 1 into this goes minus 1/30x to the 5th."},{"Start":"07:57.990 ","End":"07:59.460","Text":"I\u0027ll write it here,"},{"Start":"07:59.460 ","End":"08:03.550","Text":"minus 1/30x to the 5th."},{"Start":"08:03.550 ","End":"08:07.580","Text":"I\u0027m just putting a blank here to show there is a missing x to the 4th,"},{"Start":"08:07.580 ","End":"08:10.355","Text":"but we want non-zero term so we already have 1,"},{"Start":"08:10.355 ","End":"08:12.900","Text":"2, 3, and 4."},{"Start":"08:14.330 ","End":"08:20.550","Text":"That\u0027s basically the point at which we can stop."},{"Start":"08:20.960 ","End":"08:23.089","Text":"That\u0027s our answer."},{"Start":"08:23.089 ","End":"08:28.115","Text":"This is equal to x minus x squared"},{"Start":"08:28.115 ","End":"08:36.635","Text":"plus 1/3x cubed minus 1/30x to the 5th."},{"Start":"08:36.635 ","End":"08:41.455","Text":"Then there\u0027s more stuff, dot-dot-dot."},{"Start":"08:41.455 ","End":"08:43.610","Text":"If we wanted more places,"},{"Start":"08:43.610 ","End":"08:46.400","Text":"we would have had to make both of these a bit longer."},{"Start":"08:46.400 ","End":"08:47.420","Text":"Exactly how long?"},{"Start":"08:47.420 ","End":"08:48.950","Text":"Well, trial and error."},{"Start":"08:48.950 ","End":"08:53.135","Text":"But let me just highlight this."},{"Start":"08:53.135 ","End":"08:59.160","Text":"This is our answer to 4 places and we\u0027re done."}],"ID":6098},{"Watched":false,"Name":"Exercise 4","Duration":"11m 2s","ChapterTopicVideoID":28811,"CourseChapterTopicPlaylistID":84687,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28811.jpeg","UploadDate":"2022-04-29T12:54:49.0070000","DurationForVideoObject":"PT11M2S","Description":null,"VideoComments":[],"Subtitles":[],"ID":30308}],"Thumbnail":null,"ID":84687},{"Name":"Sum of Series Using Taylor and Maclaurin Expansions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"4m 26s","ChapterTopicVideoID":6089,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6089.jpeg","UploadDate":"2016-05-08T11:18:46.2700000","DurationForVideoObject":"PT4M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we actually have 3 parts,"},{"Start":"00:03.480 ","End":"00:05.910","Text":"but they are related."},{"Start":"00:05.910 ","End":"00:10.769","Text":"It\u0027s in the chapter on Taylor and Maclaurin series."},{"Start":"00:10.769 ","End":"00:12.930","Text":"We know that\u0027s going to come in somewhere."},{"Start":"00:12.930 ","End":"00:18.930","Text":"Now if you look at it, they all have this 1 over n factorial in them,"},{"Start":"00:18.930 ","End":"00:23.280","Text":"and it\u0027s all something to the power of n,"},{"Start":"00:23.280 ","End":"00:29.370","Text":"even this 1 is 1 to the power of n. It really rings a bell"},{"Start":"00:29.370 ","End":"00:33.090","Text":"and brings us to the Maclaurin series"},{"Start":"00:33.090 ","End":"00:37.590","Text":"for e to the power of x. I\u0027ll just bring it in and show you."},{"Start":"00:37.590 ","End":"00:41.450","Text":"Here\u0027s the Maclaurin series for e to the x and it"},{"Start":"00:41.450 ","End":"00:47.180","Text":"converges for all x. I just want the Sigma path. I don\u0027t need this."},{"Start":"00:47.180 ","End":"00:54.800","Text":"What I want to do is replace x with a number to give us these 3,"},{"Start":"00:54.800 ","End":"00:56.585","Text":"each time a different number."},{"Start":"00:56.585 ","End":"01:01.520","Text":"For example, in the 1st part,"},{"Start":"01:01.520 ","End":"01:03.935","Text":"I could let x equals 1,"},{"Start":"01:03.935 ","End":"01:08.010","Text":"and then 1 to the n will be exactly this."},{"Start":"01:09.100 ","End":"01:16.165","Text":"Look, e to the power of 1 is the sum"},{"Start":"01:16.165 ","End":"01:25.090","Text":"from 0 to infinity of 1 to the n over n factorial,"},{"Start":"01:25.090 ","End":"01:27.900","Text":"and this is exactly what\u0027s written here."},{"Start":"01:27.900 ","End":"01:32.435","Text":"I just rewrite it again without the n here,"},{"Start":"01:32.435 ","End":"01:35.390","Text":"1 over n factorial."},{"Start":"01:35.390 ","End":"01:40.455","Text":"In other words, the answer to the 1st question is just"},{"Start":"01:40.455 ","End":"01:47.040","Text":"e. I just write that e is the answer and I\u0027ll highlight it."},{"Start":"01:47.050 ","End":"01:49.910","Text":"I should have written question 1."},{"Start":"01:49.910 ","End":"01:53.060","Text":"A bit of a shift to the right and now that\u0027s number 1."},{"Start":"01:53.060 ","End":"01:56.260","Text":"Now, let\u0027s tackle number 2."},{"Start":"01:56.260 ","End":"01:58.080","Text":"Same idea."},{"Start":"01:58.080 ","End":"02:01.970","Text":"I\u0027ve got to find some x to substitute in order to get this."},{"Start":"02:01.970 ","End":"02:05.930","Text":"The n factorial and the n factorial is the same."},{"Start":"02:05.930 ","End":"02:12.215","Text":"I need something to the power of n. Now I\u0027ve got the n here and here, and this,"},{"Start":"02:12.215 ","End":"02:17.390","Text":"if you multiply it or use the algebra rule for exponents,"},{"Start":"02:17.390 ","End":"02:24.275","Text":"it\u0027s minus 2 to the power of n. What I\u0027m claiming is if I take e to the minus 2,"},{"Start":"02:24.275 ","End":"02:25.730","Text":"I\u0027ll get the same thing."},{"Start":"02:25.730 ","End":"02:30.200","Text":"Look, it\u0027s the sum and goes from 0 to infinity,"},{"Start":"02:30.200 ","End":"02:32.515","Text":"meaning x equals minus 2,"},{"Start":"02:32.515 ","End":"02:40.150","Text":"minus 2 to the n over n factorial,"},{"Start":"02:40.150 ","End":"02:46.240","Text":"and just a slight rewrite gives me the sum,"},{"Start":"02:46.250 ","End":"02:48.625","Text":"so of minus 2 to the n,"},{"Start":"02:48.625 ","End":"02:57.010","Text":"I can put minus 1 to the n separately and 2 the n separately over n factorial."},{"Start":"02:57.200 ","End":"03:02.415","Text":"The answer to number 2 is e to the minus 2,"},{"Start":"03:02.415 ","End":"03:05.820","Text":"or if you like, just rewrite it."},{"Start":"03:05.820 ","End":"03:08.715","Text":"It\u0027s 1 over e squared."},{"Start":"03:08.715 ","End":"03:13.995","Text":"Either way, either this way or this way, both are good."},{"Start":"03:13.995 ","End":"03:22.265","Text":"Next, number 3, I have to decide what is x going to be here."},{"Start":"03:22.265 ","End":"03:25.970","Text":"Well, here I have 1 over 2 to the n,"},{"Start":"03:25.970 ","End":"03:32.690","Text":"and that\u0027s the same as 1/2 to the power of n. It looks like x is going to equal a 1/2."},{"Start":"03:32.690 ","End":"03:37.560","Text":"Let\u0027s try that. E to the power of 1/2 is the sum,"},{"Start":"03:37.560 ","End":"03:41.580","Text":"n goes from 0 to infinity of,"},{"Start":"03:41.580 ","End":"03:42.990","Text":"I\u0027m just following this,"},{"Start":"03:42.990 ","End":"03:49.870","Text":"it\u0027s 1/2 to the n over n factorial."},{"Start":"03:51.140 ","End":"03:55.340","Text":"Just a slight rewrite, take 1 to the n,"},{"Start":"03:55.340 ","End":"04:00.970","Text":"leave it in the numerator as 1 and the 2 to the n in the denominator."},{"Start":"04:00.970 ","End":"04:03.720","Text":"I\u0027ve got 1 over 2 to the n,"},{"Start":"04:03.720 ","End":"04:08.310","Text":"n factorial, your n goes from 0 to infinity."},{"Start":"04:08.310 ","End":"04:11.390","Text":"It looks like this is our answer."},{"Start":"04:11.390 ","End":"04:13.235","Text":"I could leave it like this,"},{"Start":"04:13.235 ","End":"04:15.950","Text":"or I could write it as the square root of e,"},{"Start":"04:15.950 ","End":"04:19.670","Text":"whichever of these 2 forms you prefer."},{"Start":"04:19.670 ","End":"04:21.380","Text":"I\u0027ll go for this 1."},{"Start":"04:21.380 ","End":"04:24.930","Text":"It doesn\u0027t really matter. So yeah, we\u0027re done."}],"ID":6099},{"Watched":false,"Name":"Exercise 2","Duration":"3m 31s","ChapterTopicVideoID":6090,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6090.jpeg","UploadDate":"2016-05-08T11:19:12.1830000","DurationForVideoObject":"PT3M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.375","Text":"This is an important exercise because I\u0027m going to teach you a couple of tricks."},{"Start":"00:06.375 ","End":"00:09.450","Text":"We have to compute the sum of this series,"},{"Start":"00:09.450 ","End":"00:11.249","Text":"and it\u0027s not immediately obvious,"},{"Start":"00:11.249 ","End":"00:17.235","Text":"it doesn\u0027t look like any function that we can just substitute."},{"Start":"00:17.235 ","End":"00:19.605","Text":"Here\u0027s where the trick comes in."},{"Start":"00:19.605 ","End":"00:24.390","Text":"Well, the closest thing is the 1 over n factorial."},{"Start":"00:24.390 ","End":"00:27.780","Text":"That would look like e to the x,"},{"Start":"00:27.780 ","End":"00:30.660","Text":"where x is 1, but we have this n plus 1 here."},{"Start":"00:30.660 ","End":"00:32.790","Text":"So what do we do?"},{"Start":"00:32.790 ","End":"00:37.595","Text":"I copied in the expansion of e to the x,"},{"Start":"00:37.595 ","End":"00:41.945","Text":"the Maclaurin from the appendix."},{"Start":"00:41.945 ","End":"00:44.490","Text":"What we\u0027re going to do, well,"},{"Start":"00:44.490 ","End":"00:47.345","Text":"we\u0027re going to use the Sigma part."},{"Start":"00:47.345 ","End":"00:57.710","Text":"In other words, e to the x is the sum of x to the n over n factorial from 0 to infinity."},{"Start":"00:57.710 ","End":"01:02.710","Text":"The first thing we\u0027re going to do is multiply both sides by x,"},{"Start":"01:02.710 ","End":"01:05.205","Text":"and we\u0027ll see what that\u0027s good for,"},{"Start":"01:05.205 ","End":"01:08.835","Text":"xe to the x will be the sum."},{"Start":"01:08.835 ","End":"01:12.655","Text":"I can multiply the x termwise."},{"Start":"01:12.655 ","End":"01:14.975","Text":"I get x times x to the n,"},{"Start":"01:14.975 ","End":"01:20.200","Text":"which is x to the n plus 1 over n factorial."},{"Start":"01:20.200 ","End":"01:23.055","Text":"Now we have an n plus 1 here."},{"Start":"01:23.055 ","End":"01:24.965","Text":"Now, what do we do with this?"},{"Start":"01:24.965 ","End":"01:28.895","Text":"We\u0027re going to use the trick we\u0027ve used before of differentiation."},{"Start":"01:28.895 ","End":"01:31.475","Text":"If I differentiate this,"},{"Start":"01:31.475 ","End":"01:35.150","Text":"what I will get on the right-hand side,"},{"Start":"01:35.150 ","End":"01:37.280","Text":"I\u0027ll leave space here for the left-hand side."},{"Start":"01:37.280 ","End":"01:38.705","Text":"On the right-hand side,"},{"Start":"01:38.705 ","End":"01:47.535","Text":"I\u0027m going to get the sum from 0 to infinity of n plus 1,"},{"Start":"01:47.535 ","End":"01:51.980","Text":"x to the n over n factorial."},{"Start":"01:51.980 ","End":"01:55.660","Text":"In other words, I\u0027m going to differentiate."},{"Start":"01:55.660 ","End":"01:57.620","Text":"That\u0027s the right-hand side."},{"Start":"01:57.620 ","End":"02:00.065","Text":"Now I have to differentiate the left-hand side."},{"Start":"02:00.065 ","End":"02:03.680","Text":"It\u0027s a product, so we\u0027ll use the product rule."},{"Start":"02:03.680 ","End":"02:05.825","Text":"I\u0027ll do this at the side."},{"Start":"02:05.825 ","End":"02:07.850","Text":"You should have the product rule memorized,"},{"Start":"02:07.850 ","End":"02:11.190","Text":"but in case you just momentarily forgot it, here it is."},{"Start":"02:11.190 ","End":"02:12.990","Text":"In our case u will be x,"},{"Start":"02:12.990 ","End":"02:15.630","Text":"and v will be e to the x."},{"Start":"02:15.630 ","End":"02:20.390","Text":"If we take xe to the x derivative,"},{"Start":"02:20.390 ","End":"02:27.620","Text":"derivative of x is 1 times e to the x plus x times derivative of e to the x,"},{"Start":"02:27.620 ","End":"02:30.230","Text":"which is just e to the x itself."},{"Start":"02:30.230 ","End":"02:33.125","Text":"Then I can just take out brackets."},{"Start":"02:33.125 ","End":"02:40.430","Text":"So it\u0027s x plus 1 or 1 plus x times e to the x. I\u0027ll just copy that here,"},{"Start":"02:40.430 ","End":"02:45.010","Text":"x plus 1, e to the x."},{"Start":"02:45.140 ","End":"02:51.420","Text":"Now, I can substitute x equals 1."},{"Start":"02:51.420 ","End":"02:56.215","Text":"If I substitute x equals 1 in both sides,"},{"Start":"02:56.215 ","End":"02:59.480","Text":"what I get here is 1 plus 1,"},{"Start":"02:59.480 ","End":"03:02.790","Text":"without the calculator, is 2,"},{"Start":"03:02.790 ","End":"03:07.690","Text":"and e to the 1 is just e. So I get 2e,"},{"Start":"03:07.690 ","End":"03:12.309","Text":"is equal to the very series that we had to compute,"},{"Start":"03:12.309 ","End":"03:15.820","Text":"the sum n goes from 0 to infinity."},{"Start":"03:15.820 ","End":"03:18.325","Text":"X is 1, so 1 to the n is 1."},{"Start":"03:18.325 ","End":"03:22.580","Text":"So it\u0027s just n plus 1 over n factorial."},{"Start":"03:22.580 ","End":"03:24.960","Text":"So this is the answer."},{"Start":"03:24.960 ","End":"03:26.665","Text":"I\u0027ll just highlight it,"},{"Start":"03:26.665 ","End":"03:28.285","Text":"and we are done."},{"Start":"03:28.285 ","End":"03:30.650","Text":"Nice trick. Hey!"}],"ID":6100},{"Watched":false,"Name":"Exercise 3","Duration":"3m 22s","ChapterTopicVideoID":6091,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6091.jpeg","UploadDate":"2016-05-08T11:19:36.8700000","DurationForVideoObject":"PT3M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this exercise, we have this numerical series to compute,"},{"Start":"00:04.830 ","End":"00:08.970","Text":"and it\u0027s in the chapter on Taylor and Maclaurin series."},{"Start":"00:08.970 ","End":"00:16.815","Text":"We\u0027re going to use the usual method of finding a Maclaurin series similar to this,"},{"Start":"00:16.815 ","End":"00:19.080","Text":"and we\u0027re going to substitute a value of x."},{"Start":"00:19.080 ","End":"00:20.430","Text":"If we look in the appendix,"},{"Start":"00:20.430 ","End":"00:22.964","Text":"you find one that\u0027s very close."},{"Start":"00:22.964 ","End":"00:26.070","Text":"Arctangent is what we need because look,"},{"Start":"00:26.070 ","End":"00:27.990","Text":"I mean we have the 2n plus 1,"},{"Start":"00:27.990 ","End":"00:29.940","Text":"we have the minus 1 to the n."},{"Start":"00:29.940 ","End":"00:32.160","Text":"What we need is to let x equals 1."},{"Start":"00:32.160 ","End":"00:33.465","Text":"Let\u0027s see."},{"Start":"00:33.465 ","End":"00:41.160","Text":"We get that arctangent of 1 is equal to,"},{"Start":"00:41.160 ","End":"00:45.200","Text":"I should have mentioned that I can substitute 1 because"},{"Start":"00:45.200 ","End":"00:55.480","Text":"the interval of convergence is x between 1 and minus 1, but inclusive."},{"Start":"00:55.480 ","End":"01:01.090","Text":"I can actually let x equals 1 just barely."},{"Start":"01:01.460 ","End":"01:05.795","Text":"This is equal to, I\u0027m going to use the Sigma form."},{"Start":"01:05.795 ","End":"01:17.070","Text":"It\u0027s equal to the sum minus 1 to the n of 1 to the 2n plus 1."},{"Start":"01:17.070 ","End":"01:20.115","Text":"Now, 1 to the n, you think is 1."},{"Start":"01:20.115 ","End":"01:28.240","Text":"It\u0027s 1 over 2n plus 1."},{"Start":"01:28.240 ","End":"01:32.825","Text":"I mean, this is exactly the same as this,"},{"Start":"01:32.825 ","End":"01:35.540","Text":"except that this is in the numerator."},{"Start":"01:35.540 ","End":"01:39.500","Text":"Let\u0027s make it exactly the same."},{"Start":"01:39.500 ","End":"01:44.900","Text":"I\u0027ll put the minus 1 to the n on the numerator here."},{"Start":"01:44.900 ","End":"01:48.305","Text":"Then there is no doubt that this is this."},{"Start":"01:48.305 ","End":"01:50.000","Text":"But arctangent of 1,"},{"Start":"01:50.000 ","End":"01:51.050","Text":"we can do better than that."},{"Start":"01:51.050 ","End":"01:53.180","Text":"We can actually compute what it is."},{"Start":"01:53.180 ","End":"01:55.070","Text":"We don\u0027t even need a calculator."},{"Start":"01:55.070 ","End":"01:57.635","Text":"It\u0027s one of those well-known angles."},{"Start":"01:57.635 ","End":"02:02.215","Text":"The angle whose tangent is 1 is 45 degrees,"},{"Start":"02:02.215 ","End":"02:08.870","Text":"45 degrees is one of those equilateral right triangles anyway."},{"Start":"02:08.870 ","End":"02:12.290","Text":"Or you can use the calculator in degrees,"},{"Start":"02:12.290 ","End":"02:14.150","Text":"it will give you 45,"},{"Start":"02:14.150 ","End":"02:19.550","Text":"but this works in radians."},{"Start":"02:19.550 ","End":"02:25.140","Text":"This is actually equal to Pi over 4."},{"Start":"02:25.220 ","End":"02:28.240","Text":"I\u0027ll highlight the answer."},{"Start":"02:28.240 ","End":"02:30.650","Text":"We are done but don\u0027t go just yet."},{"Start":"02:30.650 ","End":"02:33.200","Text":"I just would like to I mean, if you want to stay,"},{"Start":"02:33.200 ","End":"02:36.860","Text":"I just want to point out something out that might be of interest."},{"Start":"02:36.860 ","End":"02:40.190","Text":"What we have here is actually a formula for Pi,"},{"Start":"02:40.190 ","End":"02:43.070","Text":"that doesn\u0027t rely on geometry and circles,"},{"Start":"02:43.070 ","End":"02:45.335","Text":"multiplying both sides by 4,"},{"Start":"02:45.335 ","End":"02:48.840","Text":"I can say that Pi equals 4 times."},{"Start":"02:48.840 ","End":"02:50.750","Text":"Instead of the series form,"},{"Start":"02:50.750 ","End":"02:55.415","Text":"let me write it as the expanded form like on the right,"},{"Start":"02:55.415 ","End":"02:58.010","Text":"you can put x equals 1 here,"},{"Start":"02:58.010 ","End":"03:04.755","Text":"we get 1 minus 1/3 plus a 1/5, minus 1/7."},{"Start":"03:04.755 ","End":"03:06.030","Text":"I think you get the idea,"},{"Start":"03:06.030 ","End":"03:08.535","Text":"plus 1/9 minus an 1/11,"},{"Start":"03:08.535 ","End":"03:15.930","Text":"the odd numbers, reciprocals, and alternating sums."},{"Start":"03:15.930 ","End":"03:19.985","Text":"We have a formula for Pi for what it\u0027s worth."},{"Start":"03:19.985 ","End":"03:22.620","Text":"Okay, we\u0027re done."}],"ID":6101},{"Watched":false,"Name":"Exercise 4","Duration":"2m 11s","ChapterTopicVideoID":6085,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6085.jpeg","UploadDate":"2016-05-08T11:17:12.2170000","DurationForVideoObject":"PT2M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.095","Text":"In this exercise, we have a numerical series."},{"Start":"00:04.095 ","End":"00:05.970","Text":"We have to compute its sum."},{"Start":"00:05.970 ","End":"00:08.760","Text":"I\u0027m going to use the usual technique of finding"},{"Start":"00:08.760 ","End":"00:12.705","Text":"a Maclaurin series and then substituting an appropriate value of x,"},{"Start":"00:12.705 ","End":"00:15.660","Text":"which most often turns out to be 1."},{"Start":"00:15.660 ","End":"00:18.420","Text":"When you look through the formulas,"},{"Start":"00:18.420 ","End":"00:22.780","Text":"there\u0027s one that immediately jumps out."},{"Start":"00:23.660 ","End":"00:27.420","Text":"Anyway, the formula is the one for sine x,"},{"Start":"00:27.420 ","End":"00:30.060","Text":"and I\u0027ll put it here."},{"Start":"00:30.060 ","End":"00:32.945","Text":"If you look at the Sigma part,"},{"Start":"00:32.945 ","End":"00:34.820","Text":"it\u0027s really similar to this."},{"Start":"00:34.820 ","End":"00:36.680","Text":"We have a minus 1 to the n,"},{"Start":"00:36.680 ","End":"00:39.410","Text":"we have a 2n plus 1 factorial,"},{"Start":"00:39.410 ","End":"00:41.900","Text":"and if x is 1, that will just be 1,"},{"Start":"00:41.900 ","End":"00:44.855","Text":"so there\u0027s really nothing much to do here."},{"Start":"00:44.855 ","End":"00:55.860","Text":"Sine of 1 is equal to sum"},{"Start":"00:55.860 ","End":"01:03.870","Text":"n goes from 0 to infinity minus 1 to the n, 1 to"},{"Start":"01:03.870 ","End":"01:13.930","Text":"the power of 2n plus 1 over 2n plus 1 factorial."},{"Start":"01:14.780 ","End":"01:17.775","Text":"This is not the exactly this,"},{"Start":"01:17.775 ","End":"01:20.100","Text":"so let\u0027s just rewrite it"},{"Start":"01:20.100 ","End":"01:23.100","Text":"so it will look really like that."},{"Start":"01:23.100 ","End":"01:25.925","Text":"Say, 1 to the the 2n plus 1 is 1,"},{"Start":"01:25.925 ","End":"01:31.460","Text":"then I can put this in the numerator, same denominator."},{"Start":"01:31.580 ","End":"01:35.380","Text":"Now this is exactly this. On the left-hand side,"},{"Start":"01:35.380 ","End":"01:38.440","Text":"I\u0027ll just leave it as sine 1."},{"Start":"01:38.440 ","End":"01:43.600","Text":"I\u0027ll also emphasize or put a little c circular measure to say that it\u0027s radians."},{"Start":"01:43.600 ","End":"01:47.360","Text":"If you want a numerical approximation,"},{"Start":"01:47.360 ","End":"01:50.070","Text":"I can give it to you."},{"Start":"01:50.070 ","End":"01:57.370","Text":"It\u0027s 0.84147, approximately."},{"Start":"01:59.000 ","End":"02:07.040","Text":"But I would leave the answer as sine 1 and not give a numerical approximation."},{"Start":"02:07.040 ","End":"02:11.070","Text":"That\u0027s best. We\u0027re done."}],"ID":6102},{"Watched":false,"Name":"Exercise 5","Duration":"2m 5s","ChapterTopicVideoID":6086,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6086.jpeg","UploadDate":"2016-05-08T11:17:27.5430000","DurationForVideoObject":"PT2M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.319","Text":"In this exercise, we have to compute the sum of a numerical series."},{"Start":"00:04.319 ","End":"00:08.970","Text":"We\u0027re going to do this by finding a similar Maclaurin series,"},{"Start":"00:08.970 ","End":"00:10.470","Text":"which is a series of functions,"},{"Start":"00:10.470 ","End":"00:15.810","Text":"and then substitute a value of x and it\u0027s usually 1, but we\u0027ll see."},{"Start":"00:15.810 ","End":"00:19.170","Text":"I look through the table of the Maclaurin series,"},{"Start":"00:19.170 ","End":"00:25.069","Text":"I find that the closest is the following: The cosine function."},{"Start":"00:25.069 ","End":"00:29.130","Text":"It\u0027s very similar. Look 1 minus 1 to the n minus 1 to the n,"},{"Start":"00:29.130 ","End":"00:31.895","Text":"2n factorial, 2n factorial."},{"Start":"00:31.895 ","End":"00:33.290","Text":"It\u0027s almost a giveaway,"},{"Start":"00:33.290 ","End":"00:34.790","Text":"just let x equals 1."},{"Start":"00:34.790 ","End":"00:39.900","Text":"In fact, let\u0027s see what is cosine of 1 equal to."},{"Start":"00:39.900 ","End":"00:44.000","Text":"I should have mentioned also, this converges for all x,"},{"Start":"00:44.000 ","End":"00:46.560","Text":"certainly, for x equals 1."},{"Start":"00:47.090 ","End":"00:49.980","Text":"We get the sum,"},{"Start":"00:49.980 ","End":"00:59.225","Text":"n goes from 1 to infinity minus 1 to the n. Then if x is 1,"},{"Start":"00:59.225 ","End":"01:04.370","Text":"1^2n over 2n factorial."},{"Start":"01:04.370 ","End":"01:08.065","Text":"Now, a minor rewrite."},{"Start":"01:08.065 ","End":"01:12.530","Text":"Sorry, this is 0 to infinity, excuse me."},{"Start":"01:13.060 ","End":"01:17.860","Text":"1^2n is just 1."},{"Start":"01:17.860 ","End":"01:24.690","Text":"I can put this in the numerator and I get minus 1 to the n over 2n factorial."},{"Start":"01:24.690 ","End":"01:28.340","Text":"Now, you\u0027ve got to agree that this looks very much like this."},{"Start":"01:28.340 ","End":"01:31.400","Text":"This is our answer, cosine of 1."},{"Start":"01:31.400 ","End":"01:33.750","Text":"It\u0027s in radians."},{"Start":"01:34.190 ","End":"01:37.555","Text":"I will just highlight it."},{"Start":"01:37.555 ","End":"01:41.300","Text":"If you want to emphasize that it\u0027s in radians, not degrees,"},{"Start":"01:41.300 ","End":"01:44.810","Text":"the degrees has little circle here and the radian we put little c here,"},{"Start":"01:44.810 ","End":"01:47.900","Text":"stands for circular measure."},{"Start":"01:47.900 ","End":"01:51.380","Text":"For those who like numerical values,"},{"Start":"01:51.380 ","End":"01:53.680","Text":"I\u0027ll give you an approximation."},{"Start":"01:53.680 ","End":"01:58.470","Text":"Approximately 0.5403."},{"Start":"01:58.470 ","End":"01:59.890","Text":"If you do it on the calculator,"},{"Start":"01:59.890 ","End":"02:05.760","Text":"make sure it\u0027s set to radians and not degrees. We\u0027re done here."}],"ID":6103},{"Watched":false,"Name":"Exercise 6","Duration":"3m 24s","ChapterTopicVideoID":6087,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6087.jpeg","UploadDate":"2016-05-08T11:17:52.1570000","DurationForVideoObject":"PT3M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"Here again, we have a series of numbers"},{"Start":"00:04.020 ","End":"00:08.250","Text":"and we have to compute its sum"},{"Start":"00:08.250 ","End":"00:11.160","Text":"using Maclaurin series."},{"Start":"00:11.160 ","End":"00:13.980","Text":"We go to the table of Maclaurin series"},{"Start":"00:13.980 ","End":"00:16.905","Text":"in the appendix and find something close."},{"Start":"00:16.905 ","End":"00:20.529","Text":"The 1 we came up with is the following."},{"Start":"00:20.529 ","End":"00:22.730","Text":"This is the 1 I\u0027m referring to,"},{"Start":"00:22.730 ","End":"00:24.710","Text":"natural log of 1 plus x."},{"Start":"00:24.710 ","End":"00:25.700","Text":"Look how close it is."},{"Start":"00:25.700 ","End":"00:27.260","Text":"We have minus 1 to the n,"},{"Start":"00:27.260 ","End":"00:28.775","Text":"minus 1 to the n,"},{"Start":"00:28.775 ","End":"00:31.040","Text":"n plus 1, n plus 1."},{"Start":"00:31.040 ","End":"00:32.150","Text":"It\u0027s a cinch."},{"Start":"00:32.150 ","End":"00:33.950","Text":"Just let x equals 1,"},{"Start":"00:33.950 ","End":"00:36.335","Text":"and we\u0027ll have exactly this."},{"Start":"00:36.335 ","End":"00:40.400","Text":"I just should add that the interval"},{"Start":"00:40.400 ","End":"00:42.140","Text":"of convergence for this"},{"Start":"00:42.140 ","End":"00:45.345","Text":"is minus 1 less than x,"},{"Start":"00:45.345 ","End":"00:47.125","Text":"less than or equal to 1."},{"Start":"00:47.125 ","End":"00:49.340","Text":"We can substitute x equals 1,"},{"Start":"00:49.340 ","End":"00:53.700","Text":"we are just in,1 is allowed."},{"Start":"00:53.700 ","End":"00:54.990","Text":"Let\u0027s see."},{"Start":"00:54.990 ","End":"00:56.705","Text":"Let\u0027s just do it properly."},{"Start":"00:56.705 ","End":"01:01.055","Text":"Natural log of, let\u0027s put x equal 1,"},{"Start":"01:01.055 ","End":"01:05.570","Text":"1 plus 1 is equal to the sum,"},{"Start":"01:05.570 ","End":"01:08.225","Text":"n goes from 0 to infinity,"},{"Start":"01:08.225 ","End":"01:10.535","Text":"minus 1 to the n,"},{"Start":"01:10.535 ","End":"01:16.650","Text":"1 to the power of n plus 1 over n plus 1."},{"Start":"01:16.650 ","End":"01:21.210","Text":"The natural log of 2 is equal to,"},{"Start":"01:21.210 ","End":"01:22.400","Text":"and I will just rewrite this"},{"Start":"01:22.400 ","End":"01:24.360","Text":"in a slightly friendlier form,"},{"Start":"01:24.360 ","End":"01:28.485","Text":"I mean, simplified, 0 to infinity,"},{"Start":"01:28.485 ","End":"01:31.580","Text":"1 to the power of anything is 1."},{"Start":"01:31.580 ","End":"01:33.950","Text":"I\u0027ll just put this in the numerator."},{"Start":"01:33.950 ","End":"01:39.330","Text":"I have minus 1 to the n over n plus 1,"},{"Start":"01:39.330 ","End":"01:43.020","Text":"and this is our answer."},{"Start":"01:43.020 ","End":"01:45.615","Text":"I\u0027ll highlight it."},{"Start":"01:45.615 ","End":"01:47.430","Text":"For those of you who like"},{"Start":"01:47.430 ","End":"01:51.265","Text":"numerical approximations,"},{"Start":"01:51.265 ","End":"01:56.080","Text":"this is approximately 0.693."},{"Start":"01:57.560 ","End":"01:59.420","Text":"This is the answer."},{"Start":"01:59.420 ","End":"02:01.160","Text":"But I\u0027d like to talk a little bit more"},{"Start":"02:01.160 ","End":"02:03.440","Text":"just to show you something."},{"Start":"02:03.440 ","End":"02:06.560","Text":"If I don\u0027t write it in the Sigma notation,"},{"Start":"02:06.560 ","End":"02:11.030","Text":"what I get is that the natural log of 2"},{"Start":"02:11.030 ","End":"02:12.410","Text":"is given by a series."},{"Start":"02:12.410 ","End":"02:16.100","Text":"Look, if n is 0, minus 1 to the 0 is 1,"},{"Start":"02:16.100 ","End":"02:18.605","Text":"and it\u0027s 1 over 1, so it\u0027s just 1."},{"Start":"02:18.605 ","End":"02:22.460","Text":"If n is 1, I get minus"},{"Start":"02:22.460 ","End":"02:26.415","Text":"and this is going to be over 2."},{"Start":"02:26.415 ","End":"02:29.345","Text":"If you put n equals 2,"},{"Start":"02:29.345 ","End":"02:33.290","Text":"this is positive and this is 3 minus 1/4"},{"Start":"02:33.290 ","End":"02:38.450","Text":"plus 1/5 minus 1/6 plus, and so on."},{"Start":"02:38.450 ","End":"02:40.010","Text":"That\u0027s a simple expression"},{"Start":"02:40.010 ","End":"02:42.290","Text":"for the natural log of 2."},{"Start":"02:42.290 ","End":"02:43.610","Text":"If we had all pluses here,"},{"Start":"02:43.610 ","End":"02:45.335","Text":"we know it doesn\u0027t converge."},{"Start":"02:45.335 ","End":"02:46.580","Text":"It\u0027s the harmonic series,"},{"Start":"02:46.580 ","End":"02:49.120","Text":"but alternating it, does."},{"Start":"02:49.120 ","End":"02:53.690","Text":"Again, as a bonus,"},{"Start":"02:53.690 ","End":"02:54.890","Text":"I\u0027m just showing you"},{"Start":"02:54.890 ","End":"02:56.450","Text":"they also had something similar"},{"Start":"02:56.450 ","End":"03:00.320","Text":"in an earlier exercise that Pi over 4 was"},{"Start":"03:00.320 ","End":"03:07.445","Text":"1 minus 1/3 plus 1/5 minus 1/7, and so on."},{"Start":"03:07.445 ","End":"03:09.920","Text":"Some of these expressions"},{"Start":"03:09.920 ","End":"03:12.500","Text":"like natural logarithm of something or Pi"},{"Start":"03:12.500 ","End":"03:18.865","Text":"are often expressible in terms of a simple series."},{"Start":"03:18.865 ","End":"03:21.275","Text":"But that was just extra."},{"Start":"03:21.275 ","End":"03:25.410","Text":"We really were done along while back."}],"ID":6104},{"Watched":false,"Name":"Exercise 7","Duration":"2m 54s","ChapterTopicVideoID":6088,"CourseChapterTopicPlaylistID":84688,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6088.jpeg","UploadDate":"2016-05-08T11:18:13.1130000","DurationForVideoObject":"PT2M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"We have here numerical theories."},{"Start":"00:03.675 ","End":"00:09.030","Text":"You have to compute its sum and we\u0027re going to use the method of taking"},{"Start":"00:09.030 ","End":"00:15.525","Text":"a Maclaurin series of functions and substituting an appropriate value of the variable x."},{"Start":"00:15.525 ","End":"00:17.310","Text":"It\u0027s often x equals 1,"},{"Start":"00:17.310 ","End":"00:19.725","Text":"but here it\u0027s not, as you\u0027ll see."},{"Start":"00:19.725 ","End":"00:25.040","Text":"Now, I look in the table of Maclaurin series and I see various things."},{"Start":"00:25.040 ","End":"00:27.260","Text":"I see there\u0027s an n plus 1 in the denominator."},{"Start":"00:27.260 ","End":"00:31.610","Text":"I put a minus 1 to the n. If you look through them,"},{"Start":"00:31.610 ","End":"00:38.080","Text":"you\u0027ll see it\u0027s what really is similar to this is the following."},{"Start":"00:38.080 ","End":"00:40.620","Text":"Natural log of 1 plus x."},{"Start":"00:40.620 ","End":"00:44.010","Text":"We\u0027ve had this 1 before just recently."},{"Start":"00:44.010 ","End":"00:46.490","Text":"This is what it\u0027s equal to."},{"Start":"00:46.490 ","End":"00:47.810","Text":"You see the n plus 1,"},{"Start":"00:47.810 ","End":"00:51.320","Text":"the minus 1 to the n. Here we have something to the n plus 1 in"},{"Start":"00:51.320 ","End":"00:56.165","Text":"the numerator here and the denominator. No problem."},{"Start":"00:56.165 ","End":"01:00.890","Text":"What we want to do is let x equals 1/2 and"},{"Start":"01:00.890 ","End":"01:06.260","Text":"then the half to the n plus 1 becomes 2 to the n plus 1 in the denominator."},{"Start":"01:06.260 ","End":"01:14.030","Text":"I should mention that the interval of convergence is minus 1 less than x,"},{"Start":"01:14.030 ","End":"01:16.625","Text":"less than or equal to 1."},{"Start":"01:16.625 ","End":"01:19.655","Text":"The previous time we used it, we let x equal 1."},{"Start":"01:19.655 ","End":"01:22.640","Text":"Certainly, x equals 1/2 is in this range."},{"Start":"01:22.640 ","End":"01:28.520","Text":"Let\u0027s see what is the natural log of 1 plus a half,"},{"Start":"01:28.520 ","End":"01:30.740","Text":"where I\u0027m taking x as a half."},{"Start":"01:30.740 ","End":"01:32.705","Text":"This is equal to,"},{"Start":"01:32.705 ","End":"01:34.415","Text":"according to the formula,"},{"Start":"01:34.415 ","End":"01:40.685","Text":"the sum and goes from 0 to infinity minus 1^n."},{"Start":"01:40.685 ","End":"01:51.180","Text":"1/2^n plus 1 over n plus 1."},{"Start":"01:51.300 ","End":"01:57.700","Text":"Now I just have to slightly rewrite this to get it to look like this."},{"Start":"01:57.700 ","End":"02:02.440","Text":"This is the sum from 0 to infinity."},{"Start":"02:02.440 ","End":"02:05.995","Text":"Now what I\u0027m going to do is put a bigger dividing line."},{"Start":"02:05.995 ","End":"02:10.865","Text":"We\u0027re going to put n plus 1 in the denominator."},{"Start":"02:10.865 ","End":"02:16.530","Text":"In the numerator, I\u0027ll have 1 to the n plus 1, which is just 1."},{"Start":"02:16.530 ","End":"02:21.580","Text":"The denominator, I\u0027ll get 2 to the n plus 1."},{"Start":"02:21.580 ","End":"02:22.870","Text":"But now I have to throw this in,"},{"Start":"02:22.870 ","End":"02:25.410","Text":"so I just put it into the numerator,"},{"Start":"02:25.410 ","End":"02:31.940","Text":"minus 1 to the power of n. Now this is exactly what we have here."},{"Start":"02:31.940 ","End":"02:36.250","Text":"On the left, it\u0027s the natural log of 3 over 2,"},{"Start":"02:36.250 ","End":"02:39.980","Text":"or 1.5, whichever you prefer."},{"Start":"02:39.980 ","End":"02:45.360","Text":"Or write above it, this is 3 over 2 or 1.5, take your pick."},{"Start":"02:45.580 ","End":"02:48.635","Text":"This is the answer."},{"Start":"02:48.635 ","End":"02:54.150","Text":"I\u0027ll just highlight the answer and declare that we are done."}],"ID":6105}],"Thumbnail":null,"ID":84688},{"Name":"Finding Limits Using Expansions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1 Part a","Duration":"4m 19s","ChapterTopicVideoID":6093,"CourseChapterTopicPlaylistID":84689,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6093.jpeg","UploadDate":"2016-05-08T11:22:02.2500000","DurationForVideoObject":"PT4M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we\u0027re asked to compute the following limit."},{"Start":"00:04.740 ","End":"00:14.640","Text":"Certainly, we can do it using L\u0027Hopital\u0027s rule because you see that if x goes to 0,"},{"Start":"00:14.640 ","End":"00:18.015","Text":"then this x goes to 0,"},{"Start":"00:18.015 ","End":"00:21.570","Text":"the arctangent of x also goes to 0."},{"Start":"00:21.570 ","End":"00:23.220","Text":"Arctangent of 0 is 0."},{"Start":"00:23.220 ","End":"00:26.279","Text":"X cubed also goes to 0."},{"Start":"00:26.279 ","End":"00:29.670","Text":"You could use L\u0027Hopital"},{"Start":"00:29.670 ","End":"00:33.660","Text":"by differentiating top and bottom 3 times and you\u0027d get to the solution,"},{"Start":"00:33.660 ","End":"00:38.650","Text":"but I want to show you an alternative method using the Maclaurin series."},{"Start":"00:40.250 ","End":"00:44.870","Text":"Here is the Maclaurin series for arctangent of x."},{"Start":"00:44.870 ","End":"00:49.385","Text":"We have the Sigma form or the expanded form and"},{"Start":"00:49.385 ","End":"00:57.075","Text":"the interval of convergence is minus 1,"},{"Start":"00:57.075 ","End":"01:02.220","Text":"x, 1, and including both the endpoint,"},{"Start":"01:02.220 ","End":"01:05.500","Text":"the 1 and the minus 1 are included."},{"Start":"01:07.550 ","End":"01:11.400","Text":"Certainly, 0 is included in this interval."},{"Start":"01:11.400 ","End":"01:13.730","Text":"When we take the limit as x goes to 0,"},{"Start":"01:13.730 ","End":"01:18.350","Text":"x will be in this interval because going to be very close to 0."},{"Start":"01:18.350 ","End":"01:21.970","Text":"Let\u0027s see. If we just plug it in here,"},{"Start":"01:21.970 ","End":"01:26.850","Text":"we\u0027ll get the limit as x goes to 0,"},{"Start":"01:26.850 ","End":"01:30.665","Text":"x minus, now here I\u0027m going to put the arctangent."},{"Start":"01:30.665 ","End":"01:34.355","Text":"I\u0027ll use the expanded form, not the Sigma,"},{"Start":"01:34.355 ","End":"01:41.760","Text":"and we have x minus x cubed over 3."},{"Start":"01:41.760 ","End":"01:43.785","Text":"I\u0027ll put 1 more,"},{"Start":"01:43.785 ","End":"01:48.194","Text":"plus x^5 5th over 5,"},{"Start":"01:48.194 ","End":"01:55.895","Text":"but in general, it\u0027s going to be after that plus or minus higher powers of x."},{"Start":"01:55.895 ","End":"02:03.740","Text":"It\u0027s going to be the sum of some kind of a_n,"},{"Start":"02:03.740 ","End":"02:08.865","Text":"x^n, but the n here will be bigger than 5,"},{"Start":"02:08.865 ","End":"02:10.695","Text":"bigger or equal to 7."},{"Start":"02:10.695 ","End":"02:13.955","Text":"Certainly, it will be bigger than 3."},{"Start":"02:13.955 ","End":"02:18.190","Text":"Let me just say that n is bigger than 5 for the time being."},{"Start":"02:18.190 ","End":"02:23.570","Text":"Then it\u0027s going to be over x cubed."},{"Start":"02:23.780 ","End":"02:26.580","Text":"Now, look what happens in the numerator."},{"Start":"02:26.580 ","End":"02:29.115","Text":"The x and the x cancel."},{"Start":"02:29.115 ","End":"02:32.324","Text":"So we just get the tail here,"},{"Start":"02:32.324 ","End":"02:35.205","Text":"but reversing the signs."},{"Start":"02:35.205 ","End":"02:38.180","Text":"We see this equals this, equals this."},{"Start":"02:38.180 ","End":"02:41.720","Text":"Then we\u0027ve got the limit as x goes to 0."},{"Start":"02:41.720 ","End":"02:44.330","Text":"We\u0027re going to have x cubed over"},{"Start":"02:44.330 ","End":"02:55.825","Text":"3 plus x^5 over 5, sorry, minus."},{"Start":"02:55.825 ","End":"03:00.395","Text":"Then plus x^7 over 7, but doesn\u0027t matter."},{"Start":"03:00.395 ","End":"03:03.700","Text":"The point is that these are all higher powers,"},{"Start":"03:03.700 ","End":"03:08.980","Text":"5 and upwards, over x cubed."},{"Start":"03:09.530 ","End":"03:13.180","Text":"We can divide top and bottom by x cubed,"},{"Start":"03:13.180 ","End":"03:14.965","Text":"see x tends to 0,"},{"Start":"03:14.965 ","End":"03:16.780","Text":"but it isn\u0027t equal to 0,"},{"Start":"03:16.780 ","End":"03:18.850","Text":"so we divide top and bottom by x cubed,"},{"Start":"03:18.850 ","End":"03:20.560","Text":"which is not 0."},{"Start":"03:20.560 ","End":"03:22.655","Text":"We get the limit,"},{"Start":"03:22.655 ","End":"03:28.830","Text":"as x goes to 0, we have 1/3."},{"Start":"03:28.830 ","End":"03:30.555","Text":"Well, this is going to disappear."},{"Start":"03:30.555 ","End":"03:37.995","Text":"We get 1/3 from the x cubed minus 1/5 x squared plus,"},{"Start":"03:37.995 ","End":"03:40.770","Text":"next thing is going to be 1/7 x^4,"},{"Start":"03:40.770 ","End":"03:44.055","Text":"but as I said, it\u0027s all higher powers."},{"Start":"03:44.055 ","End":"03:49.070","Text":"Here, bigger than 5, here the n will be bigger than 2,"},{"Start":"03:49.070 ","End":"03:53.715","Text":"we\u0027ll have the sum of something from 2 upwards."},{"Start":"03:53.715 ","End":"03:57.675","Text":"In any event, x appears in them."},{"Start":"03:57.675 ","End":"04:00.790","Text":"When we let x equal 0, these are all going to be 0,"},{"Start":"04:00.790 ","End":"04:08.000","Text":"and all we\u0027re left with will be the 1/3."},{"Start":"04:08.000 ","End":"04:14.825","Text":"That\u0027s a fairly straightforward way of finding the limit without using L\u0027Hopital."},{"Start":"04:14.825 ","End":"04:19.680","Text":"I just like to highlight the answer before I declare that we\u0027re done."}],"ID":6106},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"3m 47s","ChapterTopicVideoID":6094,"CourseChapterTopicPlaylistID":84689,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6094.jpeg","UploadDate":"2016-05-08T11:22:30.0870000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we have to compute the following limit."},{"Start":"00:04.350 ","End":"00:08.370","Text":"You might be wondering what this limit is doing in the middle"},{"Start":"00:08.370 ","End":"00:12.555","Text":"of the section on Taylor-Maclaurin series."},{"Start":"00:12.555 ","End":"00:14.640","Text":"You\u0027ll see in a moment."},{"Start":"00:14.640 ","End":"00:18.539","Text":"Normally, what you would do here is you try substituting"},{"Start":"00:18.539 ","End":"00:22.620","Text":"x equals 0 and you see that the numerator and denominator are both 0."},{"Start":"00:22.620 ","End":"00:24.735","Text":"Then you try L\u0027hopital\u0027s rule,"},{"Start":"00:24.735 ","End":"00:27.435","Text":"and it turns out you\u0027ll still get 0 over 0."},{"Start":"00:27.435 ","End":"00:32.985","Text":"In fact, you\u0027d have to apply L\u0027Hopital\u0027s rule 5 times and that can be tedious."},{"Start":"00:32.985 ","End":"00:36.965","Text":"I\u0027d like to show you an alternative method using Maclaurin series."},{"Start":"00:36.965 ","End":"00:40.070","Text":"What we do is replace sine x by"},{"Start":"00:40.070 ","End":"00:46.435","Text":"its power series and Maclaurin series and we look in the appendix and get the formula."},{"Start":"00:46.435 ","End":"00:50.435","Text":"This is the Maclaurin series for sine x."},{"Start":"00:50.435 ","End":"00:52.835","Text":"We won\u0027t be using the Sigma form."},{"Start":"00:52.835 ","End":"00:55.550","Text":"I\u0027ll use this expanded form."},{"Start":"00:55.550 ","End":"00:58.370","Text":"Just replace sine x with this."},{"Start":"00:58.370 ","End":"01:03.770","Text":"What we get is the limit as x goes to 0."},{"Start":"01:03.770 ","End":"01:06.485","Text":"For the sine x, I write this,"},{"Start":"01:06.485 ","End":"01:10.370","Text":"x minus x cubed."},{"Start":"01:10.370 ","End":"01:12.650","Text":"Let\u0027s expand the factorials."},{"Start":"01:12.650 ","End":"01:15.590","Text":"3 factorial is 6,"},{"Start":"01:15.590 ","End":"01:21.335","Text":"and then x to the 5 over 5 factorial is 120."},{"Start":"01:21.335 ","End":"01:27.905","Text":"Then minus x to the 7 over 5,040,"},{"Start":"01:27.905 ","End":"01:30.140","Text":"that\u0027s really all we\u0027ll need."},{"Start":"01:30.140 ","End":"01:32.555","Text":"As you see, we don\u0027t need anymore."},{"Start":"01:32.555 ","End":"01:35.895","Text":"Obviously, everything from here on which is a"},{"Start":"01:35.895 ","End":"01:40.470","Text":"power higher than 5 and that\u0027s important to us."},{"Start":"01:40.470 ","End":"01:43.545","Text":"That\u0027s the sine x, and I\u0027ll close the brackets."},{"Start":"01:43.545 ","End":"01:49.155","Text":"Then minus x plus 1/6 of x cubed,"},{"Start":"01:49.155 ","End":"01:52.565","Text":"I\u0027ll just write that as x cubed over 6."},{"Start":"01:52.565 ","End":"01:57.570","Text":"All this is going to be over x to the 5."},{"Start":"01:58.310 ","End":"02:01.260","Text":"Notice that some stuff cancels."},{"Start":"02:01.260 ","End":"02:04.815","Text":"This x cancels with this minus x,"},{"Start":"02:04.815 ","End":"02:12.285","Text":"and minus x cubed over 6 cancels with plus x cubed over 6."},{"Start":"02:12.285 ","End":"02:19.080","Text":"What we get is the limit as x goes to 0,"},{"Start":"02:19.080 ","End":"02:21.630","Text":"I\u0027ll rewrite the numerator."},{"Start":"02:21.630 ","End":"02:29.045","Text":"Let\u0027s take x to the 5 outside the brackets because everything here is of power 5 or above."},{"Start":"02:29.045 ","End":"02:36.415","Text":"We get x to the 5 times 1 over 120,"},{"Start":"02:36.415 ","End":"02:44.790","Text":"minus x squared over 5,040, plus etc."},{"Start":"02:44.790 ","End":"02:47.510","Text":"These are all powers of x."},{"Start":"02:47.510 ","End":"02:49.790","Text":"Next one will be x to the 4."},{"Start":"02:49.790 ","End":"02:58.145","Text":"They\u0027re higher powers of x, well, not yet."},{"Start":"02:58.145 ","End":"03:02.560","Text":"This is over x to the 5 and now,"},{"Start":"03:02.560 ","End":"03:06.140","Text":"we can cancel the x to the 5. It\u0027s not 0."},{"Start":"03:06.140 ","End":"03:08.135","Text":"X tends to 0,"},{"Start":"03:08.135 ","End":"03:10.070","Text":"but is not equal to 0,"},{"Start":"03:10.070 ","End":"03:12.390","Text":"so we can cancel."},{"Start":"03:12.430 ","End":"03:18.000","Text":"Now, we can get the limit just by substituting."},{"Start":"03:18.340 ","End":"03:25.260","Text":"Because all the terms from here onward are powers of x squared,"},{"Start":"03:25.260 ","End":"03:28.830","Text":"4th, 6th they\u0027re higher powers of x."},{"Start":"03:28.830 ","End":"03:31.880","Text":"All of these will become 0 if I substitute 0."},{"Start":"03:31.880 ","End":"03:35.800","Text":"In fact, what we\u0027re left with at the end is just"},{"Start":"03:35.800 ","End":"03:42.390","Text":"1 over 120 and that\u0027s the answer."},{"Start":"03:42.390 ","End":"03:46.750","Text":"I\u0027ll just highlight it and we are done."}],"ID":6107},{"Watched":false,"Name":"Exercise 1 Part c","Duration":"5m 41s","ChapterTopicVideoID":6092,"CourseChapterTopicPlaylistID":84689,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/6092.jpeg","UploadDate":"2016-05-08T11:21:30.0500000","DurationForVideoObject":"PT5M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have to compute the following limit,"},{"Start":"00:04.050 ","End":"00:07.620","Text":"and we\u0027re going to do it not the usual way,"},{"Start":"00:07.620 ","End":"00:10.455","Text":"but with Maclaurin series."},{"Start":"00:10.455 ","End":"00:15.015","Text":"The usual way would probably be L\u0027Hopital\u0027s theorem."},{"Start":"00:15.015 ","End":"00:17.980","Text":"You would try substituting x equals 0"},{"Start":"00:17.980 ","End":"00:21.920","Text":"or taking the limit of the numerator and denominator,"},{"Start":"00:21.920 ","End":"00:25.690","Text":"getting 0 over 0, applying L\u0027Hopital\u0027s Rule,"},{"Start":"00:25.690 ","End":"00:29.000","Text":"and it turns out you\u0027re going to have to apply it 3 times,"},{"Start":"00:29.000 ","End":"00:31.670","Text":"and the differentiation is not trivial."},{"Start":"00:31.670 ","End":"00:34.610","Text":"There\u0027s a product here and so on,"},{"Start":"00:34.610 ","End":"00:38.975","Text":"and the alternative method is to use Maclaurin series."},{"Start":"00:38.975 ","End":"00:41.540","Text":"I need 2 of them, I need the 1 for e to the x"},{"Start":"00:41.540 ","End":"00:44.530","Text":"emphasize and excellent me just bring them both in,"},{"Start":"00:44.530 ","End":"00:46.665","Text":"and here they are,"},{"Start":"00:46.665 ","End":"00:49.160","Text":"e to the x, and sine x."},{"Start":"00:49.160 ","End":"00:52.655","Text":"Both of them hold for all x,"},{"Start":"00:52.655 ","End":"00:57.395","Text":"and what I need is e to the x sine x."},{"Start":"00:57.395 ","End":"01:04.985","Text":"As you may recall that we had some exercises and multiplication of such series."},{"Start":"01:04.985 ","End":"01:07.290","Text":"Let\u0027s see what we can do."},{"Start":"01:07.290 ","End":"01:15.255","Text":"I went at the side to compute e to the power of x sine x,"},{"Start":"01:15.255 ","End":"01:20.250","Text":"and I\u0027m just going to compute some of the terms just as much as I need,"},{"Start":"01:20.250 ","End":"01:23.930","Text":"and you\u0027ll see what I mean, e to the x,"},{"Start":"01:23.930 ","End":"01:29.315","Text":"is 1 plus x plus from here,"},{"Start":"01:29.315 ","End":"01:33.905","Text":"x squared over 2, and from here,"},{"Start":"01:33.905 ","End":"01:38.479","Text":"x cubed over 6,"},{"Start":"01:38.479 ","End":"01:44.190","Text":"and then higher-order terms, higher than 3,"},{"Start":"01:44.190 ","End":"01:49.600","Text":"and the sine x is going to be x,"},{"Start":"01:49.600 ","End":"01:56.450","Text":"minus x cubed over 3 factorial is 6 plus, and so on."},{"Start":"01:56.450 ","End":"01:58.850","Text":"I\u0027m going to just stop at 3 and say,"},{"Start":"01:58.850 ","End":"02:04.245","Text":"the dot are terms of order higher than 3."},{"Start":"02:04.245 ","End":"02:06.510","Text":"Here the next 1 is x to the 4th,"},{"Start":"02:06.510 ","End":"02:08.520","Text":"here the next 1 is x to the 5th,"},{"Start":"02:08.520 ","End":"02:11.040","Text":"but it\u0027s in every case is higher than 3,"},{"Start":"02:11.040 ","End":"02:14.145","Text":"and now let\u0027s just multiply out,"},{"Start":"02:14.145 ","End":"02:19.730","Text":"and we\u0027re only going to take the terms that are 3 or lower."},{"Start":"02:19.730 ","End":"02:28.710","Text":"Well, let\u0027s see. 1 times x is x,"},{"Start":"02:28.710 ","End":"02:32.335","Text":"and then where can I get x squared from?"},{"Start":"02:32.335 ","End":"02:39.170","Text":"I can get x squared from x times x, and that\u0027s all."},{"Start":"02:39.170 ","End":"02:42.810","Text":"That gives me x squared."},{"Start":"02:42.920 ","End":"02:48.230","Text":"x cubed I can get from 2 places from x squared times x,"},{"Start":"02:48.230 ","End":"02:51.139","Text":"and that would give me a half x cubed."},{"Start":"02:51.139 ","End":"02:55.070","Text":"But I also have this with this,"},{"Start":"02:55.070 ","End":"02:57.050","Text":"which will give me minus a 6th."},{"Start":"02:57.050 ","End":"03:05.030","Text":"Let me just write this as 1.5 minus 6x cubed,"},{"Start":"03:05.030 ","End":"03:08.854","Text":"and everything else will be higher powers,"},{"Start":"03:08.854 ","End":"03:11.905","Text":"meaning powers higher than 3."},{"Start":"03:11.905 ","End":"03:15.005","Text":"Okay, now that we have this,"},{"Start":"03:15.005 ","End":"03:18.140","Text":"let\u0027s go back here and substitute it."},{"Start":"03:18.140 ","End":"03:28.955","Text":"What this is equal to is the limit as x goes to 0,"},{"Start":"03:28.955 ","End":"03:32.690","Text":"e to the x sine x. I\u0027ll put this instead,"},{"Start":"03:32.690 ","End":"03:38.080","Text":"which is x, plus x squared,"},{"Start":"03:38.080 ","End":"03:48.659","Text":"and then this happens to be a 3rd half minus a 6th plus 1 third x cubed plus dot,"},{"Start":"03:48.659 ","End":"03:56.495","Text":"means powers higher than 3, and then minus."},{"Start":"03:56.495 ","End":"03:59.680","Text":"If I multiply this out, it\u0027s x plus x squared,"},{"Start":"03:59.680 ","End":"04:01.345","Text":"so it\u0027s minus x,"},{"Start":"04:01.345 ","End":"04:08.155","Text":"minus x squared, and all this over x cubed."},{"Start":"04:08.155 ","End":"04:10.960","Text":"Now notice a lot of stuff\u0027s going to cancel,"},{"Start":"04:10.960 ","End":"04:14.335","Text":"x and minus x, x squared,"},{"Start":"04:14.335 ","End":"04:16.605","Text":"and minus x squared,"},{"Start":"04:16.605 ","End":"04:22.460","Text":"and from the numerator we can take x cubed outside the brackets."},{"Start":"04:22.460 ","End":"04:29.760","Text":"We will get the limit of"},{"Start":"04:29.760 ","End":"04:37.115","Text":"x cubed times now 1 third plus,"},{"Start":"04:37.115 ","End":"04:39.720","Text":"I\u0027ll also write dot, dot, dot."},{"Start":"04:39.720 ","End":"04:43.835","Text":"These things were of powers higher than 3,"},{"Start":"04:43.835 ","End":"04:45.765","Text":"meaning at least 4."},{"Start":"04:45.765 ","End":"04:48.700","Text":"These will be powers of at least 1."},{"Start":"04:48.700 ","End":"04:51.145","Text":"In other words, they will contain some power of x,"},{"Start":"04:51.145 ","End":"04:57.710","Text":"positive power over x cubed."},{"Start":"04:57.820 ","End":"05:03.380","Text":"Now the x cubed in the numerator and denominator cancel."},{"Start":"05:03.380 ","End":"05:06.620","Text":"I can cancel because x is not equal to 0,"},{"Start":"05:06.620 ","End":"05:09.065","Text":"it just tends to 0,"},{"Start":"05:09.065 ","End":"05:14.785","Text":"and what we\u0027re left with is the limit as x goes to 0,"},{"Start":"05:14.785 ","End":"05:20.040","Text":"of 1 third, plus dot, dot, dot,"},{"Start":"05:20.040 ","End":"05:24.330","Text":"meaning powers of x positive powers,"},{"Start":"05:24.330 ","End":"05:29.510","Text":"and so what we get is if we substitute x equals 0 in these powers of x,"},{"Start":"05:29.510 ","End":"05:31.310","Text":"that will just be 0."},{"Start":"05:31.310 ","End":"05:35.140","Text":"The answer will be just 1 third,"},{"Start":"05:35.140 ","End":"05:37.385","Text":"and that is the answer."},{"Start":"05:37.385 ","End":"05:41.220","Text":"Just highlight it and we are done."}],"ID":6108},{"Watched":false,"Name":"Exercise 2","Duration":"13m 21s","ChapterTopicVideoID":28812,"CourseChapterTopicPlaylistID":84689,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28812.jpeg","UploadDate":"2022-04-29T13:05:18.3800000","DurationForVideoObject":"PT13M21S","Description":null,"VideoComments":[],"Subtitles":[],"ID":30309}],"Thumbnail":null,"ID":84689},{"Name":"Computations with Taylor Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Computations with Taylor Series and a given precision","Duration":"9m 41s","ChapterTopicVideoID":10124,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10124.jpeg","UploadDate":"2020-01-16T13:03:18.4430000","DurationForVideoObject":"PT9M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In the exercises following this clip,"},{"Start":"00:02.490 ","End":"00:06.690","Text":"you\u0027ll be asked to do all kinds of computations involving Taylor series."},{"Start":"00:06.690 ","End":"00:12.660","Text":"A typical question would be something like compute the value of e to within"},{"Start":"00:12.660 ","End":"00:21.390","Text":"3 decimal places or compute 1 over e to an accuracy of 0.002 or something."},{"Start":"00:21.390 ","End":"00:24.270","Text":"That\u0027s the exercise we\u0027ll be doing."},{"Start":"00:24.270 ","End":"00:27.495","Text":"It\u0027s making a computation using Taylor series"},{"Start":"00:27.495 ","End":"00:31.020","Text":"and when you want to compute a value to a given precision."},{"Start":"00:31.020 ","End":"00:33.720","Text":"The main theorem we\u0027re going to be using is the"},{"Start":"00:33.720 ","End":"00:38.250","Text":"following: Leibniz\u0027s alternating series test."},{"Start":"00:38.250 ","End":"00:41.000","Text":"I just want to say that a Leibniz series and"},{"Start":"00:41.000 ","End":"00:44.340","Text":"an alternating series generally the same thing."},{"Start":"00:44.560 ","End":"00:49.830","Text":"What we have is we\u0027re given a Leibniz series."},{"Start":"00:50.090 ","End":"00:52.700","Text":"Given the Leibniz series,"},{"Start":"00:52.700 ","End":"00:54.574","Text":"which is an alternating series,"},{"Start":"00:54.574 ","End":"01:02.350","Text":"the following sum to infinity of minus 1 to the n makes the alternation and a_n,"},{"Start":"01:02.350 ","End":"01:05.130","Text":"well of course, a_n has to be bigger than 0,"},{"Start":"01:05.130 ","End":"01:08.795","Text":"as part of the definition of an alternating series, so it\u0027s redundant."},{"Start":"01:08.795 ","End":"01:10.460","Text":"By the way, we don\u0027t have to start from 1,"},{"Start":"01:10.460 ","End":"01:13.300","Text":"we can start from anywhere as long as we go to infinity,"},{"Start":"01:13.300 ","End":"01:17.565","Text":"because with a series the beginning is never important, but only the end."},{"Start":"01:17.565 ","End":"01:22.920","Text":"It\u0027s non-negative and I want it to be decreasing."},{"Start":"01:22.920 ","End":"01:27.165","Text":"So a_n is a decreasing series."},{"Start":"01:27.165 ","End":"01:34.780","Text":"Also, I want a_n to tend to 0 as n goes to infinity, of course."},{"Start":"01:36.500 ","End":"01:46.790","Text":"Given all those conditions, I\u0027ll stop hear a minute."},{"Start":"01:46.790 ","End":"01:52.675","Text":"Usually what we do here is we can\u0027t sum it to infinity,"},{"Start":"01:52.675 ","End":"01:56.500","Text":"so we estimate the sum by adding the first, I don\u0027t know,"},{"Start":"01:56.500 ","End":"02:01.170","Text":"a 100 terms or 5 terms, whatever."},{"Start":"02:01.170 ","End":"02:06.440","Text":"Then we use that partial sum of the first terms to estimate the total sum."},{"Start":"02:06.440 ","End":"02:09.590","Text":"But we need to know how much the error is in doing that."},{"Start":"02:09.590 ","End":"02:12.695","Text":"If we use n terms to estimate,"},{"Start":"02:12.695 ","End":"02:16.100","Text":"then the error between the estimated value and"},{"Start":"02:16.100 ","End":"02:21.815","Text":"the actual value is less than a_n plus 1, an absolute value."},{"Start":"02:21.815 ","End":"02:24.050","Text":"In other words, if I take it and make it positive,"},{"Start":"02:24.050 ","End":"02:26.120","Text":"if the error is negative and make it positive,"},{"Start":"02:26.120 ","End":"02:28.910","Text":"it\u0027s still less than a_n plus 1."},{"Start":"02:28.910 ","End":"02:33.290","Text":"I\u0027m going to explain this more because it\u0027s important to understand this theorem."},{"Start":"02:33.290 ","End":"02:36.950","Text":"It\u0027s the key to solving the exercises."},{"Start":"02:36.950 ","End":"02:39.155","Text":"Let me explain further."},{"Start":"02:39.155 ","End":"02:43.410","Text":"Suppose I want to estimate,"},{"Start":"02:43.410 ","End":"02:45.215","Text":"as part of the exercise,"},{"Start":"02:45.215 ","End":"02:50.270","Text":"1 over e. Later we\u0027ll introduce the matter of precision."},{"Start":"02:50.270 ","End":"02:53.585","Text":"1 over e is e^-1."},{"Start":"02:53.585 ","End":"02:57.170","Text":"Now, I want to get a series to converge to"},{"Start":"02:57.170 ","End":"03:03.560","Text":"1 over e. That series is going to be the Taylor series evaluated at some point."},{"Start":"03:03.560 ","End":"03:07.630","Text":"Now, e to the x in general,"},{"Start":"03:07.630 ","End":"03:12.510","Text":"is 1 plus x plus x squared over"},{"Start":"03:12.510 ","End":"03:21.485","Text":"2 factorial plus x cubed over 3 factorial plus x to the fourth over 4 factorial, etc."},{"Start":"03:21.485 ","End":"03:25.910","Text":"If we substitute x equals minus 1,"},{"Start":"03:25.910 ","End":"03:30.845","Text":"then we get that e to the minus 1 is equal."},{"Start":"03:30.845 ","End":"03:35.515","Text":"We get the infinite series 1 minus x."},{"Start":"03:35.515 ","End":"03:39.145","Text":"What\u0027s going to happen is this is going to become alternating,"},{"Start":"03:39.145 ","End":"03:40.490","Text":"because all the odd powers,"},{"Start":"03:40.490 ","End":"03:44.574","Text":"if you put x as minus 1 will be negative and the even powers will be positive,"},{"Start":"03:44.574 ","End":"03:50.540","Text":"so plus x squared over 2 factorial minus x cubed over"},{"Start":"03:50.540 ","End":"03:58.765","Text":"3 factorial plus x to the fourth over 4 factorial, etc."},{"Start":"03:58.765 ","End":"04:05.250","Text":"Now, what we have is a series that meets all these conditions for this test."},{"Start":"04:06.880 ","End":"04:10.899","Text":"Well, we know that this series converges."},{"Start":"04:10.899 ","End":"04:12.640","Text":"If it converges, then of course,"},{"Start":"04:12.640 ","End":"04:14.500","Text":"the general term goes to 0."},{"Start":"04:14.500 ","End":"04:16.345","Text":"That\u0027s a necessary condition."},{"Start":"04:16.345 ","End":"04:19.010","Text":"It is decreasing."},{"Start":"04:20.100 ","End":"04:23.600","Text":"Now, just start this again."},{"Start":"04:28.720 ","End":"04:34.450","Text":"If we let x equals minus 1 to give us 1 over e,"},{"Start":"04:34.450 ","End":"04:37.525","Text":"then we get the series for 1 over e,"},{"Start":"04:37.525 ","End":"04:44.590","Text":"which is 1 minus 1 plus 1 over 2"},{"Start":"04:44.590 ","End":"04:51.405","Text":"factorial minus 1 over 3 factorial"},{"Start":"04:51.405 ","End":"04:56.860","Text":"plus 1 over 4 factorial, etc."},{"Start":"04:57.200 ","End":"05:01.665","Text":"I could write this as the sum,"},{"Start":"05:01.665 ","End":"05:09.960","Text":"n goes from 0 to infinity of minus 1 to the n,"},{"Start":"05:09.960 ","End":"05:12.015","Text":"that\u0027s what gives me my plus or minus,"},{"Start":"05:12.015 ","End":"05:19.685","Text":"times a_n, where a_n is just the factorial."},{"Start":"05:19.685 ","End":"05:22.520","Text":"A_n is 1 over n factorial."},{"Start":"05:22.520 ","End":"05:28.605","Text":"Like when n is 3, I get minus 1 to the 3 is minus 1 over 3 factorial."},{"Start":"05:28.605 ","End":"05:30.870","Text":"This is a general term."},{"Start":"05:30.870 ","End":"05:35.630","Text":"I wrote it in this form because I wanted to check if the a_n"},{"Start":"05:35.630 ","End":"05:40.245","Text":"satisfies the alternating series test, and they do."},{"Start":"05:40.245 ","End":"05:44.340","Text":"First of all, a_ns are all non-negative."},{"Start":"05:44.340 ","End":"05:48.840","Text":"1 over n factorial is positive or non-negative."},{"Start":"05:49.370 ","End":"05:52.520","Text":"N gets bigger, n factorial gets bigger,"},{"Start":"05:52.520 ","End":"05:54.274","Text":"and so a_n gets smaller,"},{"Start":"05:54.274 ","End":"05:57.550","Text":"and it does tend to 0 because as n goes to infinity,"},{"Start":"05:57.550 ","End":"06:02.195","Text":"n factorial goes even faster to infinity and 1 over infinity is 0."},{"Start":"06:02.195 ","End":"06:10.365","Text":"So it meets all the conditions in the Leibniz alternating series test."},{"Start":"06:10.365 ","End":"06:13.055","Text":"I can also take the conclusions,"},{"Start":"06:13.055 ","End":"06:16.760","Text":"which is that if I take only n terms,"},{"Start":"06:16.760 ","End":"06:18.970","Text":"the sum of the terms up to"},{"Start":"06:18.970 ","End":"06:23.240","Text":"a_n to estimate then the arrow is going to be less than a_n plus 1."},{"Start":"06:23.240 ","End":"06:24.740","Text":"Let me show what this means."},{"Start":"06:24.740 ","End":"06:26.720","Text":"I\u0027ll give a couple of examples."},{"Start":"06:26.720 ","End":"06:32.530","Text":"Suppose I want n to take S_3,"},{"Start":"06:32.530 ","End":"06:35.490","Text":"the sum of up to 3,"},{"Start":"06:35.490 ","End":"06:36.960","Text":"I\u0027ll start counting at 0."},{"Start":"06:36.960 ","End":"06:38.925","Text":"This is 0, 1, 2, 3."},{"Start":"06:38.925 ","End":"06:47.760","Text":"S_3 would be 1 minus 1 plus a 1/2 minus 1/6."},{"Start":"06:47.760 ","End":"06:53.370","Text":"That is equal to 1/3."},{"Start":"06:53.370 ","End":"06:56.955","Text":"When I use a 1/3 to estimate 1 over e,"},{"Start":"06:56.955 ","End":"07:01.160","Text":"then the error, according to the theorem,"},{"Start":"07:01.160 ","End":"07:04.830","Text":"is less than a_n plus 1."},{"Start":"07:05.780 ","End":"07:09.045","Text":"If the arrows would be less than,"},{"Start":"07:09.045 ","End":"07:11.340","Text":"n plus 1 in this case is 4,"},{"Start":"07:11.340 ","End":"07:12.990","Text":"is less than a_4,"},{"Start":"07:12.990 ","End":"07:15.255","Text":"which is 1 over 24."},{"Start":"07:15.255 ","End":"07:18.540","Text":"Yeah, 4 factorial is 24,"},{"Start":"07:18.540 ","End":"07:22.570","Text":"5 factorial is a 120, and so on."},{"Start":"07:22.940 ","End":"07:26.585","Text":"1 over 24 is good but not great."},{"Start":"07:26.585 ","End":"07:28.920","Text":"If I take another term, S_4,"},{"Start":"07:28.920 ","End":"07:35.250","Text":"it\u0027s 1 minus 1 plus 1/2 minus a 1/6 plus 1 over 24."},{"Start":"07:35.250 ","End":"07:44.480","Text":"This plus 1 over 24 comes out to be 17 over 24."},{"Start":"07:44.480 ","End":"07:48.005","Text":"If I estimate 1 over e at this and compute it,"},{"Start":"07:48.005 ","End":"07:56.130","Text":"then the error is guaranteed to be less than 1 over the next term,"},{"Start":"07:56.130 ","End":"07:59.730","Text":"which is 5 factorial is 120,"},{"Start":"07:59.730 ","End":"08:01.860","Text":"so it\u0027s less than 1 over a 120."},{"Start":"08:01.860 ","End":"08:05.270","Text":"For example, it\u0027s already good within 2 decimal places because it\u0027s less"},{"Start":"08:05.270 ","End":"08:08.595","Text":"than a 100th and so on."},{"Start":"08:08.595 ","End":"08:10.640","Text":"So the more terms you take,"},{"Start":"08:10.640 ","End":"08:13.940","Text":"the smaller the error is going to be because the terms are tending to"},{"Start":"08:13.940 ","End":"08:17.715","Text":"0 and they\u0027re decreasing to 0."},{"Start":"08:17.715 ","End":"08:26.360","Text":"That\u0027s basically how we use Leibniz test to do estimations and to estimate the error."},{"Start":"08:26.360 ","End":"08:29.480","Text":"When I say error, I mean absolute value of the error,"},{"Start":"08:29.480 ","End":"08:34.050","Text":"which means 1 over e minus 1 over 24, for example."},{"Start":"08:34.050 ","End":"08:38.700","Text":"You just compare the 2 and take the absolute value of the difference."},{"Start":"08:50.870 ","End":"08:55.570","Text":"Let me just write down the estimates."},{"Start":"08:55.570 ","End":"08:59.430","Text":"This is 1 over e, is what we\u0027re trying to estimate."},{"Start":"08:59.430 ","End":"09:01.470","Text":"We got 2 estimates in,"},{"Start":"09:01.470 ","End":"09:08.470","Text":"1 is the estimate is 1/3,"},{"Start":"09:08.470 ","End":"09:13.640","Text":"and the other estimate we have is 17 over 24."},{"Start":"09:13.640 ","End":"09:17.015","Text":"Now, if I take the difference between this and this,"},{"Start":"09:17.015 ","End":"09:20.340","Text":"it\u0027s going to be less than and probably a lot less than 1 over 24."},{"Start":"09:20.340 ","End":"09:23.365","Text":"If I take the difference between this and this, an absolute value,"},{"Start":"09:23.365 ","End":"09:27.220","Text":"it\u0027s going to be less than a 120,"},{"Start":"09:27.220 ","End":"09:29.270","Text":"or probably a lot less."},{"Start":"09:30.730 ","End":"09:33.425","Text":"I\u0027m done here."},{"Start":"09:33.425 ","End":"09:40.890","Text":"Remember to do the exercises following this as many as you can. That\u0027s all."}],"ID":10346},{"Watched":false,"Name":"Exercise 1 Part a","Duration":"3m 47s","ChapterTopicVideoID":10084,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10084.jpeg","UploadDate":"2017-08-20T11:26:24.9670000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to evaluate 1 over the cube root of e with"},{"Start":"00:04.530 ","End":"00:09.089","Text":"a precision of less than 0.001 or 1 thousandth."},{"Start":"00:09.089 ","End":"00:12.060","Text":"The idea is to use a Taylor series."},{"Start":"00:12.060 ","End":"00:15.150","Text":"We\u0027re going to use e to the power of x."},{"Start":"00:15.150 ","End":"00:18.435","Text":"If we write this as e the to the power of something,"},{"Start":"00:18.435 ","End":"00:22.050","Text":"and I presume you know the rules of exponents,"},{"Start":"00:22.050 ","End":"00:25.815","Text":"that this will equal e to the power of minus 1/3."},{"Start":"00:25.815 ","End":"00:28.290","Text":"Then we\u0027re going to use the Taylor series for e to the"},{"Start":"00:28.290 ","End":"00:32.880","Text":"x. I\u0027ve copy pasted it from the appendix."},{"Start":"00:32.880 ","End":"00:35.820","Text":"This series converges for all x,"},{"Start":"00:35.820 ","End":"00:40.845","Text":"in particular for x equals minus 1/3,"},{"Start":"00:40.845 ","End":"00:42.615","Text":"and if we do that,"},{"Start":"00:42.615 ","End":"00:44.570","Text":"I\u0027m going to use the expanded form,"},{"Start":"00:44.570 ","End":"00:45.860","Text":"not the Sigma form."},{"Start":"00:45.860 ","End":"00:47.525","Text":"While x equal minus a 1/3,"},{"Start":"00:47.525 ","End":"00:56.490","Text":"we\u0027ll get that e to the minus 1/3 is equal to 1 plus minus 1/3 to"},{"Start":"00:56.490 ","End":"01:06.050","Text":"the power of 1 over 1 plus minus 1/3 to the power of 2 over 2 factorial,"},{"Start":"01:06.050 ","End":"01:13.880","Text":"which is 2, plus minus 1/3 to the power of 3 over 3 factorial is 6."},{"Start":"01:13.880 ","End":"01:17.305","Text":"Let\u0027s do 1 more, from experience, we\u0027ll need 1 more,"},{"Start":"01:17.305 ","End":"01:22.670","Text":"minus 1/3 to the power of 4 over 4 factorial,"},{"Start":"01:22.670 ","End":"01:26.015","Text":"which is 24, and so on."},{"Start":"01:26.015 ","End":"01:28.190","Text":"Let\u0027s see what this equals."},{"Start":"01:28.190 ","End":"01:33.505","Text":"First of all, it\u0027s 1 minus a 1/3."},{"Start":"01:33.505 ","End":"01:35.505","Text":"Now, it\u0027s going to be,"},{"Start":"01:35.505 ","End":"01:37.740","Text":"a 1/3 squared is plus,"},{"Start":"01:37.740 ","End":"01:40.215","Text":"plus 1/9 over 2,"},{"Start":"01:40.215 ","End":"01:42.825","Text":"which is 1 over 18."},{"Start":"01:42.825 ","End":"01:47.575","Text":"Then we\u0027re going to get a minus because it\u0027s minus to an odd power,"},{"Start":"01:47.575 ","End":"01:49.950","Text":"1 over 27 times 6,"},{"Start":"01:49.950 ","End":"01:54.740","Text":"27 times 6, 162,"},{"Start":"01:54.740 ","End":"01:57.380","Text":"we can use a calculator here."},{"Start":"01:57.380 ","End":"02:01.400","Text":"Then 1/3 to the fourth is 1 over 81."},{"Start":"02:01.400 ","End":"02:05.120","Text":"81 times 24, and it\u0027s a plus,"},{"Start":"02:05.120 ","End":"02:10.425","Text":"81 times 24 is 1,944."},{"Start":"02:10.425 ","End":"02:13.360","Text":"Actually, now I can stop here."},{"Start":"02:13.360 ","End":"02:16.340","Text":"Let me explain how I know to stop here."},{"Start":"02:16.340 ","End":"02:20.405","Text":"Notice that this is what is called a Leibniz series."},{"Start":"02:20.405 ","End":"02:23.210","Text":"What this means is that it has 2 properties."},{"Start":"02:23.210 ","End":"02:26.615","Text":"First of all, that it\u0027s alternating,"},{"Start":"02:26.615 ","End":"02:28.730","Text":"meaning plus, minus, plus,"},{"Start":"02:28.730 ","End":"02:30.695","Text":"minus, plus, minus, and so on."},{"Start":"02:30.695 ","End":"02:34.010","Text":"Secondly, it\u0027s decreasing, meaning"},{"Start":"02:34.010 ","End":"02:37.430","Text":"that each term in absolute value is less than the previous term."},{"Start":"02:37.430 ","End":"02:39.365","Text":"You see 1, 1/3, 18th,"},{"Start":"02:39.365 ","End":"02:42.260","Text":"1 over 162, they\u0027re all getting smaller and smaller."},{"Start":"02:42.260 ","End":"02:44.220","Text":"So if it has these 2 properties,"},{"Start":"02:44.220 ","End":"02:45.430","Text":"it\u0027s a Leibniz series,"},{"Start":"02:45.430 ","End":"02:49.070","Text":"and as a theorem on Leibniz series that if I take a partial sum,"},{"Start":"02:49.070 ","End":"02:50.765","Text":"in this case, to here,"},{"Start":"02:50.765 ","End":"02:56.640","Text":"then the error will be less than the following term in absolute value."},{"Start":"02:56.870 ","End":"03:00.120","Text":"In our case the error, if I took these 4,"},{"Start":"03:00.120 ","End":"03:04.805","Text":"would be less than this and this is less than 1 over 1000,"},{"Start":"03:04.805 ","End":"03:10.815","Text":"and 1 over 1000 is exactly the 0.001."},{"Start":"03:10.815 ","End":"03:14.085","Text":"So what I\u0027m going to do is say that this,"},{"Start":"03:14.085 ","End":"03:20.645","Text":"which is 1 over cube root of e is approximately equal to,"},{"Start":"03:20.645 ","End":"03:25.125","Text":"and I\u0027ll just copy these 4 here,"},{"Start":"03:25.125 ","End":"03:26.990","Text":"and I\u0027ll compute this."},{"Start":"03:26.990 ","End":"03:29.750","Text":"It comes out, if you do it as a fraction,"},{"Start":"03:29.750 ","End":"03:37.650","Text":"to be 58 over 81 and so we know that 1 over"},{"Start":"03:37.650 ","End":"03:41.870","Text":"the cube root of e is approximately this to within"},{"Start":"03:41.870 ","End":"03:47.430","Text":"an error of less than this. We are done."}],"ID":10347},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"3m 36s","ChapterTopicVideoID":10081,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10081.jpeg","UploadDate":"2017-08-20T11:25:47.7300000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we have to estimate the sine of 3 degrees,"},{"Start":"00:04.410 ","End":"00:06.330","Text":"and I\u0027ll just write that in radians,"},{"Start":"00:06.330 ","End":"00:09.540","Text":"that sine of Pi over 60,"},{"Start":"00:09.540 ","End":"00:11.100","Text":"because Pi is 180,"},{"Start":"00:11.100 ","End":"00:13.110","Text":"180 over 60 is 3,"},{"Start":"00:13.110 ","End":"00:17.580","Text":"and within an error of less than 0.001,"},{"Start":"00:17.580 ","End":"00:22.980","Text":"which I can write as 1 over 1,000 so I want the error to be less than."},{"Start":"00:22.980 ","End":"00:26.700","Text":"Now, this is in the section on Taylor and Maclaurin series,"},{"Start":"00:26.700 ","End":"00:32.770","Text":"so the obvious thing to do is to start off with the Maclaurin series for sine."},{"Start":"00:32.770 ","End":"00:38.270","Text":"Here it is, this was taken from the appendix in the Maclaurin table."},{"Start":"00:38.270 ","End":"00:41.150","Text":"We\u0027ll use the expanded form,"},{"Start":"00:41.150 ","End":"00:42.290","Text":"not the Sigma form."},{"Start":"00:42.290 ","End":"00:46.490","Text":"We can get the sine of Pi over 60,"},{"Start":"00:46.490 ","End":"00:52.250","Text":"which is what I get when I just put x equals Pi over 60 is going to equal x,"},{"Start":"00:52.250 ","End":"00:56.705","Text":"which is Pi over 60 minus this term is"},{"Start":"00:56.705 ","End":"01:02.825","Text":"Pi over 60 cubed over 3 factorial is 6."},{"Start":"01:02.825 ","End":"01:09.600","Text":"1 more, Pi over 60 to the 5th over 5 factorial,"},{"Start":"01:09.600 ","End":"01:12.610","Text":"which is 120, and so on."},{"Start":"01:12.610 ","End":"01:19.410","Text":"Notice that what we have here is an alternating series: plus,"},{"Start":"01:19.410 ","End":"01:23.525","Text":"minus, plus, minus, and each successive term gets smaller and smaller."},{"Start":"01:23.525 ","End":"01:26.870","Text":"In other words, it\u0027s a Leibnitz series,"},{"Start":"01:26.870 ","End":"01:30.350","Text":"sometimes called Leibnitz alternating series."},{"Start":"01:30.350 ","End":"01:34.355","Text":"Because of this, we can use the theorem,"},{"Start":"01:34.355 ","End":"01:36.935","Text":"if we take a partial sum,"},{"Start":"01:36.935 ","End":"01:42.165","Text":"the error is less than the next term in absolute value."},{"Start":"01:42.165 ","End":"01:47.600","Text":"Let\u0027s see, we\u0027ll keep going along until we get to a term that\u0027s less than 1,000th."},{"Start":"01:47.600 ","End":"01:50.300","Text":"This is Pi over 60."},{"Start":"01:50.300 ","End":"01:52.370","Text":"This is about 3 over 60,"},{"Start":"01:52.370 ","End":"01:55.504","Text":"120th, not good enough, minus,"},{"Start":"01:55.504 ","End":"02:00.700","Text":"the next term is Pi cubed over,"},{"Start":"02:00.700 ","End":"02:02.170","Text":"let\u0027s see, 60 cubed."},{"Start":"02:02.170 ","End":"02:05.260","Text":"6 cubed is 216,"},{"Start":"02:05.260 ","End":"02:12.020","Text":"but it\u0027s 60 so it\u0027s 216,000 times 6."},{"Start":"02:12.230 ","End":"02:17.095","Text":"Clearly, this is going to be less than 1,000th."},{"Start":"02:17.095 ","End":"02:18.220","Text":"You can do it on the calculator,"},{"Start":"02:18.220 ","End":"02:22.550","Text":"but Pi cubed is about 3 times 3 times 3,"},{"Start":"02:22.550 ","End":"02:26.135","Text":"and this is 100s of thousands."},{"Start":"02:26.135 ","End":"02:29.640","Text":"This is already less than 1,000th."},{"Start":"02:32.710 ","End":"02:35.410","Text":"Well, anyway, do it on the calculator,"},{"Start":"02:35.410 ","End":"02:37.225","Text":"you\u0027ll see it\u0027s much smaller."},{"Start":"02:37.225 ","End":"02:40.495","Text":"We can actually stop here, the partial sum,"},{"Start":"02:40.495 ","End":"02:44.580","Text":"and say that the sine of Pi over"},{"Start":"02:44.580 ","End":"02:52.005","Text":"60 is actually approximately equal to Pi over 60,"},{"Start":"02:52.005 ","End":"02:58.470","Text":"with an error that\u0027s less than way much smaller than 1,000th,"},{"Start":"02:58.470 ","End":"03:02.050","Text":"but certainly within 0.01."},{"Start":"03:02.600 ","End":"03:05.300","Text":"This is what I\u0027ll leave as the answer."},{"Start":"03:05.300 ","End":"03:07.160","Text":"I won\u0027t do it on the calculator."},{"Start":"03:07.160 ","End":"03:09.230","Text":"You can do it if you want."},{"Start":"03:09.230 ","End":"03:11.690","Text":"But just to fully answer the question,"},{"Start":"03:11.690 ","End":"03:17.720","Text":"I can say that sine of 3 degrees is approximately Pi over"},{"Start":"03:17.720 ","End":"03:24.680","Text":"60 with error less than this expression from here,"},{"Start":"03:24.680 ","End":"03:33.230","Text":"which is way less than 0.01."},{"Start":"03:33.230 ","End":"03:35.850","Text":"We are done."}],"ID":10348},{"Watched":false,"Name":"Exercise 1 Part c","Duration":"4m 53s","ChapterTopicVideoID":10082,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10082.jpeg","UploadDate":"2017-08-20T11:25:59.5500000","DurationForVideoObject":"PT4M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we have to estimate the arctangent of"},{"Start":"00:06.030 ","End":"00:16.860","Text":"1/4 or 0.25 within an accuracy of 0.001,"},{"Start":"00:16.860 ","End":"00:25.170","Text":"which is 1/1000 and this is in the chapter on Taylor Maclaurin series so"},{"Start":"00:25.170 ","End":"00:27.690","Text":"the obvious thing to do is to find"},{"Start":"00:27.690 ","End":"00:34.590","Text":"the Maclaurin series for arctangent x and then let x equal 0.25."},{"Start":"00:34.590 ","End":"00:39.675","Text":"Here it is, I copied it from the table in the appendix."},{"Start":"00:39.675 ","End":"00:47.955","Text":"This converges for x between 1- -1, actually inclusive."},{"Start":"00:47.955 ","End":"00:52.490","Text":"Of course, it will work for x equals 0.25,"},{"Start":"00:52.490 ","End":"00:53.900","Text":"which is in the range."},{"Start":"00:53.900 ","End":"01:00.800","Text":"So what we get is that arctangent of 0.25,"},{"Start":"01:00.800 ","End":"01:04.115","Text":"if I use this expanded form,"},{"Start":"01:04.115 ","End":"01:13.720","Text":"will equal 0.25 minus 0.25 cubed over"},{"Start":"01:13.720 ","End":"01:24.940","Text":"3 plus 0.25^5 over 5 minus and so on."},{"Start":"01:26.300 ","End":"01:29.560","Text":"We\u0027ve done this kind of thing before."},{"Start":"01:29.560 ","End":"01:34.255","Text":"We see that it\u0027s a Leibniz series, I\u0027ll just write that."},{"Start":"01:34.255 ","End":"01:40.780","Text":"Just to remind you, a Leibniz series has to be alternating,"},{"Start":"01:40.780 ","End":"01:42.730","Text":"meaning plus and minus,"},{"Start":"01:42.730 ","End":"01:47.925","Text":"plus and minus and so on and it also has to be decreasing,"},{"Start":"01:47.925 ","End":"01:52.000","Text":"meaning the sequence, each term is less than"},{"Start":"01:52.000 ","End":"01:56.095","Text":"the previous 1 and it\u0027s fairly clear that this is shrinking rapidly."},{"Start":"01:56.095 ","End":"01:58.060","Text":"Numerator\u0027s decreasing a fraction to"},{"Start":"01:58.060 ","End":"02:00.775","Text":"a higher power and denominator is increasing, so it\u0027s decreasing."},{"Start":"02:00.775 ","End":"02:05.380","Text":"Actually, it has to decrease to 0 in the limit but this of course is true"},{"Start":"02:05.380 ","End":"02:10.115","Text":"for any convergent series and so if this is the case,"},{"Start":"02:10.115 ","End":"02:14.645","Text":"then 1 of the theorems about Leibniz series is that the error"},{"Start":"02:14.645 ","End":"02:20.050","Text":"of taking a partial sum is always less than the following term in absolute value."},{"Start":"02:20.050 ","End":"02:23.240","Text":"All we have to do is go along and evaluate each 1 to"},{"Start":"02:23.240 ","End":"02:26.585","Text":"see at which point we get something less than 1/1000."},{"Start":"02:26.585 ","End":"02:30.615","Text":"Well, this is 0.25."},{"Start":"02:30.615 ","End":"02:34.465","Text":"It\u0027s not less than 1/1000 or 0.001."},{"Start":"02:34.465 ","End":"02:37.895","Text":"The next 1, if you compute it, let\u0027s see it\u0027s about,"},{"Start":"02:37.895 ","End":"02:44.970","Text":"it\u0027s 1/4 cubed, which is 1/64. That makes it 1/192."},{"Start":"02:45.290 ","End":"02:49.925","Text":"Not good enough but the next 1 is,"},{"Start":"02:49.925 ","End":"02:53.810","Text":"and in fact, if you just compute 1/4,"},{"Start":"02:53.810 ","End":"02:57.965","Text":"which is 0.25^5 times 1/5,"},{"Start":"02:57.965 ","End":"03:03.860","Text":"so we get 1/5,120."},{"Start":"03:03.860 ","End":"03:10.550","Text":"In any event this is surely less than 0.001, less than 1/1000."},{"Start":"03:10.550 ","End":"03:13.100","Text":"So we can stop here."},{"Start":"03:13.100 ","End":"03:23.840","Text":"In fact we can stop here and say that the error rather is going to be less"},{"Start":"03:23.840 ","End":"03:27.710","Text":"than the following term and so all I have to do is say"},{"Start":"03:27.710 ","End":"03:37.150","Text":"that arctangent 0.25 is approximately equal to the first 2."},{"Start":"03:37.150 ","End":"03:38.960","Text":"I prefer to work in fractions."},{"Start":"03:38.960 ","End":"03:41.060","Text":"We already said that this is 1/4,"},{"Start":"03:41.060 ","End":"03:47.780","Text":"and we already computed this as 1/192 and this happens to"},{"Start":"03:47.780 ","End":"03:55.040","Text":"equal 47/192 and I\u0027ll leave this as a fraction if you want,"},{"Start":"03:55.040 ","End":"03:58.550","Text":"you can do it on the calculator as a decimal."},{"Start":"03:58.550 ","End":"04:06.330","Text":"This is going to be our approximation for arctangent of"},{"Start":"04:06.330 ","End":"04:11.420","Text":"0.25 and the error is going to"},{"Start":"04:11.420 ","End":"04:18.460","Text":"be less than 0.001."},{"Start":"04:18.640 ","End":"04:23.270","Text":"Just let me emphasize that this is in radians so in"},{"Start":"04:23.270 ","End":"04:26.940","Text":"case you are going to check on the calculator and actually I suggest you do it,"},{"Start":"04:26.940 ","End":"04:30.890","Text":"if you plug in 0.25 and then do the arctangent,"},{"Start":"04:30.890 ","End":"04:33.200","Text":"which is often shift and then tangent,"},{"Start":"04:33.200 ","End":"04:36.650","Text":"you should get the same answer as doing"},{"Start":"04:36.650 ","End":"04:41.760","Text":"this fraction but you have to set your calculator to radians."},{"Start":"04:42.320 ","End":"04:44.450","Text":"If you want it in degrees,"},{"Start":"04:44.450 ","End":"04:49.730","Text":"you just have to multiply it by 180 over Pi or set your calculator, whatever."},{"Start":"04:49.770 ","End":"04:53.390","Text":"We are done with this 1."}],"ID":10349},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"3m 26s","ChapterTopicVideoID":10086,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10086.jpeg","UploadDate":"2017-08-20T11:26:40.2600000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to evaluate 1 over the square root of e,"},{"Start":"00:04.530 ","End":"00:11.740","Text":"which we immediately see as e^-1.5."},{"Start":"00:11.740 ","End":"00:16.680","Text":"We\u0027re going to use the first 3 non-zero elements of a Maclaurin series,"},{"Start":"00:16.680 ","End":"00:18.585","Text":"it\u0027s going to be e^x."},{"Start":"00:18.585 ","End":"00:21.540","Text":"I\u0027ll bring that in right away."},{"Start":"00:21.540 ","End":"00:28.170","Text":"Here it is, and we\u0027ll be estimating the error using Leibnitz series theorem."},{"Start":"00:28.170 ","End":"00:38.820","Text":"e^-1/2, we want to substitute x equals minus 1.5 in the e^x expansion over here."},{"Start":"00:38.820 ","End":"00:48.720","Text":"What we get is that e^-1.5 equals 1 plus,"},{"Start":"00:48.720 ","End":"00:50.160","Text":"x is minus 1/2,"},{"Start":"00:50.160 ","End":"00:57.345","Text":"so we get -1/2^1 over 1"},{"Start":"00:57.345 ","End":"01:06.505","Text":"plus minus 1/2 squared over 2 factorial is 2."},{"Start":"01:06.505 ","End":"01:11.445","Text":"Now, if we continued and didn\u0027t stop at 3 elements,"},{"Start":"01:11.445 ","End":"01:19.230","Text":"we\u0027d get plus"},{"Start":"01:19.230 ","End":"01:22.380","Text":"minus 1/2 cubed over 3 factorial,"},{"Start":"01:22.380 ","End":"01:26.160","Text":"which is 6 and so on."},{"Start":"01:26.160 ","End":"01:29.564","Text":"What we would get would be 1,"},{"Start":"01:29.564 ","End":"01:32.475","Text":"now, this is going to be minus 1/2/."},{"Start":"01:32.475 ","End":"01:41.315","Text":"Here, we\u0027ll get plus 1/2 squared over 2 and then minus,"},{"Start":"01:41.315 ","End":"01:43.085","Text":"because it\u0027s an odd number,"},{"Start":"01:43.085 ","End":"01:47.280","Text":"1/2 cubed over 6, and then plus."},{"Start":"01:47.280 ","End":"01:51.035","Text":"Notice that we have an alternating series,"},{"Start":"01:51.035 ","End":"01:53.855","Text":"plus, minus, plus, minus, plus."},{"Start":"01:53.855 ","End":"02:00.005","Text":"Also, each term in absolute value gets successively smaller and smaller,"},{"Start":"02:00.005 ","End":"02:03.095","Text":"these are decreasing, the numerator is"},{"Start":"02:03.095 ","End":"02:06.965","Text":"getting smaller, and the denominator is getting larger, and then also goes to 0."},{"Start":"02:06.965 ","End":"02:12.415","Text":"In other words, this is what we call a Leibnitz series."},{"Start":"02:12.415 ","End":"02:18.500","Text":"As such, there\u0027s a theorem that says that if you take a partial sum, in this case,"},{"Start":"02:18.500 ","End":"02:21.530","Text":"3 terms, then the error is,"},{"Start":"02:21.530 ","End":"02:25.805","Text":"at most, the absolute value of the following term, which is this."},{"Start":"02:25.805 ","End":"02:27.260","Text":"So we need to compute 2 things;"},{"Start":"02:27.260 ","End":"02:30.305","Text":"the sum of these and the value of this."},{"Start":"02:30.305 ","End":"02:35.270","Text":"So this, which is our original 1 over square root of e,"},{"Start":"02:35.270 ","End":"02:38.735","Text":"is approximately equal to the first 3,"},{"Start":"02:38.735 ","End":"02:40.700","Text":"which is 1 minus 1/2."},{"Start":"02:40.700 ","End":"02:44.430","Text":"Now, half squared is a quarter, that\u0027s plus 1/8."},{"Start":"02:44.510 ","End":"02:46.860","Text":"If we do with 1/8,"},{"Start":"02:46.860 ","End":"02:49.140","Text":"then this is 1 minus 1/2 is 1/2,"},{"Start":"02:49.140 ","End":"02:51.540","Text":"plus 1/8 is 5/8,"},{"Start":"02:51.540 ","End":"02:54.785","Text":"so that\u0027s our approximation."},{"Start":"02:54.785 ","End":"02:59.410","Text":"You could write it as a decimal, 0.625 or whatever."},{"Start":"02:59.410 ","End":"03:02.530","Text":"The error, according to the Leibnitz series,"},{"Start":"03:02.530 ","End":"03:06.295","Text":"is going to be less than the following term without the sign,"},{"Start":"03:06.295 ","End":"03:08.905","Text":"it\u0027s 1 over 8 times 6,"},{"Start":"03:08.905 ","End":"03:12.710","Text":"is 1 over 48."},{"Start":"03:13.250 ","End":"03:17.910","Text":"Just highlight that, and essentially, that answers the question,"},{"Start":"03:17.910 ","End":"03:22.570","Text":"this approximately equal to this using these first 3 elements,"},{"Start":"03:22.570 ","End":"03:26.300","Text":"and the error is less than this. We\u0027re done."}],"ID":10350},{"Watched":false,"Name":"Exercise 2 Part b","Duration":"3m 47s","ChapterTopicVideoID":10087,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10087.jpeg","UploadDate":"2017-08-20T11:26:50.1100000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.940","Text":"In this exercise, we have to estimate cosine of 4 degrees."},{"Start":"00:05.940 ","End":"00:10.080","Text":"This is the unusual part using the first non-zero element,"},{"Start":"00:10.080 ","End":"00:11.640","Text":"the first 1 element, usually,"},{"Start":"00:11.640 ","End":"00:13.080","Text":"it\u0027s 3 elements or more,"},{"Start":"00:13.080 ","End":"00:18.255","Text":"but just using 1 element of a Maclaurin series to estimate the error."},{"Start":"00:18.255 ","End":"00:21.540","Text":"Obviously, the Maclaurin series you want is"},{"Start":"00:21.540 ","End":"00:24.870","Text":"the one for cosine and we have a table in the appendix."},{"Start":"00:24.870 ","End":"00:26.890","Text":"Let me bring the one for cosine."},{"Start":"00:26.890 ","End":"00:32.710","Text":"Here it is, but it\u0027s important to note that this is for x in radians."},{"Start":"00:32.710 ","End":"00:35.950","Text":"So we have to write cosine 4 degrees in terms of"},{"Start":"00:35.950 ","End":"00:42.485","Text":"radians and we see how many times 4 goes into a 180, it\u0027s 45."},{"Start":"00:42.485 ","End":"00:47.775","Text":"So this is actually cosine of Pi/ 45."},{"Start":"00:47.775 ","End":"00:49.620","Text":"If you are not sure about that I\u0027ll show you at the side."},{"Start":"00:49.620 ","End":"00:57.740","Text":"The formula is that we take the degrees and multiply by Pi/180 and then 4 into a 180,"},{"Start":"00:57.740 ","End":"01:01.020","Text":"goes 45 times and so we get this."},{"Start":"01:01.160 ","End":"01:08.350","Text":"What we want to do now is estimate using this series."},{"Start":"01:09.380 ","End":"01:11.750","Text":"Just want the first non-zero,"},{"Start":"01:11.750 ","End":"01:21.360","Text":"I would write that cosine of 4 degrees or Pi/45 radians would"},{"Start":"01:21.360 ","End":"01:31.290","Text":"be equal to 1 minus Pi/45^2/2 factorial,"},{"Start":"01:31.290 ","End":"01:38.060","Text":"which is 2 plus Pi/45^4/4 factorial,"},{"Start":"01:38.060 ","End":"01:42.180","Text":"which is 24 minus and so on."},{"Start":"01:42.680 ","End":"01:51.390","Text":"Notice that this is a Leibnitz series, so let\u0027s just write that."},{"Start":"01:51.390 ","End":"01:56.230","Text":"We\u0027ve talked about this many times before it\u0027s alternating in sign."},{"Start":"01:56.230 ","End":"01:58.840","Text":"We have a plus, minus, plus, minus."},{"Start":"01:58.840 ","End":"02:03.970","Text":"Each term is an absolute value decreasing and it decreases to 0."},{"Start":"02:03.970 ","End":"02:06.805","Text":"Of course, it decreases to 0 because it\u0027s convergent."},{"Start":"02:06.805 ","End":"02:09.595","Text":"Series is converges for all x."},{"Start":"02:09.595 ","End":"02:13.510","Text":"Because it\u0027s alignment series there is that theorem we keep using that if"},{"Start":"02:13.510 ","End":"02:17.080","Text":"you estimate this using a partial sum,"},{"Start":"02:17.080 ","End":"02:19.645","Text":"the error is at most the following term."},{"Start":"02:19.645 ","End":"02:22.825","Text":"Here, first non-zero element."},{"Start":"02:22.825 ","End":"02:25.720","Text":"So this is the first non-zero element, just 1."},{"Start":"02:25.720 ","End":"02:27.144","Text":"So that\u0027s my estimate."},{"Start":"02:27.144 ","End":"02:31.955","Text":"I say that cosine of Pi/45,"},{"Start":"02:31.955 ","End":"02:33.670","Text":"or in the original language,"},{"Start":"02:33.670 ","End":"02:40.250","Text":"4 degrees is approximately equal to 1 term."},{"Start":"02:40.250 ","End":"02:42.035","Text":"Unusual, like I said."},{"Start":"02:42.035 ","End":"02:48.140","Text":"The error is less than the following term, which is,"},{"Start":"02:48.140 ","End":"02:57.000","Text":"comes out to be Pi^2/4,050."},{"Start":"02:57.000 ","End":"02:58.940","Text":"If you want to get an idea roughly how much this is."},{"Start":"02:58.940 ","End":"03:03.935","Text":"Pi^2 is about 10, so 10/4,000, roughly 1/400."},{"Start":"03:03.935 ","End":"03:09.010","Text":"So that\u0027s pretty close even though we weren\u0027t taking 1 term."},{"Start":"03:09.010 ","End":"03:11.070","Text":"So this is the answer,"},{"Start":"03:11.070 ","End":"03:20.175","Text":"that\u0027s cosine 4 degrees is approximately equal to 1 and the error is less than this."},{"Start":"03:20.175 ","End":"03:22.160","Text":"Just to finish things off,"},{"Start":"03:22.160 ","End":"03:25.625","Text":"let me actually do this on the calculator."},{"Start":"03:25.625 ","End":"03:35.535","Text":"Cosine 4 degrees, actually in my calculator comes out to 0.997564 and so on."},{"Start":"03:35.535 ","End":"03:38.650","Text":"It is pretty close to 1."},{"Start":"03:38.650 ","End":"03:42.440","Text":"The error is already in the third place."},{"Start":"03:42.440 ","End":"03:45.890","Text":"Like I said, it\u0027s roughly at most 1/400."},{"Start":"03:45.890 ","End":"03:48.750","Text":"This is the error. Okay. We\u0027re done."}],"ID":10351},{"Watched":false,"Name":"Exercise 2 Part c","Duration":"2m 53s","ChapterTopicVideoID":10085,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10085.jpeg","UploadDate":"2017-08-20T11:26:31.7270000","DurationForVideoObject":"PT2M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, we have to evaluate natural log of 1.5,"},{"Start":"00:03.990 ","End":"00:08.940","Text":"which I can also write as a fraction natural log of 1 and a 1/2."},{"Start":"00:08.940 ","End":"00:13.140","Text":"We\u0027re going to use the first 4 non-zero elements of Maclaurin series."},{"Start":"00:13.140 ","End":"00:15.945","Text":"After that, we also have to estimate the error."},{"Start":"00:15.945 ","End":"00:18.060","Text":"Now, which Maclaurin series are we going to use?"},{"Start":"00:18.060 ","End":"00:20.205","Text":"We need something with natural logarithm."},{"Start":"00:20.205 ","End":"00:23.415","Text":"The closest we\u0027ve got is the following."},{"Start":"00:23.415 ","End":"00:30.480","Text":"Here it is and it converges for minus 1 less than x,"},{"Start":"00:30.480 ","End":"00:35.010","Text":"less than or equal to 1."},{"Start":"00:35.010 ","End":"00:42.780","Text":"What we have is the natural log of 1 plus a 1/2."},{"Start":"00:42.780 ","End":"00:48.735","Text":"It\u0027s 1 plus x if we let x equal 1/2 here."},{"Start":"00:48.735 ","End":"00:55.835","Text":"What we get is that the natural log of 1 and a 1/2 is equal to,"},{"Start":"00:55.835 ","End":"00:58.500","Text":"I\u0027ll not take the sigma part,"},{"Start":"00:58.500 ","End":"01:00.920","Text":"I\u0027ll take the expanded part is x,"},{"Start":"01:00.920 ","End":"01:08.324","Text":"which is 1/2 minus a 1/2 squared over 2,"},{"Start":"01:08.324 ","End":"01:18.885","Text":"plus 1/2 cubed over 3 minus 1/2 to the 4 over 4."},{"Start":"01:18.885 ","End":"01:22.985","Text":"Now that\u0027s already 4 terms and that will be good for the estimation,"},{"Start":"01:22.985 ","End":"01:27.110","Text":"but for the error, we\u0027ll need the next 3 term."},{"Start":"01:27.110 ","End":"01:32.500","Text":"Obviously, the next 1 is 1/2 to the 5 over 5 and with a plus."},{"Start":"01:32.500 ","End":"01:38.070","Text":"Notice that what we have here is a Leibniz series,"},{"Start":"01:38.890 ","End":"01:44.300","Text":"which we\u0027ve seen often because it\u0027s alternating in sign;"},{"Start":"01:44.300 ","End":"01:46.295","Text":"plus, minus, plus, minus, plus."},{"Start":"01:46.295 ","End":"01:52.325","Text":"Also, the terms in absolute value get successively smaller and in fact 10-0."},{"Start":"01:52.325 ","End":"01:54.290","Text":"You see the power is bigger and bigger,"},{"Start":"01:54.290 ","End":"01:57.890","Text":"so the numerator is smaller and smaller,"},{"Start":"01:57.890 ","End":"01:59.195","Text":"denominator is bigger and bigger."},{"Start":"01:59.195 ","End":"02:02.195","Text":"It\u0027s decreasing all the time and it turns to 0."},{"Start":"02:02.195 ","End":"02:07.040","Text":"We can use the theorem that if we estimate using a certain number of terms,"},{"Start":"02:07.040 ","End":"02:12.410","Text":"in this case, 4 the error is at most the following term without the sign."},{"Start":"02:12.410 ","End":"02:14.945","Text":"We have to evaluate these 2 things."},{"Start":"02:14.945 ","End":"02:21.725","Text":"The approximation natural log of 1.5 is approximately equal to what\u0027s here."},{"Start":"02:21.725 ","End":"02:23.680","Text":"I\u0027ll tell you the answer."},{"Start":"02:23.680 ","End":"02:30.150","Text":"Comes out to 77 over 192 for the estimation"},{"Start":"02:30.150 ","End":"02:34.760","Text":"and the error is going to be less than the following term."},{"Start":"02:34.760 ","End":"02:37.220","Text":"This we can quickly do."},{"Start":"02:37.220 ","End":"02:43.130","Text":"2 to the 5 is 32, so it\u0027s 1 over 32 times 5 is 160,"},{"Start":"02:43.130 ","End":"02:46.870","Text":"so the error is less than 1 over a 160."},{"Start":"02:46.870 ","End":"02:50.960","Text":"These were what we were looking for, the estimation and the error."},{"Start":"02:50.960 ","End":"02:54.090","Text":"We are done."}],"ID":10352},{"Watched":false,"Name":"Exercise 3 Part a","Duration":"5m 15s","ChapterTopicVideoID":10089,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10089.jpeg","UploadDate":"2017-08-20T11:27:13.4630000","DurationForVideoObject":"PT5M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"This exercise is a rather strange 1 at first."},{"Start":"00:03.600 ","End":"00:05.970","Text":"Let\u0027s read it and then see what it means."},{"Start":"00:05.970 ","End":"00:11.250","Text":"What\u0027s the maximum error when we approximate sine x as x minus x cubed over"},{"Start":"00:11.250 ","End":"00:17.700","Text":"3 factorial for absolute value of x less than or equal to Pi over 6."},{"Start":"00:17.700 ","End":"00:21.210","Text":"Well, let\u0027s start somewhere."},{"Start":"00:21.210 ","End":"00:25.170","Text":"We know we\u0027re in the chapter on Taylor and Maclaurin series,"},{"Start":"00:25.170 ","End":"00:28.380","Text":"so let\u0027s see what this expression means,"},{"Start":"00:28.380 ","End":"00:29.745","Text":"where it came from."},{"Start":"00:29.745 ","End":"00:33.255","Text":"The obvious thing to do is to look for the Taylor or Maclaurin"},{"Start":"00:33.255 ","End":"00:37.310","Text":"expansion of sine x. Here it is."},{"Start":"00:37.310 ","End":"00:42.605","Text":"I copy pasted it from the appendix where we have a table of Maclaurin expansions."},{"Start":"00:42.605 ","End":"00:49.130","Text":"Look, this x minus x cubed over 3 factorial is the same as this,"},{"Start":"00:49.130 ","End":"00:51.990","Text":"and we\u0027re talking about sine x."},{"Start":"00:53.530 ","End":"00:58.160","Text":"What we\u0027ve done is we\u0027ve taken just the first 2 terms in"},{"Start":"00:58.160 ","End":"01:03.785","Text":"the Maclaurin expansion and say that\u0027s going to be an approximation."},{"Start":"01:03.785 ","End":"01:05.600","Text":"Now the question is,"},{"Start":"01:05.600 ","End":"01:07.985","Text":"how good an approximation is this?"},{"Start":"01:07.985 ","End":"01:11.540","Text":"Well, that would depend on which x,"},{"Start":"01:11.540 ","End":"01:15.860","Text":"for different x\u0027s will get a different quality of approximation."},{"Start":"01:15.860 ","End":"01:17.630","Text":"What we\u0027re saying is,"},{"Start":"01:17.630 ","End":"01:21.124","Text":"let\u0027s restrict x to this interval,"},{"Start":"01:21.124 ","End":"01:29.930","Text":"which also interprets as x is between Pi over 6 and minus Pi over 6."},{"Start":"01:29.930 ","End":"01:31.280","Text":"Or if you like it in degrees,"},{"Start":"01:31.280 ","End":"01:34.505","Text":"this is 30 degrees and this is minus 30 degrees."},{"Start":"01:34.505 ","End":"01:38.130","Text":"What we\u0027re saying is, suppose x is in this interval,"},{"Start":"01:38.130 ","End":"01:45.290","Text":"what\u0027s the worst-case scenario for an error when we approximate this by this."},{"Start":"01:45.290 ","End":"01:47.360","Text":"We\u0027re going to take all the x\u0027s in this range,"},{"Start":"01:47.360 ","End":"01:50.270","Text":"and take the worst possible approximation means"},{"Start":"01:50.270 ","End":"01:55.915","Text":"the biggest distance between these 2 and see if we can put a limit on that."},{"Start":"01:55.915 ","End":"01:58.160","Text":"That explains the question."},{"Start":"01:58.160 ","End":"02:00.715","Text":"Now let\u0027s go to it."},{"Start":"02:00.715 ","End":"02:04.640","Text":"This Maclaurin series converges for all x."},{"Start":"02:04.640 ","End":"02:06.785","Text":"There is no problem with that."},{"Start":"02:06.785 ","End":"02:09.530","Text":"If I don\u0027t do the approximation,"},{"Start":"02:09.530 ","End":"02:11.585","Text":"I get that sine x."},{"Start":"02:11.585 ","End":"02:13.265","Text":"Let me just write it again."},{"Start":"02:13.265 ","End":"02:18.530","Text":"Sine x is x minus x cubed over 3 factorial"},{"Start":"02:18.530 ","End":"02:24.425","Text":"plus x^5 over 5 factorial minus x^7 over 7 factorial,"},{"Start":"02:24.425 ","End":"02:26.705","Text":"and so on to infinity."},{"Start":"02:26.705 ","End":"02:29.195","Text":"That\u0027s exactly equal too."},{"Start":"02:29.195 ","End":"02:37.370","Text":"Now, notice that this is actually a Leibniz series on this range."},{"Start":"02:37.370 ","End":"02:45.830","Text":"In fact, it\u0027s a Leibniz series whenever absolute value of x is less than 1 because, well,"},{"Start":"02:45.830 ","End":"02:49.910","Text":"first of all, it has alternating signs everywhere,"},{"Start":"02:49.910 ","End":"02:51.980","Text":"because this is plus, minus, plus,"},{"Start":"02:51.980 ","End":"02:54.320","Text":"minus if x is positive."},{"Start":"02:54.320 ","End":"02:56.420","Text":"If x is negative, then it\u0027s minus, plus,"},{"Start":"02:56.420 ","End":"02:59.255","Text":"minus, plus and still alternating."},{"Start":"02:59.255 ","End":"03:02.690","Text":"The reason it\u0027s also decreasing,"},{"Start":"03:02.690 ","End":"03:05.255","Text":"which is another condition for Leibniz,"},{"Start":"03:05.255 ","End":"03:07.550","Text":"it\u0027s decreasing because if we take"},{"Start":"03:07.550 ","End":"03:11.780","Text":"the absolute value of each of the terms, they\u0027re getting smaller."},{"Start":"03:11.780 ","End":"03:15.935","Text":"Because whenever x is less than 1 in magnitude,"},{"Start":"03:15.935 ","End":"03:18.275","Text":"then when you take higher powers,"},{"Start":"03:18.275 ","End":"03:20.090","Text":"it gets smaller and smaller."},{"Start":"03:20.090 ","End":"03:22.880","Text":"That\u0027s the numerators and the denominators get bigger and bigger."},{"Start":"03:22.880 ","End":"03:25.420","Text":"The whole thing is decreasing."},{"Start":"03:25.420 ","End":"03:27.600","Text":"It also tends to 0,"},{"Start":"03:27.600 ","End":"03:30.290","Text":"that\u0027s because this is a convergent series,"},{"Start":"03:30.290 ","End":"03:32.360","Text":"so the terms go to 0."},{"Start":"03:32.360 ","End":"03:34.460","Text":"We do have a Leibniz series,"},{"Start":"03:34.460 ","End":"03:36.260","Text":"at least in this range."},{"Start":"03:36.260 ","End":"03:38.300","Text":"It turns out it\u0027s also Leibniz everywhere,"},{"Start":"03:38.300 ","End":"03:39.905","Text":"but never mind that."},{"Start":"03:39.905 ","End":"03:43.490","Text":"Certainly here and Pi over 6 is less than 1."},{"Start":"03:43.490 ","End":"03:45.310","Text":"That means that we can use the theorem,"},{"Start":"03:45.310 ","End":"03:50.585","Text":"if I take the partial sum and in this case I\u0027ll take the partial sum of 2 terms,"},{"Start":"03:50.585 ","End":"03:57.810","Text":"then the error is at most the magnitude of the following term."},{"Start":"03:58.570 ","End":"04:02.840","Text":"What I will have to do now is estimate what this is."},{"Start":"04:02.840 ","End":"04:04.580","Text":"This is the error."},{"Start":"04:04.580 ","End":"04:07.500","Text":"This is the approximation,"},{"Start":"04:07.500 ","End":"04:11.374","Text":"and I have to estimate the error on this interval."},{"Start":"04:11.374 ","End":"04:17.640","Text":"The absolute value of x^5 over 5 factorial, well,"},{"Start":"04:17.640 ","End":"04:25.800","Text":"first of all it\u0027s equal to the absolute value of x^5 over 5 factorial."},{"Start":"04:26.600 ","End":"04:31.185","Text":"Now, something to the 5th gets bigger as that thing gets bigger."},{"Start":"04:31.185 ","End":"04:36.530","Text":"If x is in magnitude is less than Pi over 6,"},{"Start":"04:36.530 ","End":"04:47.820","Text":"then this is certainly going to be less than or equal to Pi over 6^5 over 5 factorial."},{"Start":"04:48.730 ","End":"04:54.560","Text":"We could leave the answer like this and just leave it as an expression."},{"Start":"04:54.560 ","End":"04:56.930","Text":"I actually did it on the calculator."},{"Start":"04:56.930 ","End":"05:05.240","Text":"On the calculator it came out to be 0.0003 something,"},{"Start":"05:05.240 ","End":"05:07.550","Text":"something, and so on."},{"Start":"05:07.550 ","End":"05:16.170","Text":"It\u0027s certainly less than 1/1000 which is the 001 here. I\u0027ll leave it at that."}],"ID":10353},{"Watched":false,"Name":"Exercise 3 Part b","Duration":"3m 35s","ChapterTopicVideoID":10090,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10090.jpeg","UploadDate":"2017-08-20T11:27:23.0530000","DurationForVideoObject":"PT3M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"This exercise might appear strange at first, it asks,"},{"Start":"00:04.020 ","End":"00:07.740","Text":"what\u0027s the maximum error in approximating the natural log of"},{"Start":"00:07.740 ","End":"00:14.130","Text":"1 plus x as just x for certain x and a certain interval."},{"Start":"00:14.130 ","End":"00:16.860","Text":"To understand it, you really have to look at"},{"Start":"00:16.860 ","End":"00:21.910","Text":"the Maclaurin series for the natural log of 1 plus x."},{"Start":"00:22.010 ","End":"00:28.425","Text":"What it\u0027s asking is if instead of natural log of 1 plus x,"},{"Start":"00:28.425 ","End":"00:33.015","Text":"and I look at the expanded part to get the Sigma part,"},{"Start":"00:33.015 ","End":"00:35.370","Text":"if I, instead of taking the whole series,"},{"Start":"00:35.370 ","End":"00:41.090","Text":"just take the first term x then as an approximation,"},{"Start":"00:41.090 ","End":"00:44.645","Text":"we know there is a certain error and we\u0027re going to estimate the error,"},{"Start":"00:44.645 ","End":"00:50.420","Text":"not for all x, but for x in this range satisfies this condition."},{"Start":"00:50.420 ","End":"00:53.525","Text":"Let me just write this out longhand."},{"Start":"00:53.525 ","End":"00:59.570","Text":"Natural log of 1 plus x is x minus x squared over"},{"Start":"00:59.570 ","End":"01:07.235","Text":"2 plus x cubed over 3 minus x^4 over 4 plus, and so on."},{"Start":"01:07.235 ","End":"01:09.470","Text":"Now also from the appendix,"},{"Start":"01:09.470 ","End":"01:13.175","Text":"I see that the radius or the interval rather of convergence"},{"Start":"01:13.175 ","End":"01:18.320","Text":"is where x is between 1 and minus 1 actually includes the 1."},{"Start":"01:18.320 ","End":"01:22.490","Text":"Certainly on this interval,"},{"Start":"01:22.490 ","End":"01:29.720","Text":"which I could write as x between 0.01 and minus 0.01."},{"Start":"01:29.720 ","End":"01:30.875","Text":"This is included in this,"},{"Start":"01:30.875 ","End":"01:33.530","Text":"so it\u0027s certainly convergent here."},{"Start":"01:33.530 ","End":"01:38.630","Text":"Now what the question is asking is if I just take a partial sum, in fact,"},{"Start":"01:38.630 ","End":"01:46.940","Text":"a very partial sum of just the first term what will the arrow be and as usual,"},{"Start":"01:46.940 ","End":"01:49.025","Text":"in exercises like this,"},{"Start":"01:49.025 ","End":"01:51.620","Text":"this is a Leibnitz series."},{"Start":"01:51.620 ","End":"01:53.945","Text":"We\u0027ve seen this actually before."},{"Start":"01:53.945 ","End":"01:59.765","Text":"Le\u0027s write the word Leibnitz series."},{"Start":"01:59.765 ","End":"02:08.805","Text":"The reason is because it\u0027s alternating in sign and the terms decrease down to 0."},{"Start":"02:08.805 ","End":"02:13.340","Text":"We know that there\u0027s a theorem for this and if I take a partial sum,"},{"Start":"02:13.340 ","End":"02:21.045","Text":"then the error is at most the following term in absolute value forgetting about sign."},{"Start":"02:21.045 ","End":"02:24.660","Text":"All I have to do to estimate the error,"},{"Start":"02:24.910 ","End":"02:29.200","Text":"this is the upper bound for the error,"},{"Start":"02:29.200 ","End":"02:32.990","Text":"is to just figure out this on this range."},{"Start":"02:32.990 ","End":"02:36.190","Text":"The x squared over 2,"},{"Start":"02:36.190 ","End":"02:39.800","Text":"and I can write this as absolute value of x squared over 2,"},{"Start":"02:39.800 ","End":"02:41.780","Text":"which is really what I want,"},{"Start":"02:41.780 ","End":"02:44.855","Text":"is going to be less than,"},{"Start":"02:44.855 ","End":"02:48.460","Text":"because x is less than 0.01."},{"Start":"02:48.460 ","End":"02:51.100","Text":"For positive numbers, when you square it,"},{"Start":"02:51.100 ","End":"02:53.335","Text":"it preserves the order,"},{"Start":"02:53.335 ","End":"03:00.190","Text":"x squared is going to be less than 0.01 squared over 2."},{"Start":"03:00.190 ","End":"03:03.190","Text":"I can actually compute this without a calculator."},{"Start":"03:03.190 ","End":"03:09.625","Text":"This is 100, so squared is 1 over 10,000 that\u0027s 1 over 20,000,"},{"Start":"03:09.625 ","End":"03:14.150","Text":"1 over 20,000 is 0.000,"},{"Start":"03:14.150 ","End":"03:19.970","Text":"lets see the10,000, that would be here so I need another 0 and 5."},{"Start":"03:19.970 ","End":"03:24.175","Text":"This is a worst case and it\u0027s still an upper bound."},{"Start":"03:24.175 ","End":"03:27.705","Text":"The arrow is going to be usually quite a bit less than this even."},{"Start":"03:27.705 ","End":"03:30.505","Text":"Anyway, this is the answer."},{"Start":"03:30.505 ","End":"03:34.910","Text":"I\u0027ll highlight it and we\u0027re done."}],"ID":10354},{"Watched":false,"Name":"Exercise 3 Part c","Duration":"4m 16s","ChapterTopicVideoID":10088,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10088.jpeg","UploadDate":"2017-08-20T11:27:01.3400000","DurationForVideoObject":"PT4M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we have to figure out the maximum error when we approximate"},{"Start":"00:05.730 ","End":"00:11.610","Text":"cosine of x by the following expression,"},{"Start":"00:11.610 ","End":"00:16.665","Text":"which is actually the partial sum of a Maclaurin series."},{"Start":"00:16.665 ","End":"00:19.455","Text":"We\u0027ve seen this kind of exercise before."},{"Start":"00:19.455 ","End":"00:23.670","Text":"I\u0027ll copy the Maclaurin series in a moment,"},{"Start":"00:23.670 ","End":"00:26.370","Text":"and we have to make the estimation of the error in"},{"Start":"00:26.370 ","End":"00:29.805","Text":"a certain range of xs that were under this condition."},{"Start":"00:29.805 ","End":"00:33.870","Text":"Here\u0027s the Maclaurin series,"},{"Start":"00:33.870 ","End":"00:37.220","Text":"which I copied from the table in the appendix."},{"Start":"00:37.220 ","End":"00:43.095","Text":"Notice that what we have here is exactly the first 3 terms here. Let me write it out."},{"Start":"00:43.095 ","End":"00:51.210","Text":"Cosine x is equal to 1 minus x squared over 2 factorial,"},{"Start":"00:51.210 ","End":"00:55.245","Text":"plus x^4 over 4 factorial,"},{"Start":"00:55.245 ","End":"00:59.250","Text":"minus x^6 over 6 factorial."},{"Start":"00:59.250 ","End":"01:01.310","Text":"This will be enough,"},{"Start":"01:01.310 ","End":"01:05.990","Text":"because we want to estimate the cosine of x as approximately"},{"Start":"01:05.990 ","End":"01:11.165","Text":"equal to the first 3 terms of the series,"},{"Start":"01:11.165 ","End":"01:15.050","Text":"and as usual, we\u0027re going to show that this is a Leibniz series,"},{"Start":"01:15.050 ","End":"01:18.470","Text":"and the arrow will be the following term."},{"Start":"01:18.470 ","End":"01:21.805","Text":"So let\u0027s go in more detail."},{"Start":"01:21.805 ","End":"01:25.520","Text":"Now, I use the expression Leibnitz series."},{"Start":"01:25.520 ","End":"01:27.120","Text":"You\u0027ve done this many times before,"},{"Start":"01:27.120 ","End":"01:29.120","Text":"it means alternating in sign,"},{"Start":"01:29.120 ","End":"01:30.560","Text":"which is obvious, plus,"},{"Start":"01:30.560 ","End":"01:33.365","Text":"minus, plus, minus, and so on."},{"Start":"01:33.365 ","End":"01:37.200","Text":"Each term is decreasing,"},{"Start":"01:37.340 ","End":"01:41.160","Text":"it\u0027s certainly decreasing on this range,"},{"Start":"01:41.160 ","End":"01:45.950","Text":"because if x is less than 0.2 in absolute value,"},{"Start":"01:45.950 ","End":"01:47.375","Text":"x squared, x^4, x^6,"},{"Start":"01:47.375 ","End":"01:48.905","Text":"they keep getting smaller,"},{"Start":"01:48.905 ","End":"01:52.610","Text":"because when a number is less than 1 in magnitude,"},{"Start":"01:52.610 ","End":"01:55.480","Text":"then the powers only get smaller,"},{"Start":"01:55.480 ","End":"01:59.090","Text":"and certainly, the general term goes to 0."},{"Start":"01:59.090 ","End":"02:00.875","Text":"It\u0027s another condition for Leibnitz."},{"Start":"02:00.875 ","End":"02:05.450","Text":"That\u0027s because the series converges so the term tends to 0."},{"Start":"02:05.450 ","End":"02:09.145","Text":"We can use the theorems from Leibnitz series,"},{"Start":"02:09.145 ","End":"02:10.620","Text":"and I just forgot to mention,"},{"Start":"02:10.620 ","End":"02:13.140","Text":"this is true for all x."},{"Start":"02:13.140 ","End":"02:18.125","Text":"In particular, it converges for the x in our range,"},{"Start":"02:18.125 ","End":"02:20.965","Text":"which we could also write as,"},{"Start":"02:20.965 ","End":"02:23.010","Text":"if you want to interpret this,"},{"Start":"02:23.010 ","End":"02:26.115","Text":"it means that x is between 0.2,"},{"Start":"02:26.115 ","End":"02:30.130","Text":"and minus 0.2, just to help you visualize it."},{"Start":"02:30.200 ","End":"02:33.960","Text":"All we have to do is this is the approximation,"},{"Start":"02:33.960 ","End":"02:36.045","Text":"this is a bound on the error,"},{"Start":"02:36.045 ","End":"02:37.700","Text":"it\u0027s not the actual error,"},{"Start":"02:37.700 ","End":"02:39.770","Text":"it\u0027s an upper bound on the error,"},{"Start":"02:39.770 ","End":"02:42.410","Text":"and we want to see what\u0027s the worst-case scenario,"},{"Start":"02:42.410 ","End":"02:46.500","Text":"the maximum error on this range."},{"Start":"02:46.640 ","End":"02:49.395","Text":"What we\u0027ll do is, we\u0027ll say,"},{"Start":"02:49.395 ","End":"02:58.190","Text":"what is the value at most of x^6 over 6 factorial in absolute value?"},{"Start":"02:58.190 ","End":"03:00.515","Text":"I can put the absolute value here."},{"Start":"03:00.515 ","End":"03:03.050","Text":"That\u0027s the absolute value. This is the same as this."},{"Start":"03:03.050 ","End":"03:09.500","Text":"It\u0027s certainly going to be less than or equal to if x is less than 0.2, and to the 6,"},{"Start":"03:09.500 ","End":"03:11.360","Text":"it\u0027s still going to preserve the inequality,"},{"Start":"03:11.360 ","End":"03:13.715","Text":"it\u0027s going to be 0.2^6,"},{"Start":"03:13.715 ","End":"03:18.110","Text":"over 6 factorial, what is that?"},{"Start":"03:18.110 ","End":"03:20.455","Text":"720."},{"Start":"03:20.455 ","End":"03:24.080","Text":"Maybe I\u0027ll just repeat that this is true if"},{"Start":"03:24.080 ","End":"03:28.490","Text":"absolute value of x is less than or equal to 0.2."},{"Start":"03:28.490 ","End":"03:31.480","Text":"I could leave the answer like this,"},{"Start":"03:31.480 ","End":"03:36.650","Text":"but perhaps, we should just get an idea of what it is equal to."},{"Start":"03:36.650 ","End":"03:38.690","Text":"I\u0027ll do it on the calculator."},{"Start":"03:38.690 ","End":"03:42.080","Text":"What I did was instead of 0.2, I put 1/5,"},{"Start":"03:42.080 ","End":"03:49.825","Text":"and I got that this equals 1 over 11,250,000."},{"Start":"03:49.825 ","End":"03:52.875","Text":"It\u0027s certainly very small,"},{"Start":"03:52.875 ","End":"03:55.335","Text":"less than 1 over 10,000,000."},{"Start":"03:55.335 ","End":"03:58.610","Text":"This is a pretty good approximation for cosine."},{"Start":"03:58.610 ","End":"04:00.665","Text":"It\u0027s like these 3 terms,"},{"Start":"04:00.665 ","End":"04:06.485","Text":"at least when x is between 0.2 and minus 0.2 radians."},{"Start":"04:06.485 ","End":"04:08.900","Text":"Anyway, we can leave the answer like this."},{"Start":"04:08.900 ","End":"04:11.180","Text":"In fact, you could even leave 6 factorial in"},{"Start":"04:11.180 ","End":"04:15.900","Text":"the denominator and that would be okay, and we\u0027re done."}],"ID":10355},{"Watched":false,"Name":"Exercise 4 Part a","Duration":"2m 58s","ChapterTopicVideoID":10092,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10092.jpeg","UploadDate":"2017-08-20T11:27:39.1930000","DurationForVideoObject":"PT2M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we\u0027re approximating"},{"Start":"00:03.390 ","End":"00:07.980","Text":"the sine x by this expression."},{"Start":"00:07.980 ","End":"00:12.060","Text":"We want to know for which values of x"},{"Start":"00:12.060 ","End":"00:16.855","Text":"the error in this approximation is less than 0.001."},{"Start":"00:16.855 ","End":"00:20.240","Text":"This is similar to previous exercises"},{"Start":"00:20.240 ","End":"00:21.890","Text":"where we\u0027ve been given the reverse."},{"Start":"00:21.890 ","End":"00:24.710","Text":"We give which values of x in a certain range,"},{"Start":"00:24.710 ","End":"00:26.240","Text":"and we have to estimate the error."},{"Start":"00:26.240 ","End":"00:27.850","Text":"This is the reverse."},{"Start":"00:27.850 ","End":"00:29.615","Text":"But same idea."},{"Start":"00:29.615 ","End":"00:32.600","Text":"We start with the Maclaurin series"},{"Start":"00:32.600 ","End":"00:36.870","Text":"for sine x, which is this."},{"Start":"00:36.870 ","End":"00:39.810","Text":"It\u0027s true for all x."},{"Start":"00:39.810 ","End":"00:42.080","Text":"Let me just write some of it out."},{"Start":"00:42.080 ","End":"00:45.160","Text":"Sine x is equal to,"},{"Start":"00:45.160 ","End":"00:47.565","Text":"use this form, not the Sigma form,"},{"Start":"00:47.565 ","End":"00:51.900","Text":"x minus x cubed over 3 factorial"},{"Start":"00:51.900 ","End":"00:56.460","Text":"plus x^5th over 5 factorial minus,"},{"Start":"00:56.460 ","End":"00:59.050","Text":"this will be enough for me."},{"Start":"01:02.750 ","End":"01:04.830","Text":"I\u0027ll stop here."},{"Start":"01:04.830 ","End":"01:09.710","Text":"As usual, we note"},{"Start":"01:09.710 ","End":"01:13.265","Text":"that this is a Leibniz series."},{"Start":"01:13.265 ","End":"01:14.720","Text":"We\u0027ve seen this before,"},{"Start":"01:14.720 ","End":"01:15.980","Text":"so I won\u0027t go into too much detail."},{"Start":"01:15.980 ","End":"01:17.240","Text":"Alternating signs,"},{"Start":"01:17.240 ","End":"01:18.860","Text":"plus, minus, plus, minus,"},{"Start":"01:18.860 ","End":"01:24.060","Text":"and decreasing terms that tend to 0."},{"Start":"01:24.060 ","End":"01:27.620","Text":"There\u0027s a theorem that if I take a partial sum,"},{"Start":"01:27.620 ","End":"01:29.975","Text":"in this case, the first 2 terms,"},{"Start":"01:29.975 ","End":"01:31.640","Text":"that\u0027s the estimate,"},{"Start":"01:31.640 ","End":"01:35.480","Text":"then the error is bounded by"},{"Start":"01:35.480 ","End":"01:38.420","Text":"the absolute value of the following term."},{"Start":"01:38.420 ","End":"01:41.375","Text":"Now, if this is an upper bound on the error,"},{"Start":"01:41.375 ","End":"01:45.965","Text":"if I guarantee that x^5th over 5 factorial,"},{"Start":"01:45.965 ","End":"01:47.855","Text":"at least an absolute value,"},{"Start":"01:47.855 ","End":"01:52.130","Text":"is less than 0.001,"},{"Start":"01:52.130 ","End":"01:53.990","Text":"then that will guarantee that"},{"Start":"01:53.990 ","End":"01:57.225","Text":"this is what we\u0027re asking for."},{"Start":"01:57.225 ","End":"01:59.810","Text":"All we have to do is find the range of x"},{"Start":"01:59.810 ","End":"02:03.505","Text":"or set of values for which this is true."},{"Start":"02:03.505 ","End":"02:06.920","Text":"Now, 5 factorial is 120,"},{"Start":"02:06.920 ","End":"02:09.290","Text":"so I get absolute value of x to the 5th,"},{"Start":"02:09.290 ","End":"02:10.895","Text":"which I can write like this,"},{"Start":"02:10.895 ","End":"02:17.930","Text":"is less than 120 times this is 0.120 or 0.12."},{"Start":"02:17.930 ","End":"02:20.030","Text":"I get that the absolute value of x"},{"Start":"02:20.030 ","End":"02:26.280","Text":"is less than the 5th root of 0.12."},{"Start":"02:26.280 ","End":"02:28.430","Text":"You don\u0027t have to actually estimate this."},{"Start":"02:28.430 ","End":"02:30.890","Text":"But if you\u0027re curious on the calculator,"},{"Start":"02:30.890 ","End":"02:37.310","Text":"it gives 0.654 something,"},{"Start":"02:37.310 ","End":"02:38.810","Text":"just to give you an idea."},{"Start":"02:38.810 ","End":"02:40.010","Text":"But this is fine."},{"Start":"02:40.010 ","End":"02:42.050","Text":"If you prefer it as an interval,"},{"Start":"02:42.050 ","End":"02:45.690","Text":"you could say x is between a 5th root"},{"Start":"02:45.690 ","End":"02:51.140","Text":"of 0.12 and minus the 5th root of 0.12."},{"Start":"02:51.140 ","End":"02:52.910","Text":"But it\u0027s perfectly fine"},{"Start":"02:52.910 ","End":"02:56.869","Text":"just to leave it like this,"},{"Start":"02:56.869 ","End":"02:59.370","Text":"and we\u0027re done."}],"ID":10356},{"Watched":false,"Name":"Exercise 4 Part b","Duration":"3m 4s","ChapterTopicVideoID":10091,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10091.jpeg","UploadDate":"2017-08-20T11:27:31.5970000","DurationForVideoObject":"PT3M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.535","Text":"In this exercise, we\u0027re approximating the arctangent of x as this expression,"},{"Start":"00:07.535 ","End":"00:11.505","Text":"and we have to find out for which values of x we are guaranteed"},{"Start":"00:11.505 ","End":"00:16.390","Text":"that the error in the approximation is less than 0.01."},{"Start":"00:17.450 ","End":"00:22.730","Text":"This, in fact, turns out to be the partial sum of a Maclaurin series."},{"Start":"00:22.730 ","End":"00:25.325","Text":"Let me bring it."},{"Start":"00:25.325 ","End":"00:30.590","Text":"I got this from the table in the appendix and this thing holds"},{"Start":"00:30.590 ","End":"00:36.960","Text":"for x between minus 1 and 1 inclusive."},{"Start":"00:37.600 ","End":"00:39.980","Text":"What we\u0027re saying is,"},{"Start":"00:39.980 ","End":"00:41.375","Text":"well, let me just copy it,"},{"Start":"00:41.375 ","End":"00:47.870","Text":"that arc tangent of x is x minus x"},{"Start":"00:47.870 ","End":"00:56.115","Text":"cubed over 3 plus x to the 5th over 5,"},{"Start":"00:56.115 ","End":"00:59.760","Text":"minus x to the 7th over 7."},{"Start":"00:59.760 ","End":"01:01.710","Text":"You know what? I\u0027ll need 1 more term."},{"Start":"01:01.710 ","End":"01:07.485","Text":"Let\u0027s put x to the 9th over 9 and so on."},{"Start":"01:07.485 ","End":"01:11.620","Text":"What we\u0027re saying is, we\u0027re going to approximate the arctangent x by"},{"Start":"01:11.620 ","End":"01:17.680","Text":"just the first 4 terms it is and as usual,"},{"Start":"01:17.680 ","End":"01:22.150","Text":"we\u0027re going to say that this is a Leibniz series and the error is going to be this."},{"Start":"01:22.150 ","End":"01:25.465","Text":"In fact, why is it a Leibniz series?"},{"Start":"01:25.465 ","End":"01:31.150","Text":"Because it\u0027s alternating in sign whether x is positive or negative."},{"Start":"01:31.150 ","End":"01:32.800","Text":"In 1 case it\u0027s plus, minus plus minus,"},{"Start":"01:32.800 ","End":"01:34.900","Text":"and the other case is minus plus minus plus."},{"Start":"01:34.900 ","End":"01:39.730","Text":"The terms are decreasing in magnitude without the sign,"},{"Start":"01:39.730 ","End":"01:46.674","Text":"because the powers of x keep getting smaller when x is less than 1 in magnitude,"},{"Start":"01:46.674 ","End":"01:49.374","Text":"and it goes to 0 of course,"},{"Start":"01:49.374 ","End":"01:53.430","Text":"the x to the increasing powers,"},{"Start":"01:53.430 ","End":"01:54.735","Text":"it goes to 0."},{"Start":"01:54.735 ","End":"01:56.385","Text":"It is the Leibniz series."},{"Start":"01:56.385 ","End":"02:00.535","Text":"This is true that this is an upper bound on the error,"},{"Start":"02:00.535 ","End":"02:06.210","Text":"the following term after the approximation, the partial sum."},{"Start":"02:06.210 ","End":"02:09.730","Text":"All I have to do, is write an inequality"},{"Start":"02:09.730 ","End":"02:14.660","Text":"now that the absolute value of x to the 9th over 9,"},{"Start":"02:15.210 ","End":"02:19.310","Text":"is less than 0.01,"},{"Start":"02:19.310 ","End":"02:24.945","Text":"and find a set of values or range for x where this is true."},{"Start":"02:24.945 ","End":"02:27.005","Text":"What this actually says,"},{"Start":"02:27.005 ","End":"02:32.490","Text":"is that the absolute value of x to the power of 9 is less than,"},{"Start":"02:32.490 ","End":"02:34.590","Text":"I\u0027ll multiply by 9,"},{"Start":"02:34.590 ","End":"02:38.340","Text":"it gives me 0.09."},{"Start":"02:38.340 ","End":"02:48.195","Text":"So absolute value of x will be less than the 9th root of 0.09."},{"Start":"02:48.195 ","End":"02:50.930","Text":"We could leave the answer like this,"},{"Start":"02:50.930 ","End":"02:54.575","Text":"but if you\u0027re curious as to what this comes out to,"},{"Start":"02:54.575 ","End":"03:00.869","Text":"it\u0027s 0.765 something."},{"Start":"03:00.869 ","End":"03:03.820","Text":"Okay, we\u0027re done."}],"ID":10357},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"5m 10s","ChapterTopicVideoID":10095,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10095.jpeg","UploadDate":"2017-08-20T11:28:19.2000000","DurationForVideoObject":"PT5M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have to, not exactly compute,"},{"Start":"00:04.620 ","End":"00:07.260","Text":"but approximate or estimate the following definite"},{"Start":"00:07.260 ","End":"00:10.290","Text":"integral to within an error of less than,"},{"Start":"00:10.290 ","End":"00:14.580","Text":"this comes out 0.0001."},{"Start":"00:14.580 ","End":"00:17.520","Text":"I don\u0027t know how to do this integral as an indefinite integral,"},{"Start":"00:17.520 ","End":"00:21.525","Text":"but it\u0027s in the chapter on Taylor Maclaurin series."},{"Start":"00:21.525 ","End":"00:23.040","Text":"It does say approximate,"},{"Start":"00:23.040 ","End":"00:25.350","Text":"so that should give you a hint that we\u0027re looking for"},{"Start":"00:25.350 ","End":"00:28.020","Text":"the solution involving Maclaurin\u0027s series."},{"Start":"00:28.020 ","End":"00:30.945","Text":"The obvious thing to try is sine x."},{"Start":"00:30.945 ","End":"00:35.180","Text":"Here is the Maclaurin series for sine x,"},{"Start":"00:35.180 ","End":"00:37.730","Text":"which I got from the table in the appendix."},{"Start":"00:37.730 ","End":"00:40.430","Text":"We\u0027ll use the expanded form,"},{"Start":"00:40.430 ","End":"00:41.900","Text":"not the Sigma part."},{"Start":"00:41.900 ","End":"00:49.935","Text":"What we have is that sine x over x is equal to,"},{"Start":"00:49.935 ","End":"00:57.530","Text":"from here I get 1 minus x squared over 3 factorial,"},{"Start":"00:57.530 ","End":"01:06.285","Text":"I\u0027ll also write that as 6 plus x^4 over 5 factorial is"},{"Start":"01:06.285 ","End":"01:17.505","Text":"120 minus x^6 over 7 factorial is 5,040,"},{"Start":"01:17.505 ","End":"01:22.610","Text":"and then plus and so on and so on, alternating sines,"},{"Start":"01:22.610 ","End":"01:26.120","Text":"even powers of x and then 3 factorial,"},{"Start":"01:26.120 ","End":"01:28.850","Text":"5 factorial, 7 factorial, and so on."},{"Start":"01:28.850 ","End":"01:33.000","Text":"That\u0027s the pattern. I\u0027ve changed my mind."},{"Start":"01:33.000 ","End":"01:35.370","Text":"I\u0027ll keep these in factorial form,"},{"Start":"01:35.370 ","End":"01:38.175","Text":"just so it\u0027s easier to see the pattern."},{"Start":"01:38.175 ","End":"01:43.520","Text":"Now, the integral of sine x over x,"},{"Start":"01:43.520 ","End":"01:46.370","Text":"we want it from 0 to 0.2,"},{"Start":"01:46.370 ","End":"01:50.100","Text":"actually, I might want it as 1/5, we\u0027ll see."},{"Start":"01:50.710 ","End":"01:54.920","Text":"Another note, I\u0027m not going to go into the technical details about"},{"Start":"01:54.920 ","End":"01:58.535","Text":"the interval of convergence and dividing by x and taking the integral."},{"Start":"01:58.535 ","End":"02:00.740","Text":"It works out and it can be justified."},{"Start":"02:00.740 ","End":"02:06.150","Text":"Let\u0027s just get the technical part of it."},{"Start":"02:06.170 ","End":"02:11.150","Text":"We have, first of all, the indefinite integral is x minus, now,"},{"Start":"02:11.150 ","End":"02:14.540","Text":"we raise the power by 1 and divide by that power,"},{"Start":"02:14.540 ","End":"02:16.595","Text":"by 3 times 3 factorial,"},{"Start":"02:16.595 ","End":"02:23.240","Text":"then plus x^5 over 5 times 5 factorial and it\u0027s easier to see the pattern,"},{"Start":"02:23.240 ","End":"02:24.939","Text":"that\u0027s why I kept the factorial,"},{"Start":"02:24.939 ","End":"02:34.365","Text":"minus x^7 over 7 times 7 factorial plus etc."},{"Start":"02:34.365 ","End":"02:43.600","Text":"We want all these evaluated between 0 and 1/5."},{"Start":"02:44.110 ","End":"02:47.705","Text":"Notice that if I substitute 0,"},{"Start":"02:47.705 ","End":"02:50.135","Text":"everything is 0, so that\u0027s 0."},{"Start":"02:50.135 ","End":"02:52.625","Text":"I just have to substitute 1/5."},{"Start":"02:52.625 ","End":"02:56.015","Text":"What I get is the following series;"},{"Start":"02:56.015 ","End":"03:04.770","Text":"1/5 minus 1/5 cubed over 3 times 3 factorial plus 1/5^5"},{"Start":"03:04.770 ","End":"03:15.730","Text":"over 5 times 5 factorial minus 1/5^7 over 7 times 7 factorial, etc."},{"Start":"03:16.160 ","End":"03:19.440","Text":"Already we see the alternating sines plus,"},{"Start":"03:19.440 ","End":"03:21.450","Text":"minus, plus, minus, plus."},{"Start":"03:21.450 ","End":"03:24.460","Text":"It looks like it\u0027s going to be alive in that series,"},{"Start":"03:24.460 ","End":"03:27.970","Text":"and it certainly is."},{"Start":"03:28.220 ","End":"03:34.405","Text":"Not just alternating, but the terms are decreasing,"},{"Start":"03:34.405 ","End":"03:38.524","Text":"certainly 1/5 to each power gets smaller and smaller,"},{"Start":"03:38.524 ","End":"03:40.550","Text":"and the denominator is getting bigger and bigger,"},{"Start":"03:40.550 ","End":"03:42.430","Text":"it decreases and it tends to 0."},{"Start":"03:42.430 ","End":"03:45.230","Text":"So we definitely are with the Leibniz series."},{"Start":"03:45.230 ","End":"03:52.025","Text":"That means that if I approximate it with a partial sum,"},{"Start":"03:52.025 ","End":"03:57.185","Text":"the error is at most the following term without the sine."},{"Start":"03:57.185 ","End":"03:59.615","Text":"All I have to do is go along,"},{"Start":"03:59.615 ","End":"04:06.460","Text":"term by term until we find something that\u0027s less than 0.0001."},{"Start":"04:06.460 ","End":"04:08.315","Text":"Well, this is not it."},{"Start":"04:08.315 ","End":"04:10.835","Text":"This is not small enough."},{"Start":"04:10.835 ","End":"04:13.820","Text":"This one, if you do it on the calculator,"},{"Start":"04:13.820 ","End":"04:16.370","Text":"comes up much less than this."},{"Start":"04:16.370 ","End":"04:23.940","Text":"I can already see it\u0027s 1/5^6 times 5 factorial and 5^6 is already more than 100,000."},{"Start":"04:23.940 ","End":"04:26.960","Text":"We\u0027re going to get 1 of many million."},{"Start":"04:26.960 ","End":"04:28.280","Text":"This is certainly it,"},{"Start":"04:28.280 ","End":"04:32.360","Text":"which means that we can approximate using this partial sum,"},{"Start":"04:32.360 ","End":"04:34.235","Text":"the first 2 terms."},{"Start":"04:34.235 ","End":"04:36.875","Text":"Let\u0027s see, I\u0027ll compute this."},{"Start":"04:36.875 ","End":"04:39.155","Text":"This is 1/5 minus,"},{"Start":"04:39.155 ","End":"04:42.080","Text":"let\u0027s see, 5 cubed is 125,"},{"Start":"04:42.080 ","End":"04:45.090","Text":"3 times 6 is 18,"},{"Start":"04:45.370 ","End":"04:51.425","Text":"125 times 18 is 2,250."},{"Start":"04:51.425 ","End":"04:54.785","Text":"This is 450/2,250,"},{"Start":"04:54.785 ","End":"05:02.555","Text":"altogether we get 449/2,250."},{"Start":"05:02.555 ","End":"05:04.880","Text":"I\u0027ll leave it as a fraction,"},{"Start":"05:04.880 ","End":"05:07.115","Text":"and I\u0027ll highlight it."},{"Start":"05:07.115 ","End":"05:10.350","Text":"This is our answer, we\u0027re done."}],"ID":10358},{"Watched":false,"Name":"Exercise 5 Part b","Duration":"5m 8s","ChapterTopicVideoID":10093,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10093.jpeg","UploadDate":"2017-08-20T11:27:52.7400000","DurationForVideoObject":"PT5M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have to estimate or approximate the"},{"Start":"00:03.720 ","End":"00:07.830","Text":"following definite integral to within an error of less than,"},{"Start":"00:07.830 ","End":"00:12.760","Text":"this comes out to be, 1 over a 1000."},{"Start":"00:13.160 ","End":"00:17.058","Text":"We\u0027re going to do it with Taylor-Maclaurin series,"},{"Start":"00:17.058 ","End":"00:18.840","Text":"naturally, that\u0027s the chapter we\u0027re in."},{"Start":"00:18.840 ","End":"00:22.065","Text":"Also, there is actually no easy way to compute this"},{"Start":"00:22.065 ","End":"00:26.685","Text":"indefinite integral so we going to use an approximation."},{"Start":"00:26.685 ","End":"00:31.665","Text":"We already know what the Maclaurin series is for natural log of 1 plus x."},{"Start":"00:31.665 ","End":"00:35.085","Text":"You can find it in the table in the appendix."},{"Start":"00:35.085 ","End":"00:38.020","Text":"Here it is."},{"Start":"00:38.090 ","End":"00:46.735","Text":"What we want to do is to compute the integral from 0 to 0.1."},{"Start":"00:46.735 ","End":"00:49.470","Text":"Now write this out as 1 over x,"},{"Start":"00:49.470 ","End":"00:50.720","Text":"that\u0027s the x part,"},{"Start":"00:50.720 ","End":"00:54.170","Text":"and the natural log, I\u0027ll replace by this series."},{"Start":"00:54.170 ","End":"00:56.390","Text":"I\u0027ll use the expanded part,"},{"Start":"00:56.390 ","End":"00:57.740","Text":"not the sigma part."},{"Start":"00:57.740 ","End":"01:00.920","Text":"We have x minus x^2 over 2"},{"Start":"01:00.920 ","End":"01:05.927","Text":"plus x^3 over 3"},{"Start":"01:05.927 ","End":"01:13.870","Text":"minus x^4 over 4 plus and so on, dx."},{"Start":"01:16.150 ","End":"01:22.485","Text":"Let\u0027s first of all multiply out by 1/x so we get the integral."},{"Start":"01:22.485 ","End":"01:27.450","Text":"Now this is going to be 1 minus x/2"},{"Start":"01:27.450 ","End":"01:29.880","Text":"plus x^2 over 3"},{"Start":"01:29.880 ","End":"01:32.380","Text":"minus x^3 over 4,"},{"Start":"01:32.380 ","End":"01:34.250","Text":"plus and so on."},{"Start":"01:34.250 ","End":"01:35.510","Text":"All this, dx."},{"Start":"01:35.510 ","End":"01:38.050","Text":"I\u0027ll better put brackets here."},{"Start":"01:38.050 ","End":"01:42.435","Text":"Now let\u0027s do the actual integration."},{"Start":"01:42.435 ","End":"01:49.820","Text":"We have x and then minus x^2 over 2,"},{"Start":"01:49.820 ","End":"01:53.760","Text":"but it\u0027s over 2, so it\u0027s x^2 over 2 times 2,"},{"Start":"01:53.760 ","End":"01:57.050","Text":"and here we have x^3 over 3, but there\u0027s already a 3,"},{"Start":"01:57.050 ","End":"01:58.880","Text":"so it\u0027s 3 times 3,"},{"Start":"01:58.880 ","End":"02:04.880","Text":"minus x^4 over 4 times 4, plus and so on."},{"Start":"02:04.880 ","End":"02:10.580","Text":"But we have to take this between 0 and allow me to write this as 1/10."},{"Start":"02:10.580 ","End":"02:13.800","Text":"It might be easier to work with fractions."},{"Start":"02:14.420 ","End":"02:18.110","Text":"Now we just have to evaluate this."},{"Start":"02:18.110 ","End":"02:22.930","Text":"Notice that if we put in 0, everything is 0."},{"Start":"02:22.930 ","End":"02:25.085","Text":"I can dispense with the 0,"},{"Start":"02:25.085 ","End":"02:27.035","Text":"I\u0027ll just put it in the 1/10."},{"Start":"02:27.035 ","End":"02:30.600","Text":"What we get is 1/10"},{"Start":"02:30.600 ","End":"02:34.119","Text":"minus 1/10 squared over,"},{"Start":"02:34.119 ","End":"02:35.480","Text":"and I\u0027ll multiply it out,"},{"Start":"02:35.480 ","End":"02:36.086","Text":"this is 4"},{"Start":"02:36.086 ","End":"02:41.395","Text":"plus 1/10 cubed over 9,"},{"Start":"02:41.395 ","End":"02:46.490","Text":"minus 1/10 to the fourth over 16,"},{"Start":"02:46.490 ","End":"02:48.730","Text":"plus and so on."},{"Start":"02:48.730 ","End":"02:53.870","Text":"By the way, I\u0027m not going to get into the technical details of interval of convergence."},{"Start":"02:53.870 ","End":"02:55.355","Text":"You just trust that it works out."},{"Start":"02:55.355 ","End":"02:57.980","Text":"We just want the technical answer."},{"Start":"02:57.980 ","End":"03:04.400","Text":"Now, we use the familiar trick"},{"Start":"03:04.400 ","End":"03:09.725","Text":"of identifying it as a Leibniz Series."},{"Start":"03:09.725 ","End":"03:13.055","Text":"Remember, it\u0027s alternating in sign, that\u0027s clear."},{"Start":"03:13.055 ","End":"03:16.520","Text":"The terms are decreasing, they\u0027re shrinking,"},{"Start":"03:16.520 ","End":"03:18.260","Text":"the numerator\u0027s getting smaller,"},{"Start":"03:18.260 ","End":"03:19.910","Text":"denominator is getting bigger,"},{"Start":"03:19.910 ","End":"03:22.040","Text":"and the general term goes to 0."},{"Start":"03:22.040 ","End":"03:25.760","Text":"We can use the theorem about Leibniz series"},{"Start":"03:25.760 ","End":"03:30.245","Text":"that says that if we estimate using a partial sum,"},{"Start":"03:30.245 ","End":"03:34.750","Text":"the error is at most the following term in absolute value."},{"Start":"03:34.750 ","End":"03:36.860","Text":"What we have to do here"},{"Start":"03:36.860 ","End":"03:42.160","Text":"is to find which term is less than,"},{"Start":"03:42.160 ","End":"03:44.700","Text":"let\u0027s see what was the error that we needed,"},{"Start":"03:44.700 ","End":"03:46.740","Text":"it was 1/1000."},{"Start":"03:46.740 ","End":"03:49.364","Text":"Let\u0027s see, this is a tenth."},{"Start":"03:49.364 ","End":"03:52.840","Text":"This is 1 over 400."},{"Start":"03:54.410 ","End":"03:58.140","Text":"This one is 1 over 1,000,"},{"Start":"03:58.140 ","End":"04:00.735","Text":"so it\u0027s 1 over 9,000."},{"Start":"04:00.735 ","End":"04:10.050","Text":"We can stop already because this is already less than 1/1000 or less than 0.001."},{"Start":"04:10.050 ","End":"04:13.570","Text":"Because obviously 9,000 bigger than a thousand."},{"Start":"04:13.610 ","End":"04:17.920","Text":"That means that we can estimate"},{"Start":"04:19.820 ","End":"04:28.265","Text":"this series to within this accuracy if we just take the partial sum up to here."},{"Start":"04:28.265 ","End":"04:30.980","Text":"All I have to do now is just compute"},{"Start":"04:30.980 ","End":"04:35.270","Text":"this 1/10 minus 1/400. Let\u0027s see at the side."},{"Start":"04:35.270 ","End":"04:41.090","Text":"1/10 minus 1/400 is equal to,"},{"Start":"04:41.090 ","End":"04:43.175","Text":"let\u0027s put it all over 400,"},{"Start":"04:43.175 ","End":"04:46.310","Text":"this is going to be 40 minus 1."},{"Start":"04:46.310 ","End":"04:47.835","Text":"No need for calculator."},{"Start":"04:47.835 ","End":"04:50.190","Text":"39 over 400,"},{"Start":"04:50.190 ","End":"04:54.155","Text":"and I\u0027ll highlight it and that\u0027s the answer."},{"Start":"04:54.155 ","End":"04:57.020","Text":"That\u0027s our estimate or approximation."},{"Start":"04:57.020 ","End":"04:59.435","Text":"But for those of you who like decimals,"},{"Start":"04:59.435 ","End":"05:06.978","Text":"this comes out to be 0.9975."},{"Start":"05:06.978 ","End":"05:08.760","Text":"We\u0027re done."}],"ID":10359},{"Watched":false,"Name":"Exercise 5 Part c","Duration":"5m 26s","ChapterTopicVideoID":10094,"CourseChapterTopicPlaylistID":84690,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10094.jpeg","UploadDate":"2017-08-20T11:28:07.2230000","DurationForVideoObject":"PT5M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.930","Text":"In this exercise, we have to approximate this definite integral within the given error."},{"Start":"00:06.930 ","End":"00:09.220","Text":"It\u0027s 1000th."},{"Start":"00:10.000 ","End":"00:16.170","Text":"It\u0027s difficult, it\u0027s practically impossible to do the indefinite integral."},{"Start":"00:16.170 ","End":"00:19.980","Text":"Besides this is in the section on Taylor Maclaurin series,"},{"Start":"00:19.980 ","End":"00:23.865","Text":"so we know we have to use Maclaurin series on this."},{"Start":"00:23.865 ","End":"00:28.740","Text":"Cosine is the 1 that we\u0027re looking for, and here it is."},{"Start":"00:28.740 ","End":"00:34.035","Text":"Let\u0027s not worry about the interval of convergence in this exercise."},{"Start":"00:34.035 ","End":"00:36.540","Text":"Just take it on trust that it works."},{"Start":"00:36.540 ","End":"00:44.510","Text":"What we want is the integral from 0 to 0.5 of 1 over x squared."},{"Start":"00:44.510 ","End":"00:49.480","Text":"I\u0027ll just write that so it\u0027s easier then I put brackets, 1 minus."},{"Start":"00:49.480 ","End":"00:51.680","Text":"Here we\u0027re going to put this series."},{"Start":"00:51.680 ","End":"00:54.245","Text":"I\u0027ll use not the Sigma form, this form,"},{"Start":"00:54.245 ","End":"01:00.380","Text":"1 minus x squared over 2 factorial plus x to the 4th over 4"},{"Start":"01:00.380 ","End":"01:06.770","Text":"factorial minus x to the 6th over 6 factorial plus, and so on."},{"Start":"01:06.770 ","End":"01:08.420","Text":"Think the pattern is clear."},{"Start":"01:08.420 ","End":"01:11.135","Text":"Alternating signs even powers."},{"Start":"01:11.135 ","End":"01:15.860","Text":"The same number here and here with the factorial."},{"Start":"01:15.860 ","End":"01:21.445","Text":"Okay. All this is dx and this is equal to."},{"Start":"01:21.445 ","End":"01:24.370","Text":"Now, this part\u0027s the same."},{"Start":"01:25.190 ","End":"01:28.710","Text":"Over x squared. Let\u0027s just do the subtraction."},{"Start":"01:28.710 ","End":"01:32.340","Text":"1 minus 1 cancels and all these just change signs."},{"Start":"01:32.340 ","End":"01:38.670","Text":"It\u0027s x squared over 2 factorial minus x to the 4th over 4 factorial plus x"},{"Start":"01:38.670 ","End":"01:46.010","Text":"to the 6th over 6 factorial minus and so on, dx."},{"Start":"01:46.010 ","End":"01:51.210","Text":"Finally, we\u0027ll just divide the x squared into all these terms."},{"Start":"01:51.210 ","End":"01:53.280","Text":"It starts from x squared, so, we\u0027re all right."},{"Start":"01:53.280 ","End":"01:58.590","Text":"This is the integral from 0 to 0.5 of,"},{"Start":"01:58.590 ","End":"02:05.045","Text":"this is 1 over 2 factorial minus x squared over 4 factorial"},{"Start":"02:05.045 ","End":"02:13.510","Text":"plus x to the 4th over 6 factorial minus and so on, dx."},{"Start":"02:13.510 ","End":"02:18.220","Text":"Okay, now let\u0027s do the actual integral. Let\u0027s see."},{"Start":"02:18.220 ","End":"02:24.244","Text":"X over 2 factorial minus"},{"Start":"02:24.244 ","End":"02:30.320","Text":"x cubed over 3 times 4 factorial plus x"},{"Start":"02:30.320 ","End":"02:36.920","Text":"to the 5th over 5 times 6 factorial minus and so on."},{"Start":"02:36.920 ","End":"02:39.515","Text":"Hopefully, we won\u0027t need anymore terms."},{"Start":"02:39.515 ","End":"02:44.945","Text":"All this is evaluated between 0 and 0.5."},{"Start":"02:44.945 ","End":"02:46.280","Text":"I\u0027ll write it as a half,"},{"Start":"02:46.280 ","End":"02:49.015","Text":"I think I\u0027ll rather work with fractions."},{"Start":"02:49.015 ","End":"02:53.310","Text":"Now, all we have to do is substitute in."},{"Start":"02:54.020 ","End":"02:57.875","Text":"If we plug in 0, everything\u0027s going to be 0."},{"Start":"02:57.875 ","End":"02:59.000","Text":"We can forget about that\u0027s."},{"Start":"02:59.000 ","End":"03:02.270","Text":"All we have to do is substitute 1/2 in each of these,"},{"Start":"03:02.270 ","End":"03:04.355","Text":"and what we get is"},{"Start":"03:04.355 ","End":"03:13.110","Text":"1/2 over 2 minus half cubed is 1/8 over 3 times."},{"Start":"03:13.110 ","End":"03:16.485","Text":"Let\u0027s seem, 4 factorial is 24,"},{"Start":"03:16.485 ","End":"03:24.030","Text":"plus 1/2 to the 5 is 1 over 32 over 5 times,"},{"Start":"03:24.030 ","End":"03:30.180","Text":"now, 6 factorial is 720 minus and so on."},{"Start":"03:30.290 ","End":"03:38.534","Text":"Now, this is, I\u0027m claiming it\u0027s a Leibniz alternating series."},{"Start":"03:38.534 ","End":"03:41.310","Text":"Leibniz series for short."},{"Start":"03:41.310 ","End":"03:47.569","Text":"Clearly, the signs are alternating because we got it from an alternating sign series,"},{"Start":"03:47.569 ","End":"03:49.790","Text":"so minus, plus, minus plus, so on."},{"Start":"03:49.790 ","End":"03:54.290","Text":"The terms are getting smaller and smaller because the numerators"},{"Start":"03:54.290 ","End":"03:59.405","Text":"are decreasing and the denominators are increasing and they go down to 0."},{"Start":"03:59.405 ","End":"04:03.680","Text":"We can use the theorem on Leibniz series and estimation."},{"Start":"04:03.680 ","End":"04:06.830","Text":"We can estimate by a partial sum up to"},{"Start":"04:06.830 ","End":"04:11.540","Text":"a point and the error is at most the following term."},{"Start":"04:11.540 ","End":"04:15.500","Text":"All I have to do now is look for a term."},{"Start":"04:15.500 ","End":"04:22.965","Text":"Remember the era that we wanted was 1000th or 0.001."},{"Start":"04:22.965 ","End":"04:25.700","Text":"All we have to do is go along checking each term to"},{"Start":"04:25.700 ","End":"04:28.430","Text":"see when we get to something less than 1000th."},{"Start":"04:28.430 ","End":"04:30.775","Text":"Now, this is a quarter."},{"Start":"04:30.775 ","End":"04:33.885","Text":"This is, let\u0027s see."},{"Start":"04:33.885 ","End":"04:35.880","Text":"I think it comes out to be,"},{"Start":"04:35.880 ","End":"04:41.900","Text":"let\u0027s see 8 times 3 is 24 times 24 it\u0027s 576."},{"Start":"04:41.900 ","End":"04:45.680","Text":"Still not less than a 1000th plus 1 over,"},{"Start":"04:45.680 ","End":"04:48.080","Text":"well, this is clearly less than 1000th."},{"Start":"04:48.080 ","End":"04:50.930","Text":"I mean, already the denominator is bigger than a 1000,"},{"Start":"04:50.930 ","End":"04:53.695","Text":"it\u0027s 3600 and multiplied by 32."},{"Start":"04:53.695 ","End":"04:57.115","Text":"This is less than 1 over 1000,"},{"Start":"04:57.115 ","End":"05:01.985","Text":"so we can estimate the series by taking these 2 terms."},{"Start":"05:01.985 ","End":"05:04.385","Text":"All we have to do is compute this."},{"Start":"05:04.385 ","End":"05:08.780","Text":"Let\u0027s see, 4 into 576 goes a 144 times."},{"Start":"05:08.780 ","End":"05:14.985","Text":"We have a 144 minus 1 over 576."},{"Start":"05:14.985 ","End":"05:20.580","Text":"In short, this is 143 over 576."},{"Start":"05:20.580 ","End":"05:22.295","Text":"I\u0027ll leave it as a fraction."},{"Start":"05:22.295 ","End":"05:26.670","Text":"I\u0027ll just highlight it and we are done."}],"ID":10360}],"Thumbnail":null,"ID":84690}]

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