WEBVTT 00:00:14.640 --> 00:00:17.394 When I first learned about Taylor series, I definitely 00:00:17.394 --> 00:00:19.699 didn't appreciate just how important they are. 00:00:20.120 --> 00:00:22.828 But time and time again they come up in math, physics, 00:00:22.827 --> 00:00:25.929 and many fields of engineering because they're one of the most 00:00:25.929 --> 00:00:29.179 powerful tools that math has to offer for approximating functions. 00:00:30.000 --> 00:00:32.710 I think one of the first times this clicked for me as a 00:00:32.710 --> 00:00:35.420 student was not in a calculus class but a physics class. 00:00:35.840 --> 00:00:40.103 We were studying a certain problem that had to do with the potential energy of a 00:00:40.103 --> 00:00:44.155 pendulum, and for that you need an expression for how high the weight of the 00:00:44.155 --> 00:00:48.472 pendulum is above its lowest point, and when you work that out it comes out to be 00:00:48.472 --> 00:00:53.000 proportional to 1 minus the cosine of the angle between the pendulum and the vertical. 00:00:53.579 --> 00:00:57.875 The specifics of the problem we were trying to solve are beyond the point here, 00:00:57.875 --> 00:01:02.493 but what I'll say is that this cosine function made the problem awkward and unwieldy, 00:01:02.493 --> 00:01:06.519 and made it less clear how pendulums relate to other oscillating phenomena. 00:01:07.459 --> 00:01:12.516 But if you approximate cosine of theta as 1 minus theta squared over 2, 00:01:12.516 --> 00:01:15.959 everything just fell into place much more easily. 00:01:16.659 --> 00:01:19.213 If you've never seen anything like this before, 00:01:19.213 --> 00:01:22.780 an approximation like that might seem completely out of left field. 00:01:23.819 --> 00:01:28.803 If you graph cosine of theta along with this function, 1 minus theta squared over 2, 00:01:28.804 --> 00:01:33.203 they do seem rather close to each other, at least for small angles near 0, 00:01:33.203 --> 00:01:36.546 but how would you even think to make this approximation, 00:01:36.546 --> 00:01:39.420 and how would you find that particular quadratic? 00:01:41.219 --> 00:01:44.864 The study of Taylor series is largely about taking non-polynomial 00:01:44.864 --> 00:01:48.840 functions and finding polynomials that approximate them near some input. 00:01:48.840 --> 00:01:52.403 The motive here is that polynomials tend to be much easier to deal 00:01:52.403 --> 00:01:55.277 with than other functions, they're easier to compute, 00:01:55.277 --> 00:01:59.480 easier to take derivatives, easier to integrate, just all around more friendly. 00:02:00.680 --> 00:02:03.819 So let's take a look at that function, cosine of x, 00:02:03.819 --> 00:02:08.408 and really take a moment to think about how you might construct a quadratic 00:02:08.407 --> 00:02:10.219 approximation near x equals 0. 00:02:10.939 --> 00:02:16.444 That is, among all of the possible polynomials that look like c0 plus c1 00:02:16.444 --> 00:02:21.949 times x plus c2 times x squared, for some choice of these constants, c0, 00:02:21.949 --> 00:02:27.530 c1, and c2, find the one that most resembles cosine of x near x equals 0, 00:02:27.531 --> 00:02:32.659 whose graph kind of spoons with the graph of cosine x at that point. 00:02:33.860 --> 00:02:38.218 Well, first of all, at the input 0, the value of cosine of x is 1, 00:02:38.217 --> 00:02:41.861 so if our approximation is going to be any good at all, 00:02:41.861 --> 00:02:44.919 it should also equal 1 at the input x equals 0. 00:02:45.819 --> 00:02:50.939 Plugging in 0 just results in whatever c0 is, so we can set that equal to 1. 00:02:53.080 --> 00:02:56.608 This leaves us free to choose constants c1 and c2 to make this 00:02:56.608 --> 00:03:00.192 approximation as good as we can, but nothing we do with them is 00:03:00.192 --> 00:03:04.000 going to change the fact that the polynomial equals 1 at x equals 0. 00:03:04.960 --> 00:03:08.153 It would also be good if our approximation had the same 00:03:08.153 --> 00:03:11.120 tangent slope as cosine x at this point of interest. 00:03:11.900 --> 00:03:14.484 Otherwise the approximation drifts away from the 00:03:14.484 --> 00:03:16.700 cosine graph much faster than it needs to. 00:03:18.199 --> 00:03:22.146 The derivative of cosine is negative sine, and at x equals 0, 00:03:22.146 --> 00:03:25.840 that equals 0, meaning the tangent line is perfectly flat. 00:03:26.960 --> 00:03:31.920 On the other hand, when you work out the derivative of our quadratic, 00:03:31.919 --> 00:03:34.399 you get c1 plus 2 times c2 times x. 00:03:35.319 --> 00:03:39.419 At x equals 0, this just equals whatever we choose for c1. 00:03:40.259 --> 00:03:43.299 So this constant c1 has complete control over the 00:03:43.300 --> 00:03:46.340 derivative of our approximation around x equals 0. 00:03:47.120 --> 00:03:49.979 Setting it equal to 0 ensures that our approximation 00:03:49.979 --> 00:03:52.300 also has a flat tangent line at this point. 00:03:53.000 --> 00:03:57.810 This leaves us free to change c2, but the value and the slope of our 00:03:57.810 --> 00:04:02.620 polynomial at x equals 0 are locked in place to match that of cosine. 00:04:04.259 --> 00:04:08.378 The final thing to take advantage of is the fact that the cosine graph 00:04:08.378 --> 00:04:12.439 curves downward above x equals 0, it has a negative second derivative. 00:04:13.379 --> 00:04:17.250 Or in other words, even though the rate of change is 0 at that point, 00:04:17.250 --> 00:04:20.459 the rate of change itself is decreasing around that point. 00:04:21.278 --> 00:04:25.445 Specifically, since its derivative is negative sine of x, 00:04:25.446 --> 00:04:31.840 its second derivative is negative cosine of x, and at x equals 0, that equals negative 1. 00:04:33.079 --> 00:04:37.149 Now in the same way that we wanted the derivative of our approximation to 00:04:37.149 --> 00:04:41.933 match that of the cosine so that their values wouldn't drift apart needlessly quickly, 00:04:41.934 --> 00:04:45.784 making sure that their second derivatives match will ensure that they 00:04:45.783 --> 00:04:49.689 curve at the same rate, that the slope of our polynomial doesn't drift 00:04:49.689 --> 00:04:53.319 away from the slope of cosine x any more quickly than it needs to. 00:04:54.220 --> 00:04:59.225 Pulling up the same derivative we had before, and then taking its derivative, 00:04:59.225 --> 00:05:04.040 we see that the second derivative of this polynomial is exactly 2 times c2. 00:05:04.959 --> 00:05:10.435 So to make sure that this second derivative also equals negative 1 at x equals 0, 00:05:10.435 --> 00:05:15.579 2 times c2 has to be negative 1, meaning c2 itself should be negative 1 half. 00:05:16.379 --> 00:05:22.139 This gives us the approximation 1 plus 0x minus 1 half x squared. 00:05:23.199 --> 00:05:29.976 To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial, 00:05:29.976 --> 00:05:35.819 you'd estimate it to be 0.995, and this is the true value of cosine of 0.1. 00:05:36.639 --> 00:05:38.439 It's a really good approximation! 00:05:40.300 --> 00:05:42.520 Take a moment to reflect on what just happened. 00:05:42.519 --> 00:05:46.992 You had 3 degrees of freedom with this quadratic approximation, 00:05:46.992 --> 00:05:49.019 the constants c0, c1, and c2. 00:05:49.519 --> 00:05:55.806 c0 was responsible for making sure that the output of the approximation matches that of 00:05:55.807 --> 00:06:01.951 cosine x at x equals 0, c1 was in charge of making sure that the derivatives match at 00:06:01.951 --> 00:06:08.240 that point, and c2 was responsible for making sure that the second derivatives match up. 00:06:08.939 --> 00:06:14.271 This ensures that the way your approximation changes as you move away from x equals 0, 00:06:14.271 --> 00:06:17.459 and the way that the rate of change itself changes, 00:06:17.459 --> 00:06:20.891 is as similar as possible to the behaviour of cosine x, 00:06:20.891 --> 00:06:23.159 given the amount of control you have. 00:06:24.079 --> 00:06:27.187 You could give yourself more control by allowing more terms 00:06:27.187 --> 00:06:30.139 in your polynomial and matching higher order derivatives. 00:06:30.839 --> 00:06:36.579 For example, let's say you added on the term c3 times x cubed for some constant c3. 00:06:36.579 --> 00:06:41.479 In that case, if you take the third derivative of a cubic polynomial, 00:06:41.480 --> 00:06:44.280 anything quadratic or smaller goes to 0. 00:06:45.560 --> 00:06:50.882 As for that last term, after 3 iterations of the power rule, 00:06:50.882 --> 00:06:54.459 it looks like 1 times 2 times 3 times c3. 00:06:56.459 --> 00:07:01.339 On the other hand, the third derivative of cosine x comes out to sine x, 00:07:01.339 --> 00:07:03.279 which equals 0 at x equals 0. 00:07:03.279 --> 00:07:08.759 So to make sure that the third derivatives match, the constant c3 should be 0. 00:07:09.879 --> 00:07:14.695 Or in other words, not only is 1 minus ½ x2 the best possible quadratic 00:07:14.696 --> 00:07:19.580 approximation of cosine, it's also the best possible cubic approximation. 00:07:21.279 --> 00:07:27.059 You can make an improvement by adding on a fourth order term, c4 times x to the fourth. 00:07:27.879 --> 00:07:33.319 The fourth derivative of cosine is itself, which equals 1 at x equals 0. 00:07:34.300 --> 00:07:37.460 And what's the fourth derivative of our polynomial with this new term? 00:07:38.620 --> 00:07:42.677 Well, when you keep applying the power rule over and over, 00:07:42.677 --> 00:07:45.979 with those exponents all hopping down in front, 00:07:45.978 --> 00:07:51.000 you end up with 1 times 2 times 3 times 4 times c4, which is 24 times c4. 00:07:51.399 --> 00:07:56.130 So if we want this to match the fourth derivative of cosine x, 00:07:56.130 --> 00:07:58.759 which is 1, c4 has to be 1 over 24. 00:07:59.819 --> 00:08:05.872 And indeed, the polynomial 1 minus ½ x2 plus 1 24 times x to the fourth, 00:08:05.872 --> 00:08:12.839 which looks like this, is a very close approximation for cosine x around x equals 0. 00:08:13.740 --> 00:08:18.112 In any physics problem involving the cosine of a small angle, for example, 00:08:18.112 --> 00:08:23.127 predictions would be almost unnoticeably different if you substituted this polynomial 00:08:23.127 --> 00:08:24.060 for cosine of x. 00:08:26.100 --> 00:08:29.760 Take a step back and notice a few things happening with this process. 00:08:30.519 --> 00:08:34.199 First of all, factorial terms come up very naturally in this process. 00:08:35.019 --> 00:08:39.800 When you take n successive derivatives of the function x to the n, 00:08:39.801 --> 00:08:43.156 letting the power rule keep cascading on down, 00:08:43.155 --> 00:08:48.579 what you'll be left with is 1 times 2 times 3 on and on up to whatever n is. 00:08:49.220 --> 00:08:53.988 So you don't simply set the coefficients of the polynomial equal to whatever derivative 00:08:53.988 --> 00:08:58.540 you want, you have to divide by the appropriate factorial to cancel out this effect. 00:08:59.399 --> 00:09:05.343 For example, that x to the fourth coefficient was the fourth derivative of cosine, 00:09:05.344 --> 00:09:07.780 1, but divided by 4 factorial, 24. 00:09:09.399 --> 00:09:12.739 The second thing to notice is that adding on new terms, 00:09:12.739 --> 00:09:17.629 like this c4 times x to the fourth, doesn't mess up what the old terms should be, 00:09:17.629 --> 00:09:19.299 and that's really important. 00:09:20.100 --> 00:09:25.213 For example, the second derivative of this polynomial at x equals 0 is still equal 00:09:25.212 --> 00:09:30.079 to 2 times the second coefficient, even after you introduce higher order terms. 00:09:30.960 --> 00:09:33.879 And it's because we're plugging in x equals 0, 00:09:33.879 --> 00:09:38.537 so the second derivative of any higher order term, which all include an x, 00:09:38.537 --> 00:09:39.780 will just wash away. 00:09:40.740 --> 00:09:45.479 And the same goes for any other derivative, which is why each derivative of a 00:09:45.479 --> 00:09:50.280 polynomial at x equals 0 is controlled by one and only one of the coefficients. 00:09:52.639 --> 00:09:57.352 If instead you were approximating near an input other than 0, like x equals pi, 00:09:57.352 --> 00:10:01.771 in order to get the same effect you would have to write your polynomial in 00:10:01.772 --> 00:10:05.720 terms of powers of x minus pi, or whatever input you're looking at. 00:10:06.320 --> 00:10:09.208 This makes it look noticeably more complicated, 00:10:09.207 --> 00:10:13.961 but all we're doing is making sure that the point pi looks and behaves like 0, 00:10:13.961 --> 00:10:18.715 so that plugging in x equals pi will result in a lot of nice cancellation that 00:10:18.715 --> 00:10:20.220 leaves only one constant. 00:10:22.379 --> 00:10:27.730 And finally, on a more philosophical level, notice how what we're doing here is basically 00:10:27.730 --> 00:10:32.665 taking information about higher order derivatives of a function at a single point, 00:10:32.666 --> 00:10:37.780 and translating that into information about the value of the function near that point. 00:10:40.960 --> 00:10:44.120 You can take as many derivatives of cosine as you want. 00:10:44.600 --> 00:10:47.543 It follows this nice cyclic pattern, cosine of x, 00:10:47.543 --> 00:10:51.019 negative sine of x, negative cosine, sine, and then repeat. 00:10:52.320 --> 00:10:55.660 And the value of each one of these is easy to compute at x equals 0. 00:10:56.100 --> 00:11:01.100 It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat. 00:11:02.000 --> 00:11:07.149 And knowing the values of all those higher order derivatives is a lot of information 00:11:07.149 --> 00:11:12.480 about cosine of x, even though it only involves plugging in a single number, x equals 0. 00:11:14.259 --> 00:11:19.602 So what we're doing is leveraging that information to get an approximation around this 00:11:19.602 --> 00:11:25.130 input, and you do it by creating a polynomial whose higher order derivatives are designed 00:11:25.130 --> 00:11:30.659 to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern. 00:11:31.419 --> 00:11:35.481 And to do that, you just make each coefficient of the polynomial follow that 00:11:35.481 --> 00:11:39.439 same pattern, but you have to divide each one by the appropriate factorial. 00:11:40.120 --> 00:11:42.615 Like I mentioned before, this is what cancels out 00:11:42.615 --> 00:11:45.259 the cascading effect of many power rule applications. 00:11:47.279 --> 00:11:50.110 The polynomials you get by stopping this process at 00:11:50.110 --> 00:11:53.159 any point are called Taylor polynomials for cosine of x. 00:11:53.899 --> 00:11:58.659 More generally, and hence more abstractly, if we were dealing with some other function 00:11:58.659 --> 00:12:03.419 other than cosine, you would compute its derivative, its second derivative, and so on, 00:12:03.419 --> 00:12:08.289 getting as many terms as you'd like, and you would evaluate each one of them at x equals 00:12:08.289 --> 00:12:08.399 0. 00:12:09.580 --> 00:12:15.937 Then for the polynomial approximation, the coefficient of each x to the n term should be 00:12:15.937 --> 00:12:20.510 the value of the nth derivative of the function evaluated at 0, 00:12:20.510 --> 00:12:22.439 but divided by n factorial. 00:12:23.480 --> 00:12:27.370 This whole rather abstract formula is something you'll likely 00:12:27.370 --> 00:12:31.199 see in any text or course that touches on Taylor polynomials. 00:12:31.779 --> 00:12:36.139 And when you see it, think to yourself that the constant term ensures that 00:12:36.139 --> 00:12:39.451 the value of the polynomial matches with the value of f, 00:12:39.451 --> 00:12:43.696 the next term ensures that the slope of the polynomial matches the slope 00:12:43.696 --> 00:12:48.114 of the function at x equals 0, the next term ensures that the rate at which 00:12:48.114 --> 00:12:51.369 the slope changes is the same at that point, and so on, 00:12:51.369 --> 00:12:53.519 depending on how many terms you want. 00:12:54.620 --> 00:12:57.451 And the more terms you choose, the closer the approximation, 00:12:57.451 --> 00:13:00.980 but the tradeoff is that the polynomial you'd get would be more complicated. 00:13:02.639 --> 00:13:07.726 And to make things even more general, if you wanted to approximate near some input 00:13:07.726 --> 00:13:12.937 other than 0, which we'll call a, you would write this polynomial in terms of powers 00:13:12.937 --> 00:13:17.779 of x minus a, and you would evaluate all the derivatives of f at that input, a. 00:13:18.679 --> 00:13:23.120 This is what Taylor polynomials look like in their fullest generality. 00:13:24.000 --> 00:13:28.533 Changing the value of a changes where this approximation is hugging the original 00:13:28.533 --> 00:13:33.235 function, where its higher order derivatives will be equal to those of the original 00:13:33.235 --> 00:13:33.740 function. 00:13:35.879 --> 00:13:38.860 One of the simplest meaningful examples of this is 00:13:38.860 --> 00:13:41.899 the function e to the x around the input x equals 0. 00:13:42.759 --> 00:13:46.405 Computing the derivatives is super nice, as nice as it gets, 00:13:46.405 --> 00:13:49.274 because the derivative of e to the x is itself, 00:13:49.274 --> 00:13:53.579 so the second derivative is also e to the x, as is its third, and so on. 00:13:54.340 --> 00:13:58.240 So at the point x equals 0, all of these are equal to 1. 00:13:59.120 --> 00:14:05.720 And what that means is our polynomial approximation should look like 00:14:05.720 --> 00:14:13.947 1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed, 00:14:13.947 --> 00:14:18.539 and so on, depending on how many terms you want. 00:14:19.399 --> 00:14:22.699 These are the Taylor polynomials for e to the x. 00:14:26.379 --> 00:14:31.038 Ok, so with that as a foundation, in the spirit of showing you just how connected all 00:14:31.038 --> 00:14:34.614 the topics of calculus are, let me turn to something kind of fun, 00:14:34.614 --> 00:14:38.840 a completely different way to understand this second order term of the Taylor 00:14:38.840 --> 00:14:40.519 polynomials, but geometrically. 00:14:41.399 --> 00:14:43.903 It's related to the fundamental theorem of calculus, 00:14:43.903 --> 00:14:47.259 which I talked about in chapters 1 and 8 if you need a quick refresher. 00:14:47.980 --> 00:14:52.000 Like we did in those videos, consider a function that gives the area 00:14:52.000 --> 00:14:56.139 under some graph between a fixed left point and a variable right point. 00:14:56.980 --> 00:15:00.914 What we're going to do here is think about how to approximate this area function, 00:15:00.914 --> 00:15:04.179 not the function for the graph itself, like we've been doing before. 00:15:04.899 --> 00:15:09.439 Focusing on that area is what's going to make the second order term pop out. 00:15:10.440 --> 00:15:16.575 Remember, the fundamental theorem of calculus is that this graph itself represents the 00:15:16.575 --> 00:15:22.711 derivative of the area function, and it's because a slight nudge dx to the right bound 00:15:22.711 --> 00:15:28.988 of the area gives a new bit of area approximately equal to the height of the graph times 00:15:28.988 --> 00:15:29.200 dx. 00:15:30.039 --> 00:15:34.480 And that approximation is increasingly accurate for smaller and smaller choices of dx. 00:15:35.980 --> 00:15:39.600 But if you wanted to be more accurate about this change in area, 00:15:39.600 --> 00:15:42.666 given some change in x that isn't meant to approach 0, 00:15:42.666 --> 00:15:46.065 you would have to take into account this portion right here, 00:15:46.065 --> 00:15:47.960 which is approximately a triangle. 00:15:49.600 --> 00:15:57.460 Let's name the starting input a, and the nudged input above it x, so that change is x-a. 00:15:58.100 --> 00:16:02.985 The base of that little triangle is that change, x-a, 00:16:02.985 --> 00:16:07.600 and its height is the slope of the graph times x-a. 00:16:08.419 --> 00:16:11.986 Since this graph is the derivative of the area function, 00:16:11.986 --> 00:16:17.120 its slope is the second derivative of the area function, evaluated at the input a. 00:16:18.440 --> 00:16:22.662 So the area of this triangle, 1 half base times height, 00:16:22.662 --> 00:16:28.466 is 1 half times the second derivative of this area function, evaluated at a, 00:16:28.466 --> 00:16:29.899 multiplied by x-a2. 00:16:30.960 --> 00:16:34.379 And this is exactly what you would see with a Taylor polynomial. 00:16:34.879 --> 00:16:40.477 If you knew the various derivative information about this area function at the point a, 00:16:40.477 --> 00:16:43.659 how would you approximate the area at the point x? 00:16:45.360 --> 00:16:49.316 Well you have to include all that area up to a, f of a, 00:16:49.316 --> 00:16:54.968 plus the area of this rectangle here, which is the first derivative, times x-a, 00:16:54.967 --> 00:17:00.901 plus the area of that little triangle, which is 1 half times the second derivative, 00:17:00.902 --> 00:17:01.680 times x-a2. 00:17:02.559 --> 00:17:06.538 I really like this, because even though it looks a bit messy all written out, 00:17:06.538 --> 00:17:11.078 each one of the terms has a very clear meaning that you can just point to on the diagram. 00:17:13.400 --> 00:17:16.877 If you wanted, we could call it an end here, and you would have a 00:17:16.876 --> 00:17:20.460 phenomenally useful tool for approximating these Taylor polynomials. 00:17:21.400 --> 00:17:25.812 But if you're thinking like a mathematician, one question you might ask is 00:17:25.811 --> 00:17:30.459 whether or not it makes sense to never stop and just add infinitely many terms. 00:17:31.380 --> 00:17:35.883 In math, an infinite sum is called a series, so even though one of these 00:17:35.883 --> 00:17:40.263 approximations with finitely many terms is called a Taylor polynomial, 00:17:40.262 --> 00:17:44.519 adding all infinitely many terms gives what's called a Taylor series. 00:17:45.259 --> 00:17:48.847 You have to be really careful with the idea of an infinite series, 00:17:48.847 --> 00:17:52.597 because it doesn't actually make sense to add infinitely many things, 00:17:52.597 --> 00:17:56.079 you can only hit the plus button on the calculator so many times. 00:17:57.440 --> 00:18:01.380 But if you have a series where adding more and more of the terms, 00:18:01.380 --> 00:18:06.396 which makes sense at each step, gets you increasingly close to some specific value, 00:18:06.395 --> 00:18:09.740 what you say is that the series converges to that value. 00:18:10.319 --> 00:18:14.292 Or, if you're comfortable extending the definition of equality to 00:18:14.292 --> 00:18:19.048 include this kind of series convergence, you'd say that the series as a whole, 00:18:19.048 --> 00:18:22.359 this infinite sum, equals the value it's converging to. 00:18:23.460 --> 00:18:27.451 For example, look at the Taylor polynomial for e to the x, 00:18:27.451 --> 00:18:30.160 and plug in some input, like x equals 1. 00:18:31.140 --> 00:18:36.278 As you add more and more polynomial terms, the total sum gets closer and 00:18:36.278 --> 00:18:42.404 closer to the value e, so you say that this infinite series converges to the number e, 00:18:42.404 --> 00:18:46.700 or what's saying the same thing, that it equals the number e. 00:18:47.839 --> 00:18:53.638 In fact, it turns out that if you plug in any other value of x, like x equals 2, 00:18:53.638 --> 00:18:59.866 and look at the value of the higher and higher order Taylor polynomials at this value, 00:18:59.866 --> 00:19:04.019 they will converge towards e to the x, which is e squared. 00:19:04.680 --> 00:19:08.951 This is true for any input, no matter how far away from 0 it is, 00:19:08.951 --> 00:19:14.602 even though these Taylor polynomials are constructed only from derivative information 00:19:14.602 --> 00:19:16.180 gathered at the input 0. 00:19:18.269 --> 00:19:24.276 In a case like this, we say that e to the x equals its own Taylor series at all inputs x, 00:19:24.276 --> 00:19:27.480 which is kind of a magical thing to have happen. 00:19:28.380 --> 00:19:32.400 Even though this is also true for a couple other important functions, 00:19:32.400 --> 00:19:36.306 like sine and cosine, sometimes these series only converge within a 00:19:36.306 --> 00:19:40.500 certain range around the input whose derivative information you're using. 00:19:41.579 --> 00:19:47.272 If you work out the Taylor series for the natural log of x around the input x equals 1, 00:19:47.272 --> 00:19:51.995 which is built by evaluating the higher order derivatives of the natural 00:19:51.996 --> 00:19:55.620 log of x at x equals 1, this is what it would look like. 00:19:56.079 --> 00:20:00.799 When you plug in an input between 0 and 2, adding more and more terms of this 00:20:00.799 --> 00:20:05.519 series will indeed get you closer and closer to the natural log of that input. 00:20:06.400 --> 00:20:09.509 But outside of that range, even by just a little bit, 00:20:09.509 --> 00:20:11.700 the series fails to approach anything. 00:20:12.480 --> 00:20:17.440 As you add on more and more terms, the sum bounces back and forth wildly. 00:20:18.099 --> 00:20:22.694 It does not, as you might expect, approach the natural log of that value, 00:20:22.694 --> 00:20:27.539 even though the natural log of x is perfectly well defined for inputs above 2. 00:20:28.460 --> 00:20:31.838 In some sense, the derivative information of ln 00:20:31.838 --> 00:20:35.359 of x at x equals 1 doesn't propagate out that far. 00:20:36.579 --> 00:20:41.276 In a case like this, where adding more terms of the series doesn't approach anything, 00:20:41.277 --> 00:20:43.080 you say that the series diverges. 00:20:44.180 --> 00:20:47.952 And that maximum distance between the input you're approximating 00:20:47.952 --> 00:20:51.668 near and points where the outputs of these polynomials actually 00:20:51.669 --> 00:20:55.560 converge is called the radius of convergence for the Taylor series. 00:20:56.839 --> 00:20:59.159 There remains more to learn about Taylor series. 00:20:59.500 --> 00:21:03.241 There are many use cases, tactics for placing bounds on the error of 00:21:03.241 --> 00:21:07.635 these approximations, tests for understanding when series do and don't converge, 00:21:07.635 --> 00:21:11.759 and for that matter, there remains more to learn about calculus as a whole, 00:21:11.759 --> 00:21:14.579 and the countless topics not touched by this series. 00:21:15.319 --> 00:21:19.240 The goal with these videos is to give you the fundamental intuitions 00:21:19.240 --> 00:21:23.388 that make you feel confident and efficient in learning more on your own, 00:21:23.388 --> 00:21:27.139 and potentially even rediscovering more of the topic for yourself. 00:21:28.059 --> 00:21:32.326 In the case of Taylor series, the fundamental intuition to keep in mind 00:21:32.326 --> 00:21:36.595 as you explore more of what there is, is that they translate derivative 00:21:36.595 --> 00:21:41.160 information at a single point to approximation information around that point. 00:21:43.920 --> 00:21:46.600 Thank you once again to everybody who supported this series. 00:21:47.299 --> 00:21:49.528 The next series like it will be on probability, 00:21:49.528 --> 00:21:53.059 and if you want early access as those videos are made, you know where to go. 00:22:11.160 --> 00:22:19.060 Thank you.