WEBVTT 00:00:00.035 --> 00:00:03.622 Every function a machine can actually compute is built out 00:00:03.622 --> 00:00:05.469 of adding and multiplying. 00:00:05.921 --> 00:00:08.081 That is what a polynomial is. 00:00:08.185 --> 00:00:09.553 Sine is not one, 00:00:09.636 --> 00:00:14.444 so when your calculator reports a sine it is not doing trigonometry at all. 00:00:14.640 --> 00:00:18.981 It is evaluating a polynomial, and this lecture is about which one, 00:00:19.064 --> 00:00:21.247 and how far you can trust it. 00:00:22.644 --> 00:00:24.444 So here is the question. 00:00:24.641 --> 00:00:26.357 Sine is not a polynomial. 00:00:26.440 --> 00:00:31.712 Can we build one that behaves like it near zero, and how close can we actually get? 00:00:33.526 --> 00:00:35.672 I will tell you the destination first, 00:00:35.755 --> 00:00:39.495 because the destination is not the hard part. This is it. 00:00:39.598 --> 00:00:44.113 Sine of x equals x, minus x cubed over three factorial, 00:00:44.196 --> 00:00:48.562 plus x to the fifth over five factorial, and on forever. 00:00:49.176 --> 00:00:53.102 Every hard question in this lecture is hiding in that equals sign. 00:00:54.614 --> 00:00:56.775 So where do those numbers come from? 00:00:56.971 --> 00:01:01.730 Nowhere clever. Suppose I want a polynomial that behaves like f near zero, 00:01:01.813 --> 00:01:05.539 and the only tool I have is that I can force the two of them to agree at a single 00:01:05.539 --> 00:01:09.441 point: same value, same slope, same bend. 00:01:09.893 --> 00:01:13.841 So write the polynomial down with its coefficients still unknown. 00:01:14.637 --> 00:01:19.294 a nought, plus a one x, plus a two x squared, and so on. 00:01:19.746 --> 00:01:21.744 Now set x to zero. 00:01:21.847 --> 00:01:26.341 Every term carrying an x dies, and all that is left is a nought. 00:01:27.254 --> 00:01:28.961 Now differentiate. 00:01:29.414 --> 00:01:31.026 The constant vanishes, 00:01:31.109 --> 00:01:35.974 a one is standing there on its own, and everything after it still carries an x. 00:01:36.171 --> 00:01:39.561 Set x to zero again, and out falls a one. 00:01:40.358 --> 00:01:42.321 Differentiate once more. 00:01:42.517 --> 00:01:46.556 The square brings a two down in front, so a two ends up doubled, 00:01:46.639 --> 00:01:51.818 and setting x to zero leaves a two as the second derivative at zero, over two. 00:01:52.731 --> 00:01:54.601 And you can see the pattern. 00:01:54.797 --> 00:01:57.977 Each derivative strips one more coefficient bare, 00:01:58.060 --> 00:02:01.181 and the number it picks up in front is a factorial. 00:02:01.264 --> 00:02:05.558 So the k-th coefficient is the k-th derivative of f at zero, 00:02:05.641 --> 00:02:07.743 divided by k factorial. 00:02:08.656 --> 00:02:13.103 Stack them up and that is the Taylor polynomial of f about zero. 00:02:13.300 --> 00:02:16.537 Every coefficient is one derivative at one point. 00:02:16.620 --> 00:02:19.825 Nothing else about the function is used anywhere. 00:02:20.622 --> 00:02:22.294 Now aim it at sine. 00:02:22.398 --> 00:02:27.296 The derivatives of sine go round in a circle. Differentiate and you get cosine, 00:02:27.379 --> 00:02:32.349 then minus sine, then minus cosine, and then you are back where you started. 00:02:33.053 --> 00:02:35.608 Evaluate all of them at zero. 00:02:35.781 --> 00:02:37.799 Sine of zero is nothing. 00:02:37.882 --> 00:02:39.892 Cosine of zero is one. 00:02:40.065 --> 00:02:45.801 Minus sine of zero is nothing again, and minus cosine of zero is minus one. 00:02:46.586 --> 00:02:48.131 Look what that does. 00:02:48.304 --> 00:02:54.145 Every even coefficient has a zero on top, so every even power of x is wiped out. 00:02:54.342 --> 00:02:59.193 What survives is the odd powers, one, three, five, seven, 00:02:59.276 --> 00:03:01.436 with the signs flipping each time. 00:03:02.949 --> 00:03:06.549 So here is the machine running, against the curve itself. 00:03:06.746 --> 00:03:12.749 The crudest match uses one term, a nought, which is sine of zero, which is nothing. 00:03:13.201 --> 00:03:19.041 That polynomial is the constant zero, and on this picture it is the horizontal axis. 00:03:19.238 --> 00:03:23.139 It agrees with sine at exactly one point and nowhere else. 00:03:23.936 --> 00:03:25.609 Take one more term. 00:03:26.061 --> 00:03:29.844 The next coefficient is cosine of zero, which is one, 00:03:29.927 --> 00:03:32.667 so the first polynomial is simply x. 00:03:32.980 --> 00:03:37.602 That is the tangent line at the origin, and close in it is genuinely good. 00:03:38.085 --> 00:03:39.395 But watch it leave. 00:03:39.478 --> 00:03:44.770 By x equals two the line is still climbing and sine has already turned over. 00:03:44.853 --> 00:03:50.427 So add the next term, minus x cubed over six, and look what it does. 00:03:50.739 --> 00:03:54.536 It bends the line down, exactly where the line was going wrong. 00:03:55.240 --> 00:03:57.769 That is the whole idea in one picture. 00:03:57.852 --> 00:04:00.001 Every new term is a correction. 00:04:00.104 --> 00:04:04.839 Near zero it is negligible, because a high power of a small number is tiny. 00:04:04.922 --> 00:04:09.172 It only speaks up once you are far enough out for the previous curve to have failed. 00:04:10.085 --> 00:04:14.672 Add the fifth power term and the same thing happens one hump further out. 00:04:14.985 --> 00:04:17.482 Add the seventh and it happens again. 00:04:17.795 --> 00:04:22.509 Each polynomial tracks sine a little longer before it gives up and shoots off. 00:04:23.306 --> 00:04:24.072 Let me put a dot 00:04:24.072 --> 00:04:28.310 on sine and a dot on the seventh degree polynomial and walk them out. 00:04:28.623 --> 00:04:31.259 Near zero you cannot tell them apart. 00:04:31.456 --> 00:04:34.452 Past three they are still on top of each other. 00:04:34.648 --> 00:04:37.436 And then, quite suddenly, they are not. 00:04:38.488 --> 00:04:42.053 Which raises the only question worth asking about any of this. 00:04:42.505 --> 00:04:45.278 How wrong is the polynomial at a given x, 00:04:45.361 --> 00:04:48.995 and does taking more terms really drive that wrongness to zero? 00:04:49.192 --> 00:04:53.430 That gap has a name, and it is the real content of Taylor's theorem. 00:04:54.530 --> 00:04:56.853 So let us look at that gap directly. 00:04:57.049 --> 00:05:00.417 Here is sine again, and here is the cubic we built, 00:05:00.521 --> 00:05:03.064 x minus x cubed over six. 00:05:03.261 --> 00:05:04.815 Same picture as before, 00:05:04.898 --> 00:05:08.590 except that now I want to measure the distance between the two curves. 00:05:09.503 --> 00:05:11.640 Call that distance the remainder. 00:05:11.837 --> 00:05:16.816 R three of x is sine of x minus P three of x, 00:05:16.899 --> 00:05:20.730 and it is nothing more than the vertical gap between the blue curve and the green 00:05:20.730 --> 00:05:23.389 one at whichever x you are standing at. 00:05:24.186 --> 00:05:28.668 Walk in toward zero and it closes so fast you cannot see it happen. 00:05:28.865 --> 00:05:32.883 At a half the two curves differ by about two ten-thousandths. 00:05:33.079 --> 00:05:36.041 On this picture that is thinner than the line itself. 00:05:37.093 --> 00:05:38.510 Now walk out. 00:05:38.823 --> 00:05:41.459 At two the gap is already obvious. 00:05:41.772 --> 00:05:44.129 At three it is enormous. 00:05:44.581 --> 00:05:49.015 Sine is up near a seventh and the cubic has dived past minus one and a half. 00:05:49.098 --> 00:05:51.455 Same polynomial, same degree. 00:05:51.652 --> 00:05:54.590 All that changed is how far we walked. 00:05:55.387 --> 00:05:58.708 Raising the degree buys you distance, not safety. 00:05:58.905 --> 00:06:02.003 Here is the seventh degree polynomial on the same picture. 00:06:02.086 --> 00:06:03.688 It holds on out past four, 00:06:03.688 --> 00:06:07.590 which is much better, and then it does exactly the same thing. 00:06:07.902 --> 00:06:10.898 Every fixed polynomial eventually leaves. 00:06:11.602 --> 00:06:14.306 So here is what Taylor's theorem actually says, 00:06:14.389 --> 00:06:17.117 and notice how little of it is the series. 00:06:17.314 --> 00:06:21.030 The function equals the polynomial, plus a remainder. 00:06:21.227 --> 00:06:25.279 That is an exact statement with no approximation anywhere in it. 00:06:25.383 --> 00:06:28.437 The whole question is what the remainder is. 00:06:29.141 --> 00:06:32.857 And Lagrange gives that remainder to us in closed form. 00:06:33.054 --> 00:06:39.089 It is the n plus first derivative of f, taken at some point c between zero and x, 00:06:39.172 --> 00:06:44.049 times x to the n plus one, over n plus one factorial. 00:06:45.101 --> 00:06:47.807 Hold it next to the terms we were adding. 00:06:47.911 --> 00:06:52.637 It is exactly the shape of the next term of the series, except for one thing. 00:06:52.741 --> 00:06:57.304 The derivative is taken at c, somewhere in the middle, rather than at zero. 00:06:57.501 --> 00:07:00.346 And the theorem never tells you where c is. 00:07:01.859 --> 00:07:04.272 Now watch what that buys us for sine. 00:07:04.355 --> 00:07:08.568 Every derivative of sine is plus or minus a sine or a cosine. 00:07:08.651 --> 00:07:13.119 So whatever c turns out to be, that derivative is at most one in size, 00:07:13.202 --> 00:07:15.711 and we never need to know c at all. 00:07:16.507 --> 00:07:22.406 So the remainder is trapped below x to the n plus one, over n plus one factorial. 00:07:22.602 --> 00:07:26.028 Pure arithmetic, with no trigonometry left in it. 00:07:26.132 --> 00:07:29.441 And now look at what that quantity does as n grows. 00:07:29.983 --> 00:07:33.510 Take x equal to ten, a long way from the centre. 00:07:33.593 --> 00:07:37.518 With five terms the bound is over eight hundred, which is useless. 00:07:37.715 --> 00:07:40.781 With ten it is worse, nearly three thousand. 00:07:40.977 --> 00:07:43.962 The powers of ten are still beating the factorial. 00:07:44.387 --> 00:07:46.278 But factorials win in the end. 00:07:46.361 --> 00:07:48.811 By fifteen the bound has turned over. 00:07:48.915 --> 00:07:50.715 By twenty it is forty. 00:07:50.912 --> 00:07:55.986 By thirty it is four thousandths, and by forty it is a hundred-millionth. 00:07:56.899 --> 00:07:59.152 And ten is not special. 00:07:59.255 --> 00:08:02.237 Fix any x you like, however enormous, 00:08:02.320 --> 00:08:07.522 and x to the m over m factorial goes to zero as m grows, because eventually 00:08:07.522 --> 00:08:12.271 the factorial is multiplying by numbers larger than x at every single step. 00:08:12.975 --> 00:08:15.586 Which is the sentence we were actually after. 00:08:15.669 --> 00:08:18.607 The remainder for sine dies, everywhere. 00:08:18.803 --> 00:08:22.946 So the series really does add up to sine, at every real number. 00:08:23.029 --> 00:08:24.261 Not nearly. 00:08:24.457 --> 00:08:25.666 Exactly. 00:08:25.769 --> 00:08:29.009 And it was the remainder that proved it, not the terms. 00:08:30.018 --> 00:08:31.956 Sine was the happy case. 00:08:32.270 --> 00:08:34.628 Here is a function that is not. 00:08:34.823 --> 00:08:38.400 g of x is one over one plus x squared. 00:08:38.597 --> 00:08:43.219 Smooth everywhere, bounded, no corners, no blow-ups. 00:08:43.322 --> 00:08:46.388 There is nothing wrong with it anywhere on the real line. 00:08:47.185 --> 00:08:52.038 Its Taylor series about zero is easy to get, because you have already seen it. 00:08:52.224 --> 00:08:56.017 A geometric series sums to one over one minus r, 00:08:56.100 --> 00:08:58.772 whenever r is smaller than one in size. 00:08:58.864 --> 00:09:02.162 Now put r equal to minus x squared. 00:09:02.854 --> 00:09:05.429 Out comes one, minus x squared, 00:09:05.512 --> 00:09:10.167 plus x to the fourth, minus x to the sixth, and onward. 00:09:10.250 --> 00:09:13.572 Every even power, with the signs alternating. 00:09:13.676 --> 00:09:16.904 That is the Taylor series of g about zero. 00:09:17.608 --> 00:09:20.008 Now draw two of its partial sums. 00:09:20.092 --> 00:09:22.798 Here is one minus x squared. 00:09:22.994 --> 00:09:25.862 Here is the one that runs out to the sixth power. 00:09:25.966 --> 00:09:30.192 And here are the two walls, at minus one and at plus one. 00:09:30.897 --> 00:09:34.215 Inside those walls the partial sums close in on the curve, 00:09:34.298 --> 00:09:36.760 and taking more terms makes them better. 00:09:36.956 --> 00:09:39.292 Outside them, look what happens. 00:09:39.488 --> 00:09:43.946 The sixth power sum has already dived past minus one, while the function it came 00:09:43.946 --> 00:09:46.256 from sits calmly at four tenths. 00:09:47.308 --> 00:09:51.490 And adding more terms out there does not help. It makes things worse. 00:09:51.686 --> 00:09:53.834 Past x equals one the powers 00:09:53.834 --> 00:09:58.339 of x stop shrinking, so the series has nothing to converge to at all. 00:09:58.536 --> 00:10:00.696 It is not a bad approximation. 00:10:00.892 --> 00:10:03.006 It is not an approximation. 00:10:03.918 --> 00:10:05.507 That number one has a name, 00:10:05.590 --> 00:10:08.876 and it is the thing this lecture assumed you had not met. 00:10:09.190 --> 00:10:11.548 It is the radius of convergence. 00:10:11.744 --> 00:10:14.182 Inside it the series converges. 00:10:14.380 --> 00:10:16.212 Outside it the series diverges, 00:10:16.296 --> 00:10:19.362 and there is no argument about a remainder to be had. 00:10:20.158 --> 00:10:22.050 And here is the strange part. 00:10:22.248 --> 00:10:25.717 Nothing on the real line explains why the wall sits at one. 00:10:25.800 --> 00:10:27.911 The function is perfectly happy there. 00:10:27.994 --> 00:10:30.581 The explanation lives off the real line entirely, 00:10:30.665 --> 00:10:32.906 and it is a story for another course. 00:10:33.702 --> 00:10:35.709 So that is failure number one. 00:10:35.792 --> 00:10:39.242 The series simply stops converging past a certain distance. 00:10:39.346 --> 00:10:43.198 It is a nuisance, but it is honest, because it tells you it has given up. 00:10:43.282 --> 00:10:46.858 The second failure is far worse, because it tells you nothing at all. 00:10:47.960 --> 00:10:50.562 Here is the function that ends this story. 00:10:50.666 --> 00:10:56.864 For x away from zero, f of x is e to the minus one over x squared, 00:10:56.947 --> 00:10:59.966 and at zero we define it to be zero. 00:11:00.070 --> 00:11:02.819 It is a perfectly ordinary looking curve. 00:11:02.902 --> 00:11:07.046 It rises from nothing on the left, flattens right out at the origin, 00:11:07.129 --> 00:11:08.708 and rises again. 00:11:09.505 --> 00:11:10.850 And it is smooth. 00:11:10.933 --> 00:11:15.102 Infinitely differentiable, everywhere, including at zero. 00:11:15.298 --> 00:11:18.222 That takes a little work to prove, but it is true. 00:11:18.305 --> 00:11:22.393 So Taylor's recipe applies to it with no complaint whatsoever. 00:11:23.097 --> 00:11:27.033 But look how hard it presses itself against the axis near the origin. 00:11:27.137 --> 00:11:29.898 At one it is about zero point three seven. 00:11:29.981 --> 00:11:32.676 At a half it is down to eighteen thousandths. 00:11:32.872 --> 00:11:35.576 At a fifth it is ten to the minus eleven, 00:11:35.659 --> 00:11:38.992 and at a tenth, ten to the minus forty four. 00:11:39.788 --> 00:11:44.642 That is not merely small. That is flatter than any polynomial can manage. 00:11:44.746 --> 00:11:46.892 x to the tenth, at a tenth, 00:11:46.975 --> 00:11:52.490 is ten to the minus ten, and f is already thirty orders of magnitude below that. 00:11:52.687 --> 00:11:55.497 The exponential crushes every power. 00:11:56.294 --> 00:11:58.025 Now differentiate it. 00:11:58.128 --> 00:12:03.876 Away from zero the derivative is two over x cubed, times the same exponential. 00:12:04.073 --> 00:12:07.925 The two over x cubed blows up as x approaches the origin, 00:12:08.008 --> 00:12:12.258 but the exponential goes to zero faster than any power can blow up. 00:12:12.455 --> 00:12:14.359 So the limit is zero. 00:12:15.063 --> 00:12:19.438 Differentiate again and you get another rational function of one over x, 00:12:19.521 --> 00:12:23.470 times that same exponential, and the same thing happens. 00:12:23.666 --> 00:12:26.764 Every single derivative at zero comes out zero. 00:12:26.847 --> 00:12:28.114 Not small. 00:12:28.311 --> 00:12:29.356 Zero. 00:12:30.153 --> 00:12:32.452 So feed those into the recipe. 00:12:32.765 --> 00:12:36.910 Every coefficient is a derivative at zero over a factorial, and every one 00:12:36.910 --> 00:12:39.000 of those derivatives is zero. 00:12:39.197 --> 00:12:41.285 So every coefficient is zero, 00:12:41.368 --> 00:12:44.956 and the Taylor series of this function is the zero function. 00:12:45.753 --> 00:12:46.810 Draw it. 00:12:47.007 --> 00:12:51.036 There it is, in red, lying flat along the axis. 00:12:51.233 --> 00:12:54.752 And now ask the question we have been asking all along. 00:12:54.855 --> 00:12:56.342 How wrong is it? 00:12:56.655 --> 00:13:01.427 Put a dot on the function and a dot on its series, and walk them out. 00:13:02.038 --> 00:13:04.210 They come apart immediately. 00:13:04.407 --> 00:13:07.993 Out at two point four the gap is nearly the whole height of the picture. 00:13:08.076 --> 00:13:09.850 Walk back in and it shrinks, 00:13:09.933 --> 00:13:14.299 but it never actually closes until you are standing at the origin itself. 00:13:15.096 --> 00:13:17.836 And here is the thing that should bother you. 00:13:18.033 --> 00:13:21.830 Adding more terms does nothing at all, because every term is zero. 00:13:22.027 --> 00:13:26.091 The remainder is the whole function, at every degree, forever. 00:13:27.004 --> 00:13:29.083 So this series converges. 00:13:29.395 --> 00:13:33.619 It converges on the entire real line, beautifully, instantly, 00:13:33.702 --> 00:13:36.292 with infinite radius of convergence. 00:13:36.907 --> 00:13:39.403 And it converges to the wrong thing. 00:13:39.716 --> 00:13:42.884 It agrees with the function it came from at exactly one point, 00:13:42.967 --> 00:13:45.905 the point we built it at, and nowhere else. 00:13:47.120 --> 00:13:50.810 Which kills the comfortable picture of what a Taylor series is. 00:13:50.893 --> 00:13:53.178 It is not a decomposition of a function. 00:13:53.261 --> 00:13:56.256 It is a list of derivatives at one point, and derivatives 00:13:56.256 --> 00:14:00.994 at one point can be completely blind to what the function does anywhere else. 00:14:02.068 --> 00:14:04.634 So put the two cases next to each other. 00:14:04.831 --> 00:14:09.708 For sine we could bound the remainder, and the bound went to zero for every x. 00:14:09.812 --> 00:14:13.922 That is precisely why we are allowed to write an equals sign between the function 00:14:13.922 --> 00:14:15.292 and its series. 00:14:16.205 --> 00:14:20.327 For the flat function, the series was the zero function and the remainder 00:14:20.327 --> 00:14:23.299 was the whole of f, at every degree. 00:14:23.496 --> 00:14:24.782 It never went anywhere. 00:14:24.866 --> 00:14:30.486 So there is no equals sign to be had, except at the single point x equals zero. 00:14:31.190 --> 00:14:34.569 And notice what did the deciding in both cases. 00:14:34.673 --> 00:14:35.947 Not the terms. 00:14:36.031 --> 00:14:38.400 Not how fast the coefficients shrink. 00:14:38.597 --> 00:14:41.291 Not even whether the series converges. 00:14:41.488 --> 00:14:43.215 It was the remainder that settled it, 00:14:43.299 --> 00:14:47.050 both times, and nothing else in the argument mattered at all. 00:14:47.962 --> 00:14:50.622 So, three things to carry away. 00:14:50.819 --> 00:14:53.869 First, Taylor's theorem is not an approximation. 00:14:53.953 --> 00:14:58.865 It is an equality. The function is the polynomial plus the remainder, exactly, 00:14:58.865 --> 00:15:00.731 for every x and every n. 00:15:00.815 --> 00:15:04.472 There is nothing to be wrong about until you throw the remainder away. 00:15:05.269 --> 00:15:10.517 Second, the series represents the function only where the remainder tends to zero. 00:15:10.714 --> 00:15:14.163 That is a limit statement, and it has to be proved. 00:15:14.476 --> 00:15:17.774 For sine we proved it, with the factorial. 00:15:17.970 --> 00:15:20.827 For the flat function it is plainly false. 00:15:21.624 --> 00:15:24.550 And third, the one that catches people. 00:15:25.002 --> 00:15:28.195 A series converging is not the same thing as a series converging 00:15:28.195 --> 00:15:29.670 to the right answer. 00:15:29.867 --> 00:15:31.759 Our last example converged everywhere 00:15:31.759 --> 00:15:34.965 on the line and was wrong everywhere except one point. 00:15:35.877 --> 00:15:38.026 So here is the statement, whole. 00:15:38.339 --> 00:15:43.862 Sine of x is the sum from k equals zero to infinity of minus one to the k, 00:15:43.946 --> 00:15:48.916 times x to the two k plus one, over two k plus one factorial. 00:15:49.620 --> 00:15:52.361 And that equals sign is not notation. 00:15:52.464 --> 00:15:56.528 It is a theorem, and the thing holding it up is a bound on the remainder that goes 00:15:56.528 --> 00:15:58.534 to zero for every x. 00:15:58.618 --> 00:16:00.918 The series is the part you can see. 00:16:01.021 --> 00:16:03.762 The remainder is the part that decides.