WEBVTT 00:00:00.000 --> 00:00:01.992 align:middle line:90% [SQUEAKING] 00:00:01.992 --> 00:00:02.988 align:middle line:90% [RUSTLING] 00:00:02.988 --> 00:00:04.980 align:middle line:90% [CLICKING] 00:00:04.980 --> 00:00:12.065 align:middle line:90% 00:00:12.065 --> 00:00:13.190 align:middle line:90% CASEY RODRIGUEZ: All right. 00:00:13.190 --> 00:00:16.100 align:middle line:84% So last time, we proved the fundamental theorem 00:00:16.100 --> 00:00:16.880 align:middle line:90% of calculus. 00:00:16.880 --> 00:00:19.340 align:middle line:84% And as a consequence, integration by parts of formula 00:00:19.340 --> 00:00:23.250 align:middle line:84% and the change of variables formula or use substitution. 00:00:23.250 --> 00:00:29.590 align:middle line:84% So let me just recall, for integration by parts, 00:00:29.590 --> 00:00:36.200 align:middle line:84% it was that if I have two continuously differentiable 00:00:36.200 --> 00:00:56.670 align:middle line:84% functions, then I can shift the burden of taking a derivative. 00:00:56.670 --> 00:01:03.330 align:middle line:84% F prime times g integrate is equal to f of b, g of b, 00:01:03.330 --> 00:01:13.380 align:middle line:84% minus f of a, g of a, minus the interval from a to b of g 00:01:13.380 --> 00:01:15.810 align:middle line:90% prime f. 00:01:15.810 --> 00:01:21.540 align:middle line:84% So I can shift that derivative over from f to g. 00:01:21.540 --> 00:01:27.960 align:middle line:84% And this is apart from the triangle inequality, 00:01:27.960 --> 00:01:34.110 align:middle line:84% probably one of the most useful theorems that comes out 00:01:34.110 --> 00:01:36.070 align:middle line:84% of calculus other than the fundamental theorem 00:01:36.070 --> 00:01:36.570 align:middle line:90% of calculus. 00:01:36.570 --> 00:01:39.560 align:middle line:90% 00:01:39.560 --> 00:01:45.340 align:middle line:84% In fact, for those of you who've taken quantum mechanics, which 00:01:45.340 --> 00:01:46.810 align:middle line:90% I mean, you don't have to. 00:01:46.810 --> 00:01:50.380 align:middle line:84% But or you heard of something called the Heisenberg 00:01:50.380 --> 00:01:53.830 align:middle line:84% uncertainty principle, which says something like something 00:01:53.830 --> 00:01:58.810 align:middle line:84% to the effect that you cannot measure the position 00:01:58.810 --> 00:02:07.420 align:middle line:84% of a particle and its momentum to arbitrarily good degree, 00:02:07.420 --> 00:02:10.900 align:middle line:84% you're bound by if you can measure the position 00:02:10.900 --> 00:02:13.420 align:middle line:84% of a particle very well, then your measurement 00:02:13.420 --> 00:02:18.650 align:middle line:84% of the momentum is going to be not so great and vice versa. 00:02:18.650 --> 00:02:22.438 align:middle line:84% And that's based on an inequality, 00:02:22.438 --> 00:02:23.980 align:middle line:84% and how do you prove that inequality? 00:02:23.980 --> 00:02:25.430 align:middle line:90% Integration by parts. 00:02:25.430 --> 00:02:28.730 align:middle line:84% So integration by parts is, in fact, 00:02:28.730 --> 00:02:32.710 align:middle line:84% responsible for one of the great head-scratchers 00:02:32.710 --> 00:02:33.780 align:middle line:90% from quantum mechanics. 00:02:33.780 --> 00:02:36.837 align:middle line:84% So just to back up my claim a little bit. 00:02:36.837 --> 00:02:38.920 align:middle line:84% Now, I'm not going to give that as an application. 00:02:38.920 --> 00:02:41.510 align:middle line:84% I'm going to give a different application related to Fourier 00:02:41.510 --> 00:02:42.010 align:middle line:90% series. 00:02:42.010 --> 00:02:52.740 align:middle line:90% 00:02:52.740 --> 00:02:57.270 align:middle line:84% So the Fourier series, what are these? 00:02:57.270 --> 00:03:00.192 align:middle line:90% So suppose we have a function. 00:03:00.192 --> 00:03:01.650 align:middle line:84% And I'm not going to say what type. 00:03:01.650 --> 00:03:10.305 align:middle line:84% Suppose f from minus pi to pi to R is 2 pi periodic. 00:03:10.305 --> 00:03:15.560 align:middle line:90% 00:03:15.560 --> 00:03:22.270 align:middle line:84% And so, the question that arose due to Fourier 00:03:22.270 --> 00:03:24.990 align:middle line:90% in his study of heat transfer-- 00:03:24.990 --> 00:03:29.600 align:middle line:84% so this is Fourier, I don't know, 00:03:29.600 --> 00:03:31.040 align:middle line:90% something like 200 years ago. 00:03:31.040 --> 00:03:37.980 align:middle line:90% 00:03:37.980 --> 00:03:45.750 align:middle line:84% He made the following claim, that the function f of x 00:03:45.750 --> 00:03:59.210 align:middle line:84% can be expanded in terms of simpler building blocks, 00:03:59.210 --> 00:04:00.630 align:middle line:90% in terms of simpler functions. 00:04:00.630 --> 00:04:03.380 align:middle line:84% So we haven't talked about Taylor series. 00:04:03.380 --> 00:04:06.620 align:middle line:84% We will in just a minute, or power series, 00:04:06.620 --> 00:04:08.120 align:middle line:84% which you've come into contact with, 00:04:08.120 --> 00:04:14.870 align:middle line:84% which is a way of, if you like, expanding a function in terms 00:04:14.870 --> 00:04:17.160 align:middle line:90% of polynomials. 00:04:17.160 --> 00:04:21.050 align:middle line:90% And now-- or monomials. 00:04:21.050 --> 00:04:29.730 align:middle line:84% Now, Fourier suggested that f of x 00:04:29.730 --> 00:04:34.260 align:middle line:84% can be expanded as a superposition of functions 00:04:34.260 --> 00:04:38.790 align:middle line:84% which are 2 pi periodic and kind of the most basic 2 00:04:38.790 --> 00:04:40.260 align:middle line:90% pi periodic functions. 00:04:40.260 --> 00:04:44.710 align:middle line:84% Now, what is so special about sine x and cosine x? 00:04:44.710 --> 00:04:47.850 align:middle line:84% Well, this is a little bit deeper, 00:04:47.850 --> 00:04:50.310 align:middle line:84% the fact that they satisfy certain second order 00:04:50.310 --> 00:04:51.960 align:middle line:90% differential equation. 00:04:51.960 --> 00:04:56.670 align:middle line:84% And they are all of the solutions to the second order 00:04:56.670 --> 00:04:59.370 align:middle line:84% differential equation that are 2 pi periodic. 00:04:59.370 --> 00:05:01.880 align:middle line:90% 00:05:01.880 --> 00:05:04.550 align:middle line:84% So you should think of these as kind of being building blocks. 00:05:04.550 --> 00:05:13.955 align:middle line:84% Another way to think of this is analogous to 00:05:13.955 --> 00:05:19.940 align:middle line:84% for if you have a vector, so now this is not a partition. 00:05:19.940 --> 00:05:26.050 align:middle line:84% This is a vector, x1, x2, xn, then 00:05:26.050 --> 00:05:35.160 align:middle line:84% you can expand this vector as a sum of coefficients 00:05:35.160 --> 00:05:42.347 align:middle line:84% a sub n, a sub j, j equals 1 to n. 00:05:42.347 --> 00:05:42.930 align:middle line:90% You know what? 00:05:42.930 --> 00:05:47.880 align:middle line:84% Let me make this M. Let's make this N. Let's make this M, 00:05:47.880 --> 00:05:50.190 align:middle line:90% so it looks a little-- 00:05:50.190 --> 00:05:55.620 align:middle line:84% a sub n, e sub, where now, e sub n 00:05:55.620 --> 00:06:07.400 align:middle line:84% this is the basis vector given by 0 1 0 00:06:07.400 --> 00:06:12.260 align:middle line:90% where this is in the end spot. 00:06:12.260 --> 00:06:17.160 align:middle line:90% 00:06:17.160 --> 00:06:19.560 align:middle line:84% So you can think of this expansion 00:06:19.560 --> 00:06:21.840 align:middle line:84% in terms of sines and cosines as being 00:06:21.840 --> 00:06:29.820 align:middle line:84% analogous to expanding a vector in terms of basis elements. 00:06:29.820 --> 00:06:32.250 align:middle line:84% Or you can think of it as a different way 00:06:32.250 --> 00:06:37.980 align:middle line:84% to expand a function other than Taylor series or power series. 00:06:37.980 --> 00:06:44.790 align:middle line:84% But these components arose in a natural way 00:06:44.790 --> 00:06:48.420 align:middle line:84% if one were to study the problem of heat transfer, which 00:06:48.420 --> 00:06:57.510 align:middle line:84% is governed by the equation dtu equals dx squared u. 00:06:57.510 --> 00:07:06.530 align:middle line:84% And then, along with an initial condition, that u of 0x, 00:07:06.530 --> 00:07:09.530 align:middle line:90% so at time 0, equals f of x. 00:07:09.530 --> 00:07:13.176 align:middle line:90% 00:07:13.176 --> 00:07:23.600 align:middle line:84% So now, just like for how we expand a vector into basis 00:07:23.600 --> 00:07:28.970 align:middle line:84% elements, there's a formula for computing these coefficients. 00:07:28.970 --> 00:07:33.810 align:middle line:84% So they should be a sub n should be x sub n over here. 00:07:33.810 --> 00:07:44.100 align:middle line:84% But what's a different way of obtaining these coefficients? 00:07:44.100 --> 00:07:46.650 align:middle line:90% 00:07:46.650 --> 00:07:50.200 align:middle line:90% So this is a vector in RM. 00:07:50.200 --> 00:07:57.448 align:middle line:90% 00:07:57.448 --> 00:08:00.790 align:middle line:84% And so, how do you obtain the coefficients a sub n? 00:08:00.790 --> 00:08:05.680 align:middle line:84% Well, if I take the inner product of both sides, 00:08:05.680 --> 00:08:13.450 align:middle line:84% the dot product, say of e x dot, let's say, e sub n 00:08:13.450 --> 00:08:15.640 align:middle line:90% prime, let's say-- 00:08:15.640 --> 00:08:18.700 align:middle line:90% so I've used Mn, let's say l. 00:08:18.700 --> 00:08:25.540 align:middle line:84% This is equal to the sum from n equals 1 to the M of a sub n, e 00:08:25.540 --> 00:08:29.020 align:middle line:90% sub n dot e sub l. 00:08:29.020 --> 00:08:32.049 align:middle line:84% Now, I wrote these basis vectors this way, because that's kind 00:08:32.049 --> 00:08:34.720 align:middle line:90% of a standard choice for RM. 00:08:34.720 --> 00:08:37.840 align:middle line:84% And what makes them standard is, they have unit length. 00:08:37.840 --> 00:08:41.570 align:middle line:84% And they're orthogonal to each other. 00:08:41.570 --> 00:08:43.220 align:middle line:84% So they form in orthonormal basis. 00:08:43.220 --> 00:08:49.240 align:middle line:84% So when I take the product of e sub n with e sub l, 00:08:49.240 --> 00:08:58.650 align:middle line:84% I pick up what is usually referred to as delta in l. 00:08:58.650 --> 00:09:08.340 align:middle line:84% Where here, delta in l, this is 1 if n equals l, and 0 if n 00:09:08.340 --> 00:09:09.390 align:middle line:90% does not equal l. 00:09:09.390 --> 00:09:12.570 align:middle line:90% 00:09:12.570 --> 00:09:15.120 align:middle line:84% And therefore, this just reduces to a l. 00:09:15.120 --> 00:09:24.755 align:middle line:84% So we see that a sub l is equal to x dot e sub l. 00:09:24.755 --> 00:09:27.660 align:middle line:90% 00:09:27.660 --> 00:09:29.820 align:middle line:84% And so, all of this discussion was 00:09:29.820 --> 00:09:35.230 align:middle line:84% in the setting of a finite dimensional vector in RM. 00:09:35.230 --> 00:09:38.560 align:middle line:84% And expanding in terms of the standard basis here. 00:09:38.560 --> 00:09:41.030 align:middle line:90% But it didn't have to be. 00:09:41.030 --> 00:09:44.800 align:middle line:84% It could have been as long as it's an orthonormal basis, 00:09:44.800 --> 00:09:48.220 align:middle line:84% then I get this relation, that the coefficient that 00:09:48.220 --> 00:09:50.860 align:middle line:84% appears in front of that vector is 00:09:50.860 --> 00:09:54.400 align:middle line:84% equal to the thing I'm interested in dotted 00:09:54.400 --> 00:09:59.180 align:middle line:84% with that vector, which is written here. 00:09:59.180 --> 00:10:02.770 align:middle line:90% 00:10:02.770 --> 00:10:05.340 align:middle line:84% So let's say we try and do the same thing now with f 00:10:05.340 --> 00:10:06.900 align:middle line:90% of x, except now and say-- 00:10:06.900 --> 00:10:08.450 align:middle line:90% so these are functions. 00:10:08.450 --> 00:10:11.820 align:middle line:84% So instead of taking dot products, 00:10:11.820 --> 00:10:19.930 align:middle line:84% which is a sum of components, let's take an integral. 00:10:19.930 --> 00:10:33.310 align:middle line:84% So if I take f of x and, if you like, 00:10:33.310 --> 00:10:36.850 align:middle line:84% dot it with sine of x in sum, which 00:10:36.850 --> 00:10:40.750 align:middle line:84% is-- you can think of as I said that the integral is, 00:10:40.750 --> 00:10:44.770 align:middle line:84% you should think of as maybe a continuous sum. 00:10:44.770 --> 00:10:49.840 align:middle line:84% What do we get assuming that this expansion holds, 00:10:49.840 --> 00:10:52.450 align:middle line:84% this is equal to the sum from n equals 0. 00:10:52.450 --> 00:10:56.970 align:middle line:90% So let me make this l. 00:10:56.970 --> 00:11:04.790 align:middle line:84% This is the sum from n equals 0 to infinity of a sub n 00:11:04.790 --> 00:11:08.750 align:middle line:90% sine n, x times sine. 00:11:08.750 --> 00:11:12.080 align:middle line:84% So let me-- forgetting to write the integrals here. 00:11:12.080 --> 00:11:15.400 align:middle line:90% 00:11:15.400 --> 00:11:19.270 align:middle line:84% Skipping a point I want to make as well. 00:11:19.270 --> 00:11:23.590 align:middle line:84% Sum, and just remember the sum is starting from 0 to infinity. 00:11:23.590 --> 00:11:25.030 align:middle line:90% I don't want to keep writing it. 00:11:25.030 --> 00:11:41.510 align:middle line:84% a n sine n x sine l x plus bn cosine in x sine n l x, sine x. 00:11:41.510 --> 00:11:44.709 align:middle line:90% 00:11:44.709 --> 00:11:47.150 align:middle line:84% And then, n equals 0 to infinity. 00:11:47.150 --> 00:11:48.290 align:middle line:90% I'll just write it. 00:11:48.290 --> 00:11:49.880 align:middle line:90% Stop being lazy. 00:11:49.880 --> 00:11:52.250 align:middle line:84% Now, assuming I can do what I'm about to do, 00:11:52.250 --> 00:11:58.400 align:middle line:84% and that's actually going to be a lot of the motivation 00:11:58.400 --> 00:12:02.630 align:middle line:84% for what we're going to discuss in our final chapter, 00:12:02.630 --> 00:12:04.790 align:middle line:84% assuming I can take this infinite sum 00:12:04.790 --> 00:12:06.410 align:middle line:84% and interchange it with this integral, 00:12:06.410 --> 00:12:09.320 align:middle line:84% this is the interchanging of two limits. 00:12:09.320 --> 00:12:11.150 align:middle line:90% The sum is the infinite limit. 00:12:11.150 --> 00:12:12.770 align:middle line:90% An integration is a limit. 00:12:12.770 --> 00:12:21.240 align:middle line:84% So assuming I can switch these two limiting processes, 00:12:21.240 --> 00:12:37.060 align:middle line:84% then I pick up a sub n minus pi to pi, sine in x, sine in lx, 00:12:37.060 --> 00:12:44.650 align:middle line:84% sine lx, sorry, plus b sub n integral minus pi to pi, cosine 00:12:44.650 --> 00:12:50.446 align:middle line:90% in x, sine lx dx. 00:12:50.446 --> 00:12:53.070 align:middle line:90% 00:12:53.070 --> 00:12:58.710 align:middle line:84% Now, you can actually sit down and compute this based 00:12:58.710 --> 00:13:02.010 align:middle line:90% on trigonometric identities. 00:13:02.010 --> 00:13:07.680 align:middle line:84% And what you get is that this is always equal to 0. 00:13:07.680 --> 00:13:14.010 align:middle line:84% And that this here equals pi times delta in l. 00:13:14.010 --> 00:13:18.240 align:middle line:90% 00:13:18.240 --> 00:13:22.740 align:middle line:84% So this equals the sum from n equals 0 00:13:22.740 --> 00:13:33.300 align:middle line:84% to infinity a n pi delta in l, which equals pi times al. 00:13:33.300 --> 00:13:37.760 align:middle line:90% 00:13:37.760 --> 00:13:44.120 align:middle line:84% So we get this quantity here is assuming everything we've done 00:13:44.120 --> 00:13:46.940 align:middle line:84% is kosher equal to pi times a sub l. 00:13:46.940 --> 00:13:49.490 align:middle line:84% And then, to pick up the b sub l is the same, 00:13:49.490 --> 00:13:51.970 align:middle line:84% except now you integrate against cosine of lx. 00:13:51.970 --> 00:13:54.590 align:middle line:90% 00:13:54.590 --> 00:14:03.830 align:middle line:84% So similarly, pi times b sub l is 00:14:03.830 --> 00:14:06.680 align:middle line:84% equal to the integral from minus pi 00:14:06.680 --> 00:14:15.674 align:middle line:90% to pi of f of x cosine lx dx. 00:14:15.674 --> 00:14:27.140 align:middle line:90% 00:14:27.140 --> 00:14:29.810 align:middle line:84% So the b sub l's and a sub l's are referred to as a Fourier 00:14:29.810 --> 00:14:33.845 align:middle line:90% coefficients of the function f. 00:14:33.845 --> 00:14:37.900 align:middle line:90% 00:14:37.900 --> 00:15:01.430 align:middle line:84% So if f from minus pi to pi to R is continuous, 00:15:01.430 --> 00:15:20.450 align:middle line:84% and 2 pi periodic, the numbers a sub n equals 1 00:15:20.450 --> 00:15:29.655 align:middle line:84% over pi, integral from minus pi to pi of f of x sine nx, dx, 00:15:29.655 --> 00:15:37.950 align:middle line:84% b sub n equals 1 over pi, integral from minus pi to pi, 00:15:37.950 --> 00:15:43.490 align:middle line:84% f of x cosine in x, dx are referred to as the Fourier 00:15:43.490 --> 00:15:44.600 align:middle line:90% coefficients of f. 00:15:44.600 --> 00:16:07.115 align:middle line:90% 00:16:07.115 --> 00:16:12.250 align:middle line:84% And so, just using integration by parts, 00:16:12.250 --> 00:16:14.460 align:middle line:84% so what's the first question one should 00:16:14.460 --> 00:16:19.230 align:middle line:84% ask if it's even possible, or in what sense does 00:16:19.230 --> 00:16:21.815 align:middle line:90% f of x equal this infinite sum? 00:16:21.815 --> 00:16:23.440 align:middle line:84% Well, we haven't even gotten into that. 00:16:23.440 --> 00:16:27.240 align:middle line:84% But one question you can ask is, do these coefficients that 00:16:27.240 --> 00:16:29.280 align:middle line:84% come in front of these basic building blocks, 00:16:29.280 --> 00:16:32.640 align:middle line:84% sine nx and cosine nx, do they converge to 0? 00:16:32.640 --> 00:16:35.310 align:middle line:84% I mean, if we expect f of x to be 00:16:35.310 --> 00:16:38.280 align:middle line:84% equal to the sum of these basic parts, 00:16:38.280 --> 00:16:40.920 align:middle line:84% then the contributions from each should 00:16:40.920 --> 00:16:43.500 align:middle line:90% be getting smaller and smaller. 00:16:43.500 --> 00:16:48.520 align:middle line:84% So does a n and bn tend to 0 as n goes to infinity? 00:16:48.520 --> 00:16:50.400 align:middle line:84% And this is the content of what's 00:16:50.400 --> 00:16:52.800 align:middle line:84% usually referred to as the Riemann-Lebesque lemma. 00:16:52.800 --> 00:16:56.370 align:middle line:90% 00:16:56.370 --> 00:16:58.990 align:middle line:84% But it's usually stated in a different way. 00:16:58.990 --> 00:17:02.460 align:middle line:84% I'm just going to state it this way right now. 00:17:02.460 --> 00:17:12.859 align:middle line:90% 00:17:12.859 --> 00:17:17.220 align:middle line:90% And which is the following. 00:17:17.220 --> 00:17:34.850 align:middle line:84% If f from ab to R is continuously differentiable, 00:17:34.850 --> 00:17:44.530 align:middle line:84% then limit as n goes to infinity of a sub n 00:17:44.530 --> 00:17:50.410 align:middle line:84% equals the limit as n goes to infinity of b sub n equals 0. 00:17:50.410 --> 00:17:52.900 align:middle line:84% Now, the actual way the Riemann-Lebesque lemma 00:17:52.900 --> 00:17:55.250 align:middle line:84% is typically stated is, in fact, I 00:17:55.250 --> 00:17:57.250 align:middle line:84% don't need it to be continuously differentiable. 00:17:57.250 --> 00:17:58.640 align:middle line:90% I just need it to be continuous. 00:17:58.640 --> 00:18:01.430 align:middle line:90% This is still true. 00:18:01.430 --> 00:18:03.500 align:middle line:90% But we haven't done-- 00:18:03.500 --> 00:18:08.740 align:middle line:84% or won't do in this class approximation 00:18:08.740 --> 00:18:11.260 align:middle line:84% theorems for continuous functions. 00:18:11.260 --> 00:18:12.820 align:middle line:84% Which says that if you can do this 00:18:12.820 --> 00:18:14.650 align:middle line:84% for continuously differentiable functions, 00:18:14.650 --> 00:18:18.200 align:middle line:84% then basically you can do it for continuous functions. 00:18:18.200 --> 00:18:20.440 align:middle line:90% But this will suffice. 00:18:20.440 --> 00:18:24.250 align:middle line:84% So what this says is that the contributions coming 00:18:24.250 --> 00:18:26.470 align:middle line:84% from these building blocks is getting smaller, 00:18:26.470 --> 00:18:28.510 align:middle line:84% at least in the sense that the coefficients are 00:18:28.510 --> 00:18:29.410 align:middle line:90% getting smaller. 00:18:29.410 --> 00:18:34.240 align:middle line:84% But it says nothing about if that sum up there 00:18:34.240 --> 00:18:36.760 align:middle line:84% with the a n's and bn's defined this way actually 00:18:36.760 --> 00:18:38.170 align:middle line:90% converges to f. 00:18:38.170 --> 00:18:42.430 align:middle line:84% I do want to emphasize that, in fact, trying to straighten out 00:18:42.430 --> 00:18:46.450 align:middle line:84% this question, in what sense this series converges 00:18:46.450 --> 00:18:51.340 align:middle line:84% to f is really the motivation for a lot of analysis 00:18:51.340 --> 00:19:00.560 align:middle line:84% developed past in the first part of the last century 00:19:00.560 --> 00:19:03.200 align:middle line:84% and the last part of the century before that 00:19:03.200 --> 00:19:07.280 align:middle line:84% and forms the basis of what's called harmonic analysis. 00:19:07.280 --> 00:19:11.870 align:middle line:90% 00:19:11.870 --> 00:19:13.540 align:middle line:84% Which is a really beautiful subject 00:19:13.540 --> 00:19:16.540 align:middle line:84% and still an active area of ongoing research. 00:19:16.540 --> 00:19:24.030 align:middle line:90% 00:19:24.030 --> 00:19:25.720 align:middle line:90% So how do we prove this? 00:19:25.720 --> 00:19:29.170 align:middle line:84% Well, I stated the integration by parts formula earlier. 00:19:29.170 --> 00:19:35.730 align:middle line:84% So in fact, it'll follow pretty easily from that. 00:19:35.730 --> 00:19:41.340 align:middle line:84% Let's prove that the limit as n goes to infinity 00:19:41.340 --> 00:19:42.360 align:middle line:90% of b sub n equals 0. 00:19:42.360 --> 00:19:44.013 align:middle line:90% The one for a sub n is similar. 00:19:44.013 --> 00:19:45.180 align:middle line:90% There's just an extra piece. 00:19:45.180 --> 00:19:48.540 align:middle line:84% But I'm going to be a little bit lazy and do the easier one. 00:19:48.540 --> 00:20:06.320 align:middle line:84% We'll show-- so let's look at b sub n. 00:20:06.320 --> 00:20:15.140 align:middle line:84% This is equal to the integral from minus pi to pi of f of x. 00:20:15.140 --> 00:20:23.295 align:middle line:84% And in fact, let me write cosine nx times f of x, dx, dx. 00:20:23.295 --> 00:20:24.295 align:middle line:90% Why am I writing dx, dx? 00:20:24.295 --> 00:20:27.940 align:middle line:90% 00:20:27.940 --> 00:20:32.590 align:middle line:84% And now what I do is, cosine nx, I 00:20:32.590 --> 00:20:35.470 align:middle line:84% can write as the derivative of something. 00:20:35.470 --> 00:20:39.225 align:middle line:90% 1 over n times sine nx. 00:20:39.225 --> 00:20:41.350 align:middle line:84% If I take the derivative of that with respect to x, 00:20:41.350 --> 00:20:42.505 align:middle line:90% I get cosine nx. 00:20:42.505 --> 00:20:46.230 align:middle line:90% 00:20:46.230 --> 00:20:47.670 align:middle line:90% I didn't actually prove that. 00:20:47.670 --> 00:20:52.890 align:middle line:84% But you can look back in your calculus textbooks. 00:20:52.890 --> 00:20:57.620 align:middle line:84% We've proven enough to be able to make that precise. 00:20:57.620 --> 00:21:02.990 align:middle line:84% So by integration by parts, I can now shift the blame, 00:21:02.990 --> 00:21:05.927 align:middle line:84% or shift the burden of this derivative onto f. 00:21:05.927 --> 00:21:07.010 align:middle line:90% But look what I've gained. 00:21:07.010 --> 00:21:08.940 align:middle line:90% I've gained 1 over n here. 00:21:08.940 --> 00:21:14.840 align:middle line:84% So now, this is equal to 1 over n sine n pie, 00:21:14.840 --> 00:21:26.420 align:middle line:84% f of pi minus sine of n minus pi times f of pi-- 00:21:26.420 --> 00:21:38.190 align:middle line:84% minus pi, sorry-- minus 1 over n sine nx minus pi to pi, f 00:21:38.190 --> 00:21:39.470 align:middle line:90% prime of x, dx. 00:21:39.470 --> 00:21:43.290 align:middle line:90% 00:21:43.290 --> 00:21:47.580 align:middle line:84% And really, what this competition is showing is-- 00:21:47.580 --> 00:21:51.270 align:middle line:84% illustrating is the oscillatory nature of what's going on. 00:21:51.270 --> 00:21:55.440 align:middle line:84% Cosine of nx is oscillating as n gets very large between minus 1 00:21:55.440 --> 00:21:59.320 align:middle line:90% and 1 and equal footing. 00:21:59.320 --> 00:22:02.190 align:middle line:84% So on average, you're getting the same amount 00:22:02.190 --> 00:22:03.900 align:middle line:90% of positive f as minus-- 00:22:03.900 --> 00:22:04.920 align:middle line:90% as negative f. 00:22:04.920 --> 00:22:09.960 align:middle line:84% So or you're weighting f in such a way 00:22:09.960 --> 00:22:14.055 align:middle line:84% that it's both positive and negative in equal amounts. 00:22:14.055 --> 00:22:16.560 align:middle line:90% 00:22:16.560 --> 00:22:19.710 align:middle line:84% Now, sine of n pi, no matter what n is, I get 0. 00:22:19.710 --> 00:22:21.970 align:middle line:90% Sine of n of minus pi, I get 0. 00:22:21.970 --> 00:22:23.460 align:middle line:90% So this first part drops off. 00:22:23.460 --> 00:22:29.250 align:middle line:84% So this is equal to minus 1 over n, minus pi, pi, sine 00:22:29.250 --> 00:22:33.510 align:middle line:90% in x, times f prime of x, dx. 00:22:33.510 --> 00:22:38.220 align:middle line:84% And therefore, if I take the absolute value of b sub n, 00:22:38.220 --> 00:22:46.320 align:middle line:84% this is less than or equal to 1 over n minus pi over pi, 00:22:46.320 --> 00:22:51.180 align:middle line:90% sine in x f prime of x, dx. 00:22:51.180 --> 00:22:56.430 align:middle line:84% If I bring the absolute value inside, 00:22:56.430 --> 00:23:03.520 align:middle line:84% so I can bring the absolute value inside and still get 00:23:03.520 --> 00:23:04.020 align:middle line:90% this. 00:23:04.020 --> 00:23:06.270 align:middle line:84% So in fact, before when I had the absolute value outside, 00:23:06.270 --> 00:23:07.080 align:middle line:90% it's the equality. 00:23:07.080 --> 00:23:08.730 align:middle line:84% But now it's a less than or equality-- 00:23:08.730 --> 00:23:11.600 align:middle line:90% 00:23:11.600 --> 00:23:12.840 align:middle line:90% I mean, an inequality. 00:23:12.840 --> 00:23:18.950 align:middle line:84% So now, sign of nx is always bounded by 1. 00:23:18.950 --> 00:23:23.680 align:middle line:84% So this is less than or equal to the integral of f 00:23:23.680 --> 00:23:24.790 align:middle line:90% prime of x, dx. 00:23:24.790 --> 00:23:30.250 align:middle line:84% So this is equal to 1 over n integral a, b, f. 00:23:30.250 --> 00:23:32.530 align:middle line:90% Now, this is just f prime. 00:23:32.530 --> 00:23:35.260 align:middle line:84% This is just a fixed number times 1 over n. 00:23:35.260 --> 00:23:40.140 align:middle line:84% So this converges to 0 as n goes to infinity. 00:23:40.140 --> 00:23:42.247 align:middle line:84% And b sub n, an absolute value, of course, 00:23:42.247 --> 00:23:43.830 align:middle line:84% it's always bigger than or equal to 0. 00:23:43.830 --> 00:23:45.570 align:middle line:84% And it's bounded by something converging 00:23:45.570 --> 00:23:47.560 align:middle line:90% to 0 as n goes to infinity. 00:23:47.560 --> 00:23:56.560 align:middle line:84% So by the squeeze theorem, we conclude 00:23:56.560 --> 00:23:59.915 align:middle line:90% b sub n converges to 0. 00:23:59.915 --> 00:24:00.790 align:middle line:90% And that's the proof. 00:24:00.790 --> 00:24:05.080 align:middle line:84% The proof for the a sub n's is similar, 00:24:05.080 --> 00:24:11.790 align:middle line:84% except now you can't throw away necessarily the endpoints. 00:24:11.790 --> 00:24:13.635 align:middle line:84% But it's still not a very big deal. 00:24:13.635 --> 00:24:26.300 align:middle line:90% 00:24:26.300 --> 00:24:29.460 align:middle line:84% And in fact, if we have time, I'll show you how much-- 00:24:29.460 --> 00:24:38.650 align:middle line:84% so in fact, one can prove-- and this is proven in classes on-- 00:24:38.650 --> 00:24:41.290 align:middle line:84% courses on harmonic analysis, that in fact, 00:24:41.290 --> 00:24:45.430 align:middle line:84% for a function which is continuously differentiable, 00:24:45.430 --> 00:24:48.310 align:middle line:84% this-- and 2 pi periodic-- this series 00:24:48.310 --> 00:24:54.388 align:middle line:84% actually does converge uniformly to f on this interval. 00:24:54.388 --> 00:24:56.680 align:middle line:84% And I haven't even said what uniform convergence means. 00:24:56.680 --> 00:25:00.720 align:middle line:84% But actually does converge to the function f of x. 00:25:00.720 --> 00:25:07.760 align:middle line:84% So this is the case for continuously differentiable 00:25:07.760 --> 00:25:08.260 align:middle line:90% functions. 00:25:08.260 --> 00:25:12.970 align:middle line:84% I'll give a proof later that, in fact, this series 00:25:12.970 --> 00:25:17.140 align:middle line:84% converges if f is twice continuously differentiable. 00:25:17.140 --> 00:25:21.690 align:middle line:84% We can actually do that using the fundamental-- 00:25:21.690 --> 00:25:27.050 align:middle line:84% the integration by parts again, essentially. 00:25:27.050 --> 00:25:31.130 align:middle line:84% But so, but there are a few things here 00:25:31.130 --> 00:25:35.540 align:middle line:84% that are behind the scene that are kind of swept away. 00:25:35.540 --> 00:25:40.880 align:middle line:84% First off, when we computed these formulas-- formulae, 00:25:40.880 --> 00:25:42.980 align:middle line:90% I guess-- 00:25:42.980 --> 00:25:46.280 align:middle line:84% we interchanged summation, infinite summation of functions 00:25:46.280 --> 00:25:47.780 align:middle line:90% with integration. 00:25:47.780 --> 00:25:49.910 align:middle line:90% When can we do that? 00:25:49.910 --> 00:25:56.780 align:middle line:84% In what sense does if this Fourier series converges, 00:25:56.780 --> 00:26:00.620 align:middle line:84% in what sense does it converge to f? 00:26:00.620 --> 00:26:04.280 align:middle line:84% For convergence of real numbers, there 00:26:04.280 --> 00:26:07.080 align:middle line:84% was just one sense of the convergence of real numbers. 00:26:07.080 --> 00:26:10.010 align:middle line:84% Now, when we have a sequence of functions, which is now 00:26:10.010 --> 00:26:15.640 align:middle line:84% what we're going to turn to, we'll 00:26:15.640 --> 00:26:17.410 align:middle line:84% have different notions of convergence 00:26:17.410 --> 00:26:20.110 align:middle line:90% to another function. 00:26:20.110 --> 00:26:25.990 align:middle line:84% And depending on in what sense that conversion-- 00:26:25.990 --> 00:26:30.250 align:middle line:84% that convergence takes place, some 00:26:30.250 --> 00:26:35.710 align:middle line:84% of these limiting operations may not interchange. 00:26:35.710 --> 00:26:39.190 align:middle line:84% So now, we're going to move on to the final chapter 00:26:39.190 --> 00:26:41.620 align:middle line:90% of this class. 00:26:41.620 --> 00:26:43.510 align:middle line:84% And I know it seems like we're kind 00:26:43.510 --> 00:26:45.010 align:middle line:84% of hitting a lot of different things 00:26:45.010 --> 00:26:46.385 align:middle line:84% now towards the end of the class. 00:26:46.385 --> 00:26:48.790 align:middle line:84% And we took it slow during the first part of class. 00:26:48.790 --> 00:26:50.710 align:middle line:84% But that's, like I said, that's-- 00:26:50.710 --> 00:26:52.960 align:middle line:84% I think I even said this at the start of class. 00:26:52.960 --> 00:26:55.732 align:middle line:84% We didn't have very much to go off of. 00:26:55.732 --> 00:26:57.190 align:middle line:84% We built things from the ground up. 00:26:57.190 --> 00:26:59.810 align:middle line:84% And the more technology you have, 00:26:59.810 --> 00:27:02.950 align:middle line:84% the more things you can prove, the more interesting questions 00:27:02.950 --> 00:27:04.000 align:middle line:90% you can ask. 00:27:04.000 --> 00:27:08.110 align:middle line:84% So now, we're going to go on to sequences of functions. 00:27:08.110 --> 00:27:13.580 align:middle line:90% 00:27:13.580 --> 00:27:15.830 align:middle line:84% And you could also put sequences and series. 00:27:15.830 --> 00:27:17.840 align:middle line:84% Because a series is just a special type 00:27:17.840 --> 00:27:20.150 align:middle line:90% of sequence of functions. 00:27:20.150 --> 00:27:23.070 align:middle line:90% 00:27:23.070 --> 00:27:29.760 align:middle line:84% So I motivated a little bit of why 00:27:29.760 --> 00:27:34.990 align:middle line:84% we would be interested in functions converging 00:27:34.990 --> 00:27:37.390 align:middle line:84% to other functions or sequences of functions converging 00:27:37.390 --> 00:27:39.730 align:middle line:90% to a function. 00:27:39.730 --> 00:27:42.400 align:middle line:84% But we could look at something much more basic. 00:27:42.400 --> 00:27:48.550 align:middle line:84% So let's take a step back and look at power series. 00:27:48.550 --> 00:27:50.590 align:middle line:84% And this should be thought of as motivation 00:27:50.590 --> 00:27:56.450 align:middle line:84% for what's to come, just like our discussion about Fourier 00:27:56.450 --> 00:27:56.950 align:middle line:90% series. 00:27:56.950 --> 00:27:59.710 align:middle line:90% 00:27:59.710 --> 00:28:03.970 align:middle line:84% And again, I'm not going to ask any questions about Fourier 00:28:03.970 --> 00:28:05.810 align:middle line:84% series on the homework or on the exam. 00:28:05.810 --> 00:28:08.590 align:middle line:90% So a lot of this is just-- 00:28:08.590 --> 00:28:15.490 align:middle line:84% this discussion was to motivate this theorem here. 00:28:15.490 --> 00:28:20.830 align:middle line:84% But now, I'm going to make a kind of a more 00:28:20.830 --> 00:28:24.470 align:middle line:84% precise motivation, I guess, for what's to come. 00:28:24.470 --> 00:28:29.020 align:middle line:84% So although we've had series forever, 00:28:29.020 --> 00:28:30.400 align:middle line:90% I never brought up power series. 00:28:30.400 --> 00:28:31.340 align:middle line:90% And it's for a reason. 00:28:31.340 --> 00:28:34.060 align:middle line:84% It's because I didn't think they belonged anywhere until we 00:28:34.060 --> 00:28:35.540 align:middle line:90% got to sequences of functions. 00:28:35.540 --> 00:28:49.140 align:middle line:84% So a power series about a point x0 00:28:49.140 --> 00:29:03.080 align:middle line:84% is a series of the form sum from j equals 0 to infinity of a sub 00:29:03.080 --> 00:29:07.350 align:middle line:90% j, x minus x0 to the j. 00:29:07.350 --> 00:29:09.375 align:middle line:90% So the x0 is given. 00:29:09.375 --> 00:29:12.240 align:middle line:90% 00:29:12.240 --> 00:29:16.200 align:middle line:84% And the things that could change are the coefficients 00:29:16.200 --> 00:29:17.880 align:middle line:90% or this number x here. 00:29:17.880 --> 00:29:23.400 align:middle line:90% 00:29:23.400 --> 00:29:28.170 align:middle line:84% So theorem, which immediately follows 00:29:28.170 --> 00:29:31.000 align:middle line:90% from essentially the root test. 00:29:31.000 --> 00:29:38.320 align:middle line:84% Suppose, this number R, which is the limit as j 00:29:38.320 --> 00:29:46.030 align:middle line:84% goes to infinity of a sub j, 1 over j exists. 00:29:46.030 --> 00:29:50.200 align:middle line:84% So it's a finite number, positive, not negative number. 00:29:50.200 --> 00:29:53.860 align:middle line:90% 00:29:53.860 --> 00:30:05.730 align:middle line:84% And define rho to be 1 over R if R is bigger than 0, 00:30:05.730 --> 00:30:09.180 align:middle line:90% and infinity if R equals 0. 00:30:09.180 --> 00:30:15.780 align:middle line:90% 00:30:15.780 --> 00:30:18.930 align:middle line:84% Then we have the following conclusion, 00:30:18.930 --> 00:30:36.331 align:middle line:84% that this power series a sub j converges absolutely 00:30:36.331 --> 00:30:48.890 align:middle line:84% if x minus x0 is less than rho, and diverges 00:30:48.890 --> 00:30:52.780 align:middle line:84% if x minus x0 in absolute value is bigger than rho. 00:30:52.780 --> 00:30:55.460 align:middle line:90% 00:30:55.460 --> 00:31:04.610 align:middle line:84% And this number rho, we refer to as radius of convergence. 00:31:04.610 --> 00:31:14.440 align:middle line:90% 00:31:14.440 --> 00:31:19.360 align:middle line:84% So again, the proof follows immediately 00:31:19.360 --> 00:31:31.380 align:middle line:84% from the root test because if we take a limit as j 00:31:31.380 --> 00:31:34.920 align:middle line:84% goes to infinity of a sub j, absolute value 00:31:34.920 --> 00:31:45.900 align:middle line:84% x minus x0 j, 1 over j, this is equal to x minus x0. 00:31:45.900 --> 00:31:46.980 align:middle line:90% This kills that j. 00:31:46.980 --> 00:31:48.250 align:middle line:90% This is just a fixed number. 00:31:48.250 --> 00:31:54.110 align:middle line:90% So this pops out of the limit. 00:31:54.110 --> 00:31:55.200 align:middle line:90% And this limit exists. 00:31:55.200 --> 00:32:02.690 align:middle line:84% So this equals x minus x0 times R. 00:32:02.690 --> 00:32:06.170 align:middle line:84% And we have two things happening. 00:32:06.170 --> 00:32:13.370 align:middle line:84% This is less than 1 if x minus x0 is less than rho, 00:32:13.370 --> 00:32:18.110 align:middle line:84% bigger than 1 if x minus x0 is bigger than rho-- 00:32:18.110 --> 00:32:20.860 align:middle line:90% 00:32:20.860 --> 00:32:23.110 align:middle line:90% this number here. 00:32:23.110 --> 00:32:37.420 align:middle line:84% And therefore, by the root test, the theorem holds. 00:32:37.420 --> 00:32:48.150 align:middle line:90% 00:32:48.150 --> 00:32:53.610 align:middle line:84% So we see that this series, where the, if you like, 00:32:53.610 --> 00:32:57.100 align:middle line:84% what's given are the coefficients a sub j and x0, 00:32:57.100 --> 00:33:01.920 align:middle line:84% and what could change is x, that this series converges as long 00:33:01.920 --> 00:33:03.962 align:middle line:90% as x minus x0 is less than rho. 00:33:03.962 --> 00:33:04.920 align:middle line:90% So as long as we stay-- 00:33:04.920 --> 00:33:21.200 align:middle line:90% 00:33:21.200 --> 00:33:23.900 align:middle line:90% [SNEEZES] excuse me. 00:33:23.900 --> 00:33:26.510 align:middle line:84% As long as we stay in that interval, a symmetric interval 00:33:26.510 --> 00:33:32.180 align:middle line:84% about x0, then this series converges, absolutely. 00:33:32.180 --> 00:33:34.430 align:middle line:84% So we can define a function where 00:33:34.430 --> 00:33:38.810 align:middle line:84% if I take x in this interval, stick it into this series, 00:33:38.810 --> 00:33:39.980 align:middle line:90% I get out a number. 00:33:39.980 --> 00:33:52.670 align:middle line:84% So define function f now going from this interval. 00:33:52.670 --> 00:34:02.810 align:middle line:84% So x0 minus rho, x0 plus rho to R by f of x 00:34:02.810 --> 00:34:13.770 align:middle line:84% equals the number that gets spat out by this power series. 00:34:13.770 --> 00:34:14.850 align:middle line:90% [INAUDIBLE] j there. 00:34:14.850 --> 00:34:22.020 align:middle line:90% 00:34:22.020 --> 00:34:27.260 align:middle line:90% So for example, what is f of x? 00:34:27.260 --> 00:34:32.114 align:middle line:90% 00:34:32.114 --> 00:34:36.250 align:middle line:84% Let's say I take all of the coefficients to be-- 00:34:36.250 --> 00:34:39.040 align:middle line:84% let's say, x0 is 0, and all the coefficients are 1. 00:34:39.040 --> 00:34:43.960 align:middle line:84% So let's say I look at sum of x sub j. 00:34:43.960 --> 00:34:46.750 align:middle line:84% Then f of x, so we've already computed 00:34:46.750 --> 00:34:48.650 align:middle line:90% for a geometric series. 00:34:48.650 --> 00:34:54.370 align:middle line:84% This is equal to 1 over 1 minus x for x in minus 1, 1. 00:34:54.370 --> 00:34:57.150 align:middle line:90% 00:34:57.150 --> 00:35:01.910 align:middle line:84% So in the simple setting, 1 over 1 minus x is equal x to the j. 00:35:01.910 --> 00:35:05.930 align:middle line:90% 00:35:05.930 --> 00:35:11.100 align:middle line:84% Another example is, that if I take 00:35:11.100 --> 00:35:16.950 align:middle line:84% a sub j equals 1 over j factorial, x0 equals 0, 00:35:16.950 --> 00:35:23.190 align:middle line:84% then you've done this in an exercise, 00:35:23.190 --> 00:35:34.390 align:middle line:84% that this series here converges absolutely for all x. 00:35:34.390 --> 00:35:37.300 align:middle line:90% And meaning rho equals infinity. 00:35:37.300 --> 00:35:51.030 align:middle line:90% 00:35:51.030 --> 00:35:54.785 align:middle line:84% And this is how we define the exponential function. 00:35:54.785 --> 00:36:00.060 align:middle line:90% 00:36:00.060 --> 00:36:01.950 align:middle line:84% Exponential function, exponential 00:36:01.950 --> 00:36:06.465 align:middle line:84% of x to be what comes out of this series. 00:36:06.465 --> 00:36:17.350 align:middle line:90% 00:36:17.350 --> 00:36:19.280 align:middle line:84% And then, simply from this definition, 00:36:19.280 --> 00:36:22.600 align:middle line:84% you can show things that an exponential function should 00:36:22.600 --> 00:36:30.610 align:middle line:84% satisfy, x e to the n is equal to n e to the e of 1 00:36:30.610 --> 00:36:32.410 align:middle line:84% to the n-th power, and so on, and so on. 00:36:32.410 --> 00:36:36.340 align:middle line:84% So that this really does obey what you believe 00:36:36.340 --> 00:36:38.450 align:middle line:84% an exponential function should look like. 00:36:38.450 --> 00:36:41.830 align:middle line:84% And also, it grows faster than any power 00:36:41.830 --> 00:36:45.963 align:middle line:84% of x as x goes to infinity, goes to 0, those types of things. 00:36:45.963 --> 00:36:48.130 align:middle line:84% But this is how the exponential function is defined. 00:36:48.130 --> 00:36:50.830 align:middle line:90% 00:36:50.830 --> 00:36:58.150 align:middle line:84% So we have this function that's defined by whatever this power 00:36:58.150 --> 00:37:06.660 align:middle line:84% series spits out, for x inside this interval of convergence. 00:37:06.660 --> 00:37:14.600 align:middle line:84% So then, I could write f of x as the limit 00:37:14.600 --> 00:37:19.250 align:middle line:84% as n goes to infinity of a sequence of functions. 00:37:19.250 --> 00:37:25.890 align:middle line:84% Because this is just how it's defined, where fn of x 00:37:25.890 --> 00:37:28.110 align:middle line:90% is the partial sum. 00:37:28.110 --> 00:37:34.490 align:middle line:90% 00:37:34.490 --> 00:37:36.560 align:middle line:90% It's just a polynomial. 00:37:36.560 --> 00:37:41.240 align:middle line:84% So for power series, the limiting function, 00:37:41.240 --> 00:37:44.960 align:middle line:84% you can write it as the limit of the partial sums, which 00:37:44.960 --> 00:37:46.160 align:middle line:90% are just polynomials. 00:37:46.160 --> 00:37:48.860 align:middle line:90% 00:37:48.860 --> 00:38:01.785 align:middle line:84% And so, I should say, for all x in this interval, we have this. 00:38:01.785 --> 00:38:04.300 align:middle line:90% 00:38:04.300 --> 00:38:10.080 align:middle line:90% So now, some questions arise. 00:38:10.080 --> 00:38:13.970 align:middle line:84% So what this function is equal to, the limit of these 00:38:13.970 --> 00:38:15.230 align:middle line:90% maybe simpler functions. 00:38:15.230 --> 00:38:17.330 align:middle line:84% These simpler functions are just polynomials 00:38:17.330 --> 00:38:19.170 align:middle line:90% for the case of power series. 00:38:19.170 --> 00:38:21.470 align:middle line:84% So like I said, 1 over 1 minus x is 00:38:21.470 --> 00:38:23.600 align:middle line:84% equal to the limit of these polynomials. 00:38:23.600 --> 00:38:25.580 align:middle line:90% Some questions should arise. 00:38:25.580 --> 00:38:29.990 align:middle line:84% I mean, analysis is about limits. 00:38:29.990 --> 00:38:31.700 align:middle line:84% You can think of that as half the story. 00:38:31.700 --> 00:38:35.160 align:middle line:90% First off, what is the limit? 00:38:35.160 --> 00:38:40.430 align:middle line:84% What are important limiting processes that we consider? 00:38:40.430 --> 00:38:48.420 align:middle line:84% The second question is, how do different limits interact? 00:38:48.420 --> 00:38:54.290 align:middle line:84% So let's pose that as a question now 00:38:54.290 --> 00:38:58.610 align:middle line:84% for a three-parted question for power series. 00:38:58.610 --> 00:39:00.080 align:middle line:90% And this will motivate-- 00:39:00.080 --> 00:39:03.470 align:middle line:84% and this is, again, motivating all of what 00:39:03.470 --> 00:39:06.230 align:middle line:90% we're going to be doing now. 00:39:06.230 --> 00:39:20.450 align:middle line:84% So is the function that I get as this limit of polynomials 00:39:20.450 --> 00:39:25.700 align:middle line:84% as the output from a power series, is it continuous? 00:39:25.700 --> 00:39:27.740 align:middle line:84% The individual pieces that I take 00:39:27.740 --> 00:39:31.910 align:middle line:84% a limit to get f of x, these polynomials or the partial sums 00:39:31.910 --> 00:39:32.960 align:middle line:90% are certainly continuous. 00:39:32.960 --> 00:39:34.410 align:middle line:90% They're just polynomials. 00:39:34.410 --> 00:39:37.160 align:middle line:84% So is the limiting thing continuous? 00:39:37.160 --> 00:39:40.300 align:middle line:90% 00:39:40.300 --> 00:39:51.260 align:middle line:90% Now, if so, is f differentiable? 00:39:51.260 --> 00:39:53.990 align:middle line:84% And in particular, since f is equal to the limit 00:39:53.990 --> 00:39:59.360 align:middle line:84% as n goes to infinity of the fn's, is f prime equal 00:39:59.360 --> 00:40:03.470 align:middle line:84% to the limit as n goes to infinity of the fn prime? 00:40:03.470 --> 00:40:07.850 align:middle line:90% 00:40:07.850 --> 00:40:10.800 align:middle line:84% So the derivative is a limiting process. 00:40:10.800 --> 00:40:13.740 align:middle line:84% So I'm taking the derivative of the limit. 00:40:13.740 --> 00:40:16.490 align:middle line:84% So I'm asking, can I take that derivative inside the limit? 00:40:16.490 --> 00:40:20.060 align:middle line:90% Can I swap the two processes? 00:40:20.060 --> 00:40:23.540 align:middle line:90% And the same with integration. 00:40:23.540 --> 00:40:36.300 align:middle line:84% If one, does the integral of f equal limit as n goes 00:40:36.300 --> 00:40:40.700 align:middle line:84% to infinity of the integrals of the fn's? 00:40:40.700 --> 00:40:51.000 align:middle line:90% 00:40:51.000 --> 00:40:52.650 align:middle line:84% So again, this is a limiting process 00:40:52.650 --> 00:40:56.580 align:middle line:84% that we're asking us to flip, because f is equal to the limit 00:40:56.580 --> 00:40:58.140 align:middle line:90% as n goes to infinity of fn. 00:40:58.140 --> 00:41:00.450 align:middle line:84% And what I'm asking is, can I take this integral 00:41:00.450 --> 00:41:01.590 align:middle line:90% inside that limit? 00:41:01.590 --> 00:41:09.240 align:middle line:90% 00:41:09.240 --> 00:41:11.970 align:middle line:84% Now, you can ask these questions not just for power series, 00:41:11.970 --> 00:41:13.920 align:middle line:84% but in a more general setting, which is what 00:41:13.920 --> 00:41:16.140 align:middle line:90% we're going to turn to now. 00:41:16.140 --> 00:41:19.010 align:middle line:84% But this should be in the back of your mind 00:41:19.010 --> 00:41:20.925 align:middle line:84% as the motivation for what we're doing. 00:41:20.925 --> 00:41:23.970 align:middle line:90% 00:41:23.970 --> 00:41:35.770 align:middle line:84% And apart from being just an academic question, 00:41:35.770 --> 00:41:38.860 align:middle line:84% it's also somehow giving you information 00:41:38.860 --> 00:41:41.170 align:middle line:84% over whether the formal manipulations that you're 00:41:41.170 --> 00:41:46.780 align:middle line:84% doing with Fourier series that are actually somehow modeling 00:41:46.780 --> 00:41:50.110 align:middle line:84% some physical phenomenon, are these formal manipulations even 00:41:50.110 --> 00:41:52.360 align:middle line:90% meaningful? 00:41:52.360 --> 00:41:56.220 align:middle line:84% So these are the three questions that 00:41:56.220 --> 00:41:59.310 align:middle line:84% motivate what we're going to do going forward. 00:41:59.310 --> 00:42:02.220 align:middle line:84% But we don't have to just stick to the setting 00:42:02.220 --> 00:42:05.460 align:middle line:90% of being in power series. 00:42:05.460 --> 00:42:10.410 align:middle line:84% This should be a very important example 00:42:10.410 --> 00:42:13.590 align:middle line:84% of a sequence of functions converging 00:42:13.590 --> 00:42:17.878 align:middle line:84% to a function, a limiting function. 00:42:17.878 --> 00:42:19.420 align:middle line:84% And then, we can ask these questions. 00:42:19.420 --> 00:42:21.420 align:middle line:84% But we don't have to just stick to power series. 00:42:21.420 --> 00:42:27.090 align:middle line:84% So let me move on to a more general setting in which we'll 00:42:27.090 --> 00:42:28.590 align:middle line:90% answer these three questions. 00:42:28.590 --> 00:42:33.180 align:middle line:90% 00:42:33.180 --> 00:42:44.200 align:middle line:84% And two modes of convergence for limits of functions 00:42:44.200 --> 00:42:47.630 align:middle line:90% or sequences of functions. 00:42:47.630 --> 00:42:56.230 align:middle line:84% So first definition, this is, in fact, what we showed 00:42:56.230 --> 00:43:05.350 align:middle line:84% or what we were talking about before for N natural number, 00:43:05.350 --> 00:43:08.895 align:middle line:84% let fn be a function from S to R. S 00:43:08.895 --> 00:43:11.185 align:middle line:84% is some non-empty subset of the real numbers. 00:43:11.185 --> 00:43:14.070 align:middle line:90% 00:43:14.070 --> 00:43:21.180 align:middle line:90% And let f be from S to R. 00:43:21.180 --> 00:43:34.460 align:middle line:84% We say, fn-- so the sequence of functions fn converges 00:43:34.460 --> 00:43:48.030 align:middle line:84% point-wise to the function f if for all x in S, 00:43:48.030 --> 00:43:54.430 align:middle line:84% by sticking x into S, so for each fixed x in S, 00:43:54.430 --> 00:43:57.730 align:middle line:84% if I stick this into fn of x, I get a limit. 00:43:57.730 --> 00:44:01.160 align:middle line:90% And this limit is f of x. 00:44:01.160 --> 00:44:06.070 align:middle line:84% So for example, going back to power series, 00:44:06.070 --> 00:44:07.360 align:middle line:90% if we defined f of x-- 00:44:07.360 --> 00:44:10.360 align:middle line:90% 00:44:10.360 --> 00:44:15.855 align:middle line:84% so let me just rewrite that example that I had up there. 00:44:15.855 --> 00:44:18.600 align:middle line:90% 00:44:18.600 --> 00:44:20.670 align:middle line:84% If I define f of x equals 1 over 1 00:44:20.670 --> 00:44:29.070 align:middle line:84% minus x, fn of x to be in sum from j equals 0 n end of xj, 00:44:29.070 --> 00:44:39.650 align:middle line:84% then for all xn minus 1 to 1, limit 00:44:39.650 --> 00:44:44.690 align:middle line:84% as n goes to infinity of fn of x equals f of x. 00:44:44.690 --> 00:44:45.560 align:middle line:90% I.e. 00:44:45.560 --> 00:44:50.090 align:middle line:84% this, the sequence of partial sums corresponding 00:44:50.090 --> 00:44:52.370 align:middle line:84% to this power series, converges to 1 over 1 00:44:52.370 --> 00:44:57.110 align:middle line:84% minus x point-wise on minus 1, 1. 00:44:57.110 --> 00:45:07.658 align:middle line:90% 00:45:07.658 --> 00:45:09.700 align:middle line:84% So I said, whenever you come across a definition, 00:45:09.700 --> 00:45:10.840 align:middle line:90% you should negate it. 00:45:10.840 --> 00:45:13.870 align:middle line:84% But the negation of this definition 00:45:13.870 --> 00:45:20.020 align:middle line:84% is not too difficult. A sequence of functions 00:45:20.020 --> 00:45:23.530 align:middle line:84% does not converge to another function point-wise 00:45:23.530 --> 00:45:25.090 align:middle line:84% if there exists some point, so that 00:45:25.090 --> 00:45:28.870 align:middle line:84% when I stick them into fn of x, fn of x 00:45:28.870 --> 00:45:30.580 align:middle line:90% does not converge to f of x. 00:45:30.580 --> 00:45:37.940 align:middle line:90% 00:45:37.940 --> 00:45:39.920 align:middle line:84% So let's look at another example, which 00:45:39.920 --> 00:45:41.690 align:middle line:90% is not a power series. 00:45:41.690 --> 00:45:48.730 align:middle line:84% Let's say, we take fn of x to be x to the n, 00:45:48.730 --> 00:45:54.560 align:middle line:84% where x is in the closed interval 0, 1. 00:45:54.560 --> 00:45:57.912 align:middle line:84% So what's happening here, as n gets very large, 00:45:57.912 --> 00:46:08.480 align:middle line:84% there's 1, 1, there's, I don't know, f 5 of x. 00:46:08.480 --> 00:46:10.610 align:middle line:84% And then, as n gets very large, these guys 00:46:10.610 --> 00:46:18.520 align:middle line:84% are dropping down even more and more. 00:46:18.520 --> 00:46:22.320 align:middle line:84% And what is-- are we picking up something in the limit? 00:46:22.320 --> 00:46:25.290 align:middle line:90% Well, let's look. 00:46:25.290 --> 00:46:28.860 align:middle line:84% Well, if x equals 1, and it's pretty clear 00:46:28.860 --> 00:46:32.940 align:middle line:84% that the limit as n goes to infinity of fn of 1, 00:46:32.940 --> 00:46:34.530 align:middle line:90% this equals 1. 00:46:34.530 --> 00:46:37.890 align:middle line:84% If I stick in 1 here, I get 1 for all n. 00:46:37.890 --> 00:46:42.630 align:middle line:84% And therefore, the limit as n goes to infinity of fn1 is 1. 00:46:42.630 --> 00:46:48.820 align:middle line:84% Now, if x is in 0, 1, then, I mean, 00:46:48.820 --> 00:46:52.710 align:middle line:90% we've done this limit before. 00:46:52.710 --> 00:46:55.320 align:middle line:84% The limit as n goes to infinity of fn of x, this 00:46:55.320 --> 00:46:59.430 align:middle line:84% is equal to the limit as n goes to infinity of x to the n. 00:46:59.430 --> 00:47:00.840 align:middle line:90% Now, x is less than 1. 00:47:00.840 --> 00:47:04.000 align:middle line:84% So x is being raised to a higher and higher power. 00:47:04.000 --> 00:47:04.710 align:middle line:90% This equals 0. 00:47:04.710 --> 00:47:08.310 align:middle line:90% 00:47:08.310 --> 00:47:18.710 align:middle line:90% Thus, what do we conclude? 00:47:18.710 --> 00:47:27.200 align:middle line:84% For all x in 0, 1, this sequence of functions x to the n 00:47:27.200 --> 00:47:29.930 align:middle line:84% converges point-wise to the function 00:47:29.930 --> 00:47:36.020 align:middle line:84% f of x, which is equal to 0 if x is 00:47:36.020 --> 00:47:44.990 align:middle line:90% in 0, 1, and 1 if x equals 1. 00:47:44.990 --> 00:47:50.255 align:middle line:84% So I draw another picture here of what the limit looks like. 00:47:50.255 --> 00:48:00.830 align:middle line:90% 00:48:00.830 --> 00:48:03.670 align:middle line:84% And you can start to see this, as m gets large, 00:48:03.670 --> 00:48:06.070 align:middle line:84% again, this is becoming more vertical there. 00:48:06.070 --> 00:48:07.720 align:middle line:90% But then going to 0. 00:48:07.720 --> 00:48:11.020 align:middle line:84% So for any fixed n, it's converging 00:48:11.020 --> 00:48:13.480 align:middle line:90% to this picture on the right. 00:48:13.480 --> 00:48:19.060 align:middle line:84% And so, we can already pick up something, or at least 00:48:19.060 --> 00:48:25.030 align:middle line:84% answer one of these questions, if we take them 00:48:25.030 --> 00:48:32.760 align:middle line:84% as a question about convergence or functions 00:48:32.760 --> 00:48:35.610 align:middle line:90% in the point-wise sense. 00:48:35.610 --> 00:48:37.860 align:middle line:84% So we could have asked this question now. 00:48:37.860 --> 00:48:40.260 align:middle line:84% After having this definition, suppose 00:48:40.260 --> 00:48:44.250 align:middle line:84% fn's are continuous converging point-wise to a function f. 00:48:44.250 --> 00:48:46.950 align:middle line:90% Is the function f continuous? 00:48:46.950 --> 00:48:52.140 align:middle line:84% And what this example shows is that, no, that's not the case. 00:48:52.140 --> 00:48:55.020 align:middle line:90% x to the n is always continuous. 00:48:55.020 --> 00:48:56.970 align:middle line:84% Yet, the limit as n goes to infinity, 00:48:56.970 --> 00:48:59.700 align:middle line:84% the point-wise limit is given by this function, which 00:48:59.700 --> 00:49:03.690 align:middle line:84% is 0 from 0 to 1, and 1 at x equals 1, which 00:49:03.690 --> 00:49:05.070 align:middle line:90% is not a continuous function. 00:49:05.070 --> 00:49:10.730 align:middle line:90% 00:49:10.730 --> 00:49:14.190 align:middle line:84% So already, we're kind of seeing that point-wise convergence, 00:49:14.190 --> 00:49:19.910 align:middle line:84% which is this first weakest mode of convergence 00:49:19.910 --> 00:49:24.990 align:middle line:84% that we as of now can say about power series 00:49:24.990 --> 00:49:28.170 align:middle line:84% and is not good enough to ensure that the limit is 00:49:28.170 --> 00:49:29.010 align:middle line:90% even continuous. 00:49:29.010 --> 00:49:30.630 align:middle line:84% Because this example shows that we 00:49:30.630 --> 00:49:32.880 align:middle line:84% have a sequence of continuous functions 00:49:32.880 --> 00:49:35.580 align:middle line:84% whose point-wise limit is not continuous. 00:49:35.580 --> 00:49:50.950 align:middle line:90% 00:49:50.950 --> 00:49:55.540 align:middle line:84% So as another example, you should always-- 00:49:55.540 --> 00:49:58.090 align:middle line:84% I like this kind of last chapter, 00:49:58.090 --> 00:50:01.540 align:middle line:84% because you can draw a lot of pictures. 00:50:01.540 --> 00:50:04.960 align:middle line:84% So I'm going to draw pictures of fn. 00:50:04.960 --> 00:50:08.230 align:middle line:90% So it's piece-wise linear. 00:50:08.230 --> 00:50:11.117 align:middle line:84% So I can write down what the function is, 00:50:11.117 --> 00:50:11.950 align:middle line:90% but I don't want to. 00:50:11.950 --> 00:50:13.325 align:middle line:84% I'm just going to draw a picture. 00:50:13.325 --> 00:50:19.090 align:middle line:90% So fn of x from 0, 1 to R. 00:50:19.090 --> 00:50:21.380 align:middle line:90% This is how it looks. 00:50:21.380 --> 00:50:24.230 align:middle line:90% So there's 1, 1. 00:50:24.230 --> 00:50:29.390 align:middle line:84% And what I do is, I go to the point, 1 over n. 00:50:29.390 --> 00:50:34.730 align:middle line:84% And the function fn of x is 0 up until then. 00:50:34.730 --> 00:50:40.160 align:middle line:84% And then, it's a linear function connecting 1 over n to 2n here. 00:50:40.160 --> 00:50:44.280 align:middle line:90% 00:50:44.280 --> 00:50:49.380 align:middle line:84% And then it's connects this point to n0, so 00:50:49.380 --> 00:50:56.000 align:middle line:84% or I should say, 1 over 2n, 2n, connects that to the origin. 00:50:56.000 --> 00:50:59.260 align:middle line:90% So that's f sub n of x. 00:50:59.260 --> 00:51:00.450 align:middle line:90% It's just piece-wise linear. 00:51:00.450 --> 00:51:06.520 align:middle line:84% So for example, if I want to draw f1, 00:51:06.520 --> 00:51:13.480 align:middle line:84% f1 would look like there's 1, 0, 2, 1/2. 00:51:13.480 --> 00:51:20.520 align:middle line:90% 00:51:20.520 --> 00:51:23.460 align:middle line:84% And let's say I wanted to draw f 100. 00:51:23.460 --> 00:51:24.555 align:middle line:90% What does that look like? 00:51:24.555 --> 00:51:27.280 align:middle line:90% 00:51:27.280 --> 00:51:29.790 align:middle line:84% So maybe I should make this one a little bit bigger. 00:51:29.790 --> 00:51:37.100 align:middle line:84% There's 1, 1 over 100, and then that should be 1 over 200. 00:51:37.100 --> 00:51:56.780 align:middle line:84% And then if I go up to 200, is this piece-wise linear 00:51:56.780 --> 00:52:01.208 align:middle line:90% function, which is getting-- 00:52:01.208 --> 00:52:04.890 align:middle line:90% 00:52:04.890 --> 00:52:08.250 align:middle line:90% it's 0 from 1 up to 1 over 100. 00:52:08.250 --> 00:52:09.660 align:middle line:90% So it's 0 most of the time. 00:52:09.660 --> 00:52:11.880 align:middle line:90% And it's 0 at the origin. 00:52:11.880 --> 00:52:14.745 align:middle line:84% But in between, it's very tall and very slim. 00:52:14.745 --> 00:52:19.080 align:middle line:90% 00:52:19.080 --> 00:52:29.975 align:middle line:84% And so, my claim is that for all x in 0, 00:52:29.975 --> 00:52:40.480 align:middle line:84% 1, limit as n goes to infinity of fn of x equals 0. 00:52:40.480 --> 00:52:44.530 align:middle line:84% So this sequence of functions converges point-wise to 0. 00:52:44.530 --> 00:52:51.570 align:middle line:90% 00:52:51.570 --> 00:52:53.160 align:middle line:90% So why is this? 00:52:53.160 --> 00:52:56.090 align:middle line:90% 00:52:56.090 --> 00:53:00.100 align:middle line:84% Well, let's just give a full proof of this 00:53:00.100 --> 00:53:02.470 align:middle line:90% rather than me talking it out. 00:53:02.470 --> 00:53:05.655 align:middle line:84% I mean, I'll talk it out and give a full proof. 00:53:05.655 --> 00:53:11.960 align:middle line:90% 00:53:11.960 --> 00:53:15.185 align:middle line:84% So let's look at the easiest spot first. 00:53:15.185 --> 00:53:18.200 align:middle line:90% 00:53:18.200 --> 00:53:21.540 align:middle line:84% And I don't even need the formula for these guys. 00:53:21.540 --> 00:53:24.710 align:middle line:84% I just need to know that they have this basic characteristic 00:53:24.710 --> 00:53:26.720 align:middle line:84% that their point-wise linear, I mean, 00:53:26.720 --> 00:53:28.970 align:middle line:84% that they're piece-wise linear connecting 00:53:28.970 --> 00:53:36.000 align:middle line:84% 0 to 1 over 2n and 2n here, and then down to 1 over n 0. 00:53:36.000 --> 00:53:40.490 align:middle line:84% And then, there's 0 between that and 1. 00:53:40.490 --> 00:53:48.050 align:middle line:84% So first off, if x equals 0, then all these functions 00:53:48.050 --> 00:53:52.915 align:middle line:90% are 0 at the origin. 00:53:52.915 --> 00:53:57.394 align:middle line:90% 00:53:57.394 --> 00:53:58.600 align:middle line:90% So they equals 0. 00:53:58.600 --> 00:54:00.400 align:middle line:90% So that equals 0. 00:54:00.400 --> 00:54:02.150 align:middle line:90% So that's fine. 00:54:02.150 --> 00:54:07.960 align:middle line:90% So now suppose x is in 0, 1. 00:54:07.960 --> 00:54:11.750 align:middle line:90% 00:54:11.750 --> 00:54:13.175 align:middle line:90% And so, what do we want to show? 00:54:13.175 --> 00:54:22.300 align:middle line:90% 00:54:22.300 --> 00:54:26.070 align:middle line:84% I want to show limit as n goes to infinity 00:54:26.070 --> 00:54:31.100 align:middle line:90% of fn of x equals 0. 00:54:31.100 --> 00:54:34.020 align:middle line:84% And here, so what's the point here? 00:54:34.020 --> 00:54:41.350 align:middle line:90% Now, there's 1, there's x. 00:54:41.350 --> 00:54:42.970 align:middle line:90% So I have to give a-- 00:54:42.970 --> 00:54:47.790 align:middle line:90% 00:54:47.790 --> 00:54:52.730 align:middle line:84% well, I'm not even going to do an epsilon delta epsilon M 00:54:52.730 --> 00:54:53.230 align:middle line:90% argument. 00:54:53.230 --> 00:54:56.480 align:middle line:84% I'm just going to show you what happens. 00:54:56.480 --> 00:55:01.650 align:middle line:90% So there's x between 0 and 1. 00:55:01.650 --> 00:55:10.790 align:middle line:84% Now, let's choose a very large integer so that 1 over M 00:55:10.790 --> 00:55:13.980 align:middle line:90% is less than x. 00:55:13.980 --> 00:55:18.315 align:middle line:84% So here's 1 over M to strictly to the left of x. 00:55:18.315 --> 00:55:21.270 align:middle line:90% 00:55:21.270 --> 00:55:23.940 align:middle line:84% Now, what is the graph of fn of x 00:55:23.940 --> 00:55:28.380 align:middle line:84% look like for n bigger than or equal to M? 00:55:28.380 --> 00:55:33.750 align:middle line:84% It's 0 from x equals 1 to x equal 1 over M, 00:55:33.750 --> 00:55:38.700 align:middle line:84% and then shoots up, and then comes back down over here to 0. 00:55:38.700 --> 00:55:40.020 align:middle line:90% But here's the point. 00:55:40.020 --> 00:55:43.620 align:middle line:84% It's 0 all the way from 1 to 1 over M. So in particular, 00:55:43.620 --> 00:55:47.850 align:middle line:90% at x, fn of x is 0. 00:55:47.850 --> 00:55:55.960 align:middle line:84% So if I look at this sequence, fn 00:55:55.960 --> 00:56:00.400 align:middle line:84% of x, which I'm trying to show converges to 0, it is 0-- 00:56:00.400 --> 00:56:10.590 align:middle line:84% no, it is a f1 of x, f2 of x, up to fn minus 1 of x. 00:56:10.590 --> 00:56:18.300 align:middle line:84% And then, at fM of x, so at the-- 00:56:18.300 --> 00:56:25.570 align:middle line:84% now, so this is fM of x spot, it's 0. 00:56:25.570 --> 00:56:27.770 align:middle line:84% And this point is only going to the left. 00:56:27.770 --> 00:56:31.450 align:middle line:84% So for all n bigger than or equal to M, 00:56:31.450 --> 00:56:34.960 align:middle line:84% now this will be 1 over n will be to the left of x. 00:56:34.960 --> 00:56:38.710 align:middle line:84% And therefore, fn of x will be 0. 00:56:38.710 --> 00:56:43.780 align:middle line:84% So this is 0, 0, 0, 0, and so on. 00:56:43.780 --> 00:56:51.160 align:middle line:90% 00:56:51.160 --> 00:56:55.810 align:middle line:84% So I have the sequence is eventually 0 00:56:55.810 --> 00:57:06.750 align:middle line:84% for all n bigger than or equal to capital M. So 00:57:06.750 --> 00:57:08.160 align:middle line:84% and therefore, the limit, I mean, 00:57:08.160 --> 00:57:10.493 align:middle line:84% it's pretty easy to take a limit of a constant sequence. 00:57:10.493 --> 00:57:16.050 align:middle line:90% 00:57:16.050 --> 00:57:20.130 align:middle line:84% And which proves that this sequence of functions 00:57:20.130 --> 00:57:21.885 align:middle line:90% converges point-wise to 0. 00:57:21.885 --> 00:57:24.945 align:middle line:90% 00:57:24.945 --> 00:57:26.820 align:middle line:84% Now, I didn't come up with this fancy example 00:57:26.820 --> 00:57:28.782 align:middle line:90% for just any old reason. 00:57:28.782 --> 00:57:30.240 align:middle line:84% It'll come back in a minute when we 00:57:30.240 --> 00:57:31.907 align:middle line:84% start answering some of these questions, 00:57:31.907 --> 00:57:37.140 align:middle line:84% or asking them, again, within the context of these two 00:57:37.140 --> 00:57:38.280 align:middle line:90% convergence-- 00:57:38.280 --> 00:57:43.750 align:middle line:90% 00:57:43.750 --> 00:57:45.160 align:middle line:90% ways of converging. 00:57:45.160 --> 00:57:48.140 align:middle line:90% 00:57:48.140 --> 00:57:51.170 align:middle line:84% So, so far, I've only given you one definition 00:57:51.170 --> 00:57:53.000 align:middle line:84% of convergence-- point-wise convergence 00:57:53.000 --> 00:57:54.205 align:middle line:90% of a sequence of functions. 00:57:54.205 --> 00:57:55.830 align:middle line:84% And now, I'm going to give a slightly-- 00:57:55.830 --> 00:57:57.872 align:middle line:84% I'm going to give a stronger-- it's not slightly, 00:57:57.872 --> 00:58:00.020 align:middle line:90% it's much stronger-- 00:58:00.020 --> 00:58:05.820 align:middle line:84% definition of convergence of a sequence of functions. 00:58:05.820 --> 00:58:21.050 align:middle line:84% So we have a sequence of functions and a given function 00:58:21.050 --> 00:58:24.290 align:middle line:84% from S to R. S is a non-empty subset of R. 00:58:24.290 --> 00:58:39.580 align:middle line:84% Then we say the sequence fn converges point-wise 00:58:39.580 --> 00:58:42.760 align:middle line:90% or uniformly to 0-- 00:58:42.760 --> 00:58:45.810 align:middle line:90% uniformly to 0-- uniformly to-- 00:58:45.810 --> 00:58:46.810 align:middle line:90% it's the end of the day. 00:58:46.810 --> 00:58:49.148 align:middle line:84% If I start mixing up some of my words, 00:58:49.148 --> 00:58:50.440 align:middle line:90% I'm always going to correct it. 00:58:50.440 --> 00:58:51.880 align:middle line:84% But the first word out of my mouth 00:58:51.880 --> 00:58:53.860 align:middle line:90% may not be the correct one-- 00:58:53.860 --> 00:59:03.170 align:middle line:84% converges uniformly to f of x to f if-- 00:59:03.170 --> 00:59:06.710 align:middle line:84% now we have an epsilon in statement. 00:59:06.710 --> 00:59:12.380 align:middle line:84% For all epsilon positive, there exists an M natural number 00:59:12.380 --> 00:59:21.152 align:middle line:84% such that for all n bigger than or equal to M, for all x in S, 00:59:21.152 --> 00:59:26.905 align:middle line:84% fn of x minus f of x is less than epsilon. 00:59:26.905 --> 00:59:31.580 align:middle line:90% 00:59:31.580 --> 00:59:34.400 align:middle line:84% Now, I want to make a brief comment. 00:59:34.400 --> 00:59:37.670 align:middle line:84% This looks suspiciously like point-wise convergence, 00:59:37.670 --> 00:59:41.750 align:middle line:84% if you just wrote down what it means for the limit 00:59:41.750 --> 00:59:45.560 align:middle line:84% as n goes to infinity of fn of x goes to f of x. 00:59:45.560 --> 00:59:49.550 align:middle line:84% Except, there's a very subtle and important point. 00:59:49.550 --> 00:59:53.450 align:middle line:84% And that is, where does for all x and S appear? 00:59:53.450 --> 00:59:56.150 align:middle line:84% For point-wise convergence, you can 00:59:56.150 --> 00:59:59.990 align:middle line:84% state point-wise convergence as this 00:59:59.990 --> 01:00:02.660 align:middle line:84% being at the start of the line, for all x 01:00:02.660 --> 01:00:08.100 align:middle line:84% in S, for all epsilon positive, blah, blah, blah. 01:00:08.100 --> 01:00:12.540 align:middle line:84% Here, it appears at the end of the line of the quantifiers. 01:00:12.540 --> 01:00:15.450 align:middle line:84% And this makes a very important difference 01:00:15.450 --> 01:00:18.870 align:middle line:84% between point-wise convergence and uniform convergence. 01:00:18.870 --> 01:00:22.860 align:middle line:84% Point-wise convergence means, I take a point x, 01:00:22.860 --> 01:00:24.570 align:middle line:90% I stick it into fn of x. 01:00:24.570 --> 01:00:27.020 align:middle line:84% That gives me a sequence of numbers. 01:00:27.020 --> 01:00:29.870 align:middle line:84% And point-wise convergence says that sequence of numbers 01:00:29.870 --> 01:00:31.910 align:middle line:90% converges to f of x. 01:00:31.910 --> 01:00:34.800 align:middle line:84% For each x, I get a sequence of numbers, 01:00:34.800 --> 01:00:36.230 align:middle line:90% which converges to f of x. 01:00:36.230 --> 01:00:40.458 align:middle line:90% 01:00:40.458 --> 01:00:42.530 align:middle line:84% Now, uniform convergence is actually 01:00:42.530 --> 01:00:43.820 align:middle line:90% saying something stronger. 01:00:43.820 --> 01:00:46.190 align:middle line:90% And I'll say that. 01:00:46.190 --> 01:00:51.650 align:middle line:84% So let me, in fact, let me draw a picture that 01:00:51.650 --> 01:00:53.990 align:middle line:90% goes with this definition. 01:00:53.990 --> 01:00:57.245 align:middle line:90% Let's make S to be an interval. 01:00:57.245 --> 01:01:01.560 align:middle line:90% 01:01:01.560 --> 01:01:06.730 align:middle line:84% Let's say my limiting function is f is given by this graph. 01:01:06.730 --> 01:01:16.570 align:middle line:84% And so, what I'm going to do is basically 01:01:16.570 --> 01:01:31.790 align:middle line:84% shift the graph up and down by epsilon, 01:01:31.790 --> 01:01:36.520 align:middle line:84% meaning this length is epsilon, and so is this length, 01:01:36.520 --> 01:01:37.760 align:middle line:90% all the way across. 01:01:37.760 --> 01:01:46.950 align:middle line:84% So let me shade this in and re-outline-- 01:01:46.950 --> 01:01:47.450 align:middle line:90% oh. 01:01:47.450 --> 01:01:55.070 align:middle line:90% 01:01:55.070 --> 01:02:02.750 align:middle line:84% So I get this little, if you like, shaded area 01:02:02.750 --> 01:02:05.390 align:middle line:90% around my function f. 01:02:05.390 --> 01:02:08.990 align:middle line:90% So this is f, f of x. 01:02:08.990 --> 01:02:10.010 align:middle line:90% That's the graph. 01:02:10.010 --> 01:02:15.920 align:middle line:84% And the shaded part, this is the set of all x and y, 01:02:15.920 --> 01:02:23.090 align:middle line:84% such that f of x minus y is less than epsilon. 01:02:23.090 --> 01:02:27.290 align:middle line:84% So I get a little tube snaking with f. 01:02:27.290 --> 01:02:29.690 align:middle line:84% Now, what uniform convergence says, 01:02:29.690 --> 01:02:35.510 align:middle line:84% is that for all n bigger than or equal to some M, so given 01:02:35.510 --> 01:02:38.360 align:middle line:84% epsilon, for all n bigger than or equal to M, if I were 01:02:38.360 --> 01:02:43.810 align:middle line:84% to draw the graph of fn of x, it better 01:02:43.810 --> 01:02:51.375 align:middle line:84% fall inside this tube across all of a, b. 01:02:51.375 --> 01:02:54.660 align:middle line:84% See, this tube is defined for all x between a, b. 01:02:54.660 --> 01:03:03.080 align:middle line:84% So it's making a statement about how close fn of x 01:03:03.080 --> 01:03:08.150 align:middle line:84% is to f of x across the entire set. 01:03:08.150 --> 01:03:11.630 align:middle line:84% Point-wise convergence just says, if I put an x into fn 01:03:11.630 --> 01:03:14.990 align:middle line:84% of x, then eventually those numbers 01:03:14.990 --> 01:03:16.590 align:middle line:90% are getting close to f of x. 01:03:16.590 --> 01:03:19.220 align:middle line:84% Uniform convergence is a global property. 01:03:19.220 --> 01:03:23.960 align:middle line:84% It's saying, across the entire set, as n is getting large, 01:03:23.960 --> 01:03:26.510 align:middle line:84% the graph of this fn is getting very 01:03:26.510 --> 01:03:28.780 align:middle line:90% close to the graph of f of x. 01:03:28.780 --> 01:03:35.430 align:middle line:84% Not just if I fix an x, the fn of x at that point 01:03:35.430 --> 01:03:38.730 align:middle line:90% are converging to f of x. 01:03:38.730 --> 01:03:42.300 align:middle line:84% So I've said a couple of times that uniform convergence 01:03:42.300 --> 01:03:44.230 align:middle line:84% is stronger than point-wise convergence. 01:03:44.230 --> 01:03:49.740 align:middle line:90% Let me actually prove this now. 01:03:49.740 --> 01:03:52.860 align:middle line:84% So let me prove that following theorem. 01:03:52.860 --> 01:03:59.850 align:middle line:90% 01:03:59.850 --> 01:04:08.250 align:middle line:84% So if I have a sequence of functions from S to R, 01:04:08.250 --> 01:04:12.990 align:middle line:84% and fn, rather than write converges 01:04:12.990 --> 01:04:17.190 align:middle line:84% to f point-wise or uniformly, I'm going to put an arrow. 01:04:17.190 --> 01:04:28.790 align:middle line:84% And then, with the description afterwards, uniformly on S. 01:04:28.790 --> 01:04:39.290 align:middle line:84% And then this implies fn converges to f point-wise on S. 01:04:39.290 --> 01:04:42.020 align:middle line:90% So it's very simple. 01:04:42.020 --> 01:04:44.720 align:middle line:84% Again, what is the picture that's 01:04:44.720 --> 01:04:47.090 align:middle line:84% going on for uniform convergence is 01:04:47.090 --> 01:04:51.433 align:middle line:84% that fn is getting close to f across the entire set 01:04:51.433 --> 01:04:52.350 align:middle line:90% that we're looking at. 01:04:52.350 --> 01:04:54.710 align:middle line:84% So certainly at one point, which is all 01:04:54.710 --> 01:04:56.240 align:middle line:84% you need for point-wise convergence. 01:04:56.240 --> 01:04:58.790 align:middle line:84% Each fixed point we should be getting close. 01:04:58.790 --> 01:05:03.600 align:middle line:90% So let epsilon be positive. 01:05:03.600 --> 01:05:12.330 align:middle line:84% So first off, let's fix a number in the set S. 01:05:12.330 --> 01:05:18.380 align:middle line:84% So now we want to prove that the limit as n 01:05:18.380 --> 01:05:22.370 align:middle line:84% goes to infinity of fn of c equals f of c. 01:05:22.370 --> 01:05:26.860 align:middle line:90% 01:05:26.860 --> 01:05:30.370 align:middle line:90% Let epsilon be positive. 01:05:30.370 --> 01:05:36.280 align:middle line:84% Since fn converges to f uniformly, 01:05:36.280 --> 01:05:44.560 align:middle line:84% there exists a natural number M0 such that 01:05:44.560 --> 01:05:49.920 align:middle line:84% for all n bigger than or equal to M0, for all x in S, 01:05:49.920 --> 01:05:55.480 align:middle line:84% fn of x minus f of x is less than epsilon. 01:05:55.480 --> 01:05:58.220 align:middle line:90% 01:05:58.220 --> 01:06:06.740 align:middle line:84% So choose M to be this M0, the M that is for this epsilon. 01:06:06.740 --> 01:06:12.940 align:middle line:84% Then for all n bigger than or equal to M, 01:06:12.940 --> 01:06:21.710 align:middle line:84% let's call this equation star, star with at the single point 01:06:21.710 --> 01:06:28.910 align:middle line:84% x equals at c, implies fn of c minus f of c 01:06:28.910 --> 01:06:31.700 align:middle line:90% is less than epsilon. 01:06:31.700 --> 01:06:34.930 align:middle line:84% And thus, the limit as n goes to infinity of f of c, fn of c 01:06:34.930 --> 01:06:35.860 align:middle line:90% equals f of c. 01:06:35.860 --> 01:06:51.650 align:middle line:90% 01:06:51.650 --> 01:06:56.090 align:middle line:84% So I don't think I have enough time to do the example 01:06:56.090 --> 01:06:56.910 align:middle line:90% that I want to do. 01:06:56.910 --> 01:06:58.910 align:middle line:84% So I'm just going to leave you on the edge of your seat 01:06:58.910 --> 01:07:00.500 align:middle line:84% by stating the following theorem. 01:07:00.500 --> 01:07:04.760 align:middle line:84% That in fact, so this is a one-way street. 01:07:04.760 --> 01:07:08.060 align:middle line:84% Meaning point-wise convergence does not 01:07:08.060 --> 01:07:09.350 align:middle line:90% imply uniform convergence. 01:07:09.350 --> 01:07:11.870 align:middle line:84% So we just proved uniform implies point-wise. 01:07:11.870 --> 01:07:14.420 align:middle line:90% But the converse does not hold. 01:07:14.420 --> 01:07:16.250 align:middle line:84% And what we'll prove next time is 01:07:16.250 --> 01:07:23.080 align:middle line:84% for in the setting of this simple example of x to the n. 01:07:23.080 --> 01:07:52.240 align:middle line:84% So-- and so, what we'll prove next time is the following. 01:07:52.240 --> 01:07:55.350 align:middle line:90% 01:07:55.350 --> 01:08:02.630 align:middle line:84% If I take any b between 0 and 1, then 01:08:02.630 --> 01:08:09.680 align:middle line:84% fn convergence to f uniformly on the set 0, b. 01:08:09.680 --> 01:08:11.630 align:middle line:84% So these functions are defined on 0, 1. 01:08:11.630 --> 01:08:15.910 align:middle line:84% So they're certainly defined on 0, b for b less than 1. 01:08:15.910 --> 01:08:30.330 align:middle line:84% But however, this sequence of functions 01:08:30.330 --> 01:08:44.830 align:middle line:84% does not converge uniformly to f on 0,1. 01:08:44.830 --> 01:08:50.200 align:middle line:84% So here, this second part, since the fn's converge 01:08:50.200 --> 01:08:53.170 align:middle line:84% to point-wise on 0, 1, the second part 01:08:53.170 --> 01:08:55.600 align:middle line:84% says that this is a one-way street. 01:08:55.600 --> 01:08:57.910 align:middle line:90% This is not a two-way street. 01:08:57.910 --> 01:09:03.029 align:middle line:84% These two modes of convergence-- uniform 01:09:03.029 --> 01:09:07.229 align:middle line:84% convergence and point-wise are not equivalent. 01:09:07.229 --> 01:09:07.729 align:middle line:90% All right. 01:09:07.729 --> 01:09:09.880 align:middle line:90% I think we'll stop there. 01:09:09.880 --> 01:09:16.000 align:middle line:90%