WEBVTT 00:00:00.000 --> 00:00:02.450 align:middle line:90% [SQUEAKING] 00:00:02.450 --> 00:00:04.410 align:middle line:90% [RUSTLING] 00:00:04.410 --> 00:00:07.840 align:middle line:90% [CLICKING] 00:00:07.840 --> 00:00:11.100 align:middle line:90% 00:00:11.100 --> 00:00:15.540 align:middle line:84% TOBIAS COLDING: OK, so last time, we talked about-- 00:00:15.540 --> 00:00:19.580 align:middle line:84% so we had a function or sequence of function. 00:00:19.580 --> 00:00:24.200 align:middle line:84% We had an integral, say, I, and we had a sequence 00:00:24.200 --> 00:00:30.880 align:middle line:84% of function fn and a function f all defined on this interval. 00:00:30.880 --> 00:00:35.740 align:middle line:84% And then we talked about pointwise convergence. 00:00:35.740 --> 00:00:42.760 align:middle line:90% 00:00:42.760 --> 00:00:46.800 align:middle line:84% So pointwise convergence means that for each-- 00:00:46.800 --> 00:00:54.740 align:middle line:84% so this is for each x when you're looking at this. 00:00:54.740 --> 00:00:57.280 align:middle line:90% So this is for each x fixed. 00:00:57.280 --> 00:00:58.880 align:middle line:90% This is a sequence. 00:00:58.880 --> 00:01:02.000 align:middle line:84% This is supposed to convert to f of x. 00:01:02.000 --> 00:01:04.560 align:middle line:84% So this was pointwise convergence. 00:01:04.560 --> 00:01:08.100 align:middle line:84% And then we also talked about uniform convergence. 00:01:08.100 --> 00:01:15.160 align:middle line:90% 00:01:15.160 --> 00:01:23.320 align:middle line:84% And this meant that for all epsilon greater than 0, 00:01:23.320 --> 00:01:31.960 align:middle line:84% there exists a capital N such that if little n is bigger 00:01:31.960 --> 00:01:37.080 align:middle line:84% or equal to this capital N, then the difference between fn 00:01:37.080 --> 00:01:42.500 align:middle line:84% of x and f of x, this here, is smaller than epsilon, 00:01:42.500 --> 00:01:48.320 align:middle line:84% and this here have to hold for all x. 00:01:48.320 --> 00:01:51.780 align:middle line:84% So uniform, again, refer to that as uniform in x. 00:01:51.780 --> 00:01:55.600 align:middle line:90% The convergence is uniform in x. 00:01:55.600 --> 00:01:58.840 align:middle line:90% Then so we talk about that. 00:01:58.840 --> 00:02:00.840 align:middle line:84% And then one of the theorems we proved 00:02:00.840 --> 00:02:08.520 align:middle line:84% was that if you take a sequence, fn and [INAUDIBLE] 00:02:08.520 --> 00:02:18.800 align:middle line:84% was integrable, and these converge to some function 00:02:18.800 --> 00:02:24.800 align:middle line:90% f, then f is integrable. 00:02:24.800 --> 00:02:26.980 align:middle line:90% So this is uniform convergence. 00:02:26.980 --> 00:02:33.560 align:middle line:90% 00:02:33.560 --> 00:02:38.080 align:middle line:84% Then f here is integrable and the integral 00:02:38.080 --> 00:02:41.040 align:middle line:90% is the limit of the integrals. 00:02:41.040 --> 00:02:44.666 align:middle line:84% So the integral from a to b of fn 00:02:44.666 --> 00:02:51.000 align:middle line:84% x dx is equal as the limit here of this. 00:02:51.000 --> 00:02:58.300 align:middle line:84% So the limit exists and it is equal to the integral of f. 00:02:58.300 --> 00:03:06.160 align:middle line:90% 00:03:06.160 --> 00:03:13.410 align:middle line:84% So this is how uniform convergence and integrals 00:03:13.410 --> 00:03:13.910 align:middle line:90% interact. 00:03:13.910 --> 00:03:17.700 align:middle line:90% 00:03:17.700 --> 00:03:20.040 align:middle line:84% Suppose you take a sequence here. 00:03:20.040 --> 00:03:21.490 align:middle line:90% So this is a theorem. 00:03:21.490 --> 00:03:24.760 align:middle line:90% 00:03:24.760 --> 00:03:32.320 align:middle line:84% Suppose you take a sequence fn of functions on some interval 00:03:32.320 --> 00:03:45.160 align:middle line:84% into R. You have x0 is in this interval. 00:03:45.160 --> 00:03:47.200 align:middle line:84% And then you have the following if you 00:03:47.200 --> 00:03:51.080 align:middle line:84% have the following properties, that one 00:03:51.080 --> 00:04:00.720 align:middle line:84% is that this sequence here converged to some number C. 00:04:00.720 --> 00:04:04.600 align:middle line:84% So the sequence converged to some number c. 00:04:04.600 --> 00:04:12.160 align:middle line:84% Two is that these functions here, fn, are differentiable. 00:04:12.160 --> 00:04:17.041 align:middle line:84% So these are maybe, I'll just say here, differentiable. 00:04:17.041 --> 00:04:21.200 align:middle line:90% 00:04:21.200 --> 00:04:24.640 align:middle line:84% And these here converge to some function g. 00:04:24.640 --> 00:04:27.880 align:middle line:84% And then 3 is that these functions here-- 00:04:27.880 --> 00:04:31.400 align:middle line:84% sorry the derivative converge, and I'm also 00:04:31.400 --> 00:04:34.620 align:middle line:84% requiring that the derivative are continuous. 00:04:34.620 --> 00:04:38.440 align:middle line:90% 00:04:38.440 --> 00:04:39.950 align:middle line:84% Suppose we have these three things. 00:04:39.950 --> 00:04:42.640 align:middle line:90% 00:04:42.640 --> 00:04:46.440 align:middle line:84% So we have the sequence of differentiable functions. 00:04:46.440 --> 00:04:49.480 align:middle line:84% If you just fix this one particular point, 00:04:49.480 --> 00:04:53.360 align:middle line:84% x0, then the values of these functions converge. 00:04:53.360 --> 00:04:56.080 align:middle line:84% And then the functions-- their derivatives 00:04:56.080 --> 00:04:58.480 align:middle line:84% are continuous, and the derivative converges 00:04:58.480 --> 00:05:00.600 align:middle line:90% to some function. 00:05:00.600 --> 00:05:02.610 align:middle line:84% And this convergence here is uniform. 00:05:02.610 --> 00:05:06.400 align:middle line:90% 00:05:06.400 --> 00:05:22.460 align:middle line:84% Then the claim is that then there exists f on the interval 00:05:22.460 --> 00:05:30.580 align:middle line:90% so that f is differentiable. 00:05:30.580 --> 00:05:36.120 align:middle line:90% 00:05:36.120 --> 00:05:38.940 align:middle line:84% The derivative here is continuous. 00:05:38.940 --> 00:05:43.680 align:middle line:90% 00:05:43.680 --> 00:05:48.640 align:middle line:84% And you have the following two properties 00:05:48.640 --> 00:05:51.320 align:middle line:90% that the function here-- 00:05:51.320 --> 00:05:55.960 align:middle line:84% these functions converge to f uniformly. 00:05:55.960 --> 00:05:59.520 align:middle line:90% 00:05:59.520 --> 00:06:03.160 align:middle line:84% I'm going to restrict my interval, which 00:06:03.160 --> 00:06:09.640 align:middle line:84% would take it just to be some interval from A to B. 00:06:09.640 --> 00:06:17.840 align:middle line:84% And I want it to be bounded, like that. 00:06:17.840 --> 00:06:21.920 align:middle line:84% And so these converge uniformly, and two 00:06:21.920 --> 00:06:29.800 align:middle line:84% is that their derivative converges to the derivative of f 00:06:29.800 --> 00:06:30.740 align:middle line:90% uniformly. 00:06:30.740 --> 00:06:34.800 align:middle line:90% 00:06:34.800 --> 00:06:40.760 align:middle line:90% So we want to prove that. 00:06:40.760 --> 00:06:47.760 align:middle line:84% And then so we're going to define a function f. 00:06:47.760 --> 00:06:53.480 align:middle line:90% So g here is a proof of this. 00:06:53.480 --> 00:06:56.640 align:middle line:90% Since this sequence here-- 00:06:56.640 --> 00:07:08.960 align:middle line:84% since these are continuous and these functions here 00:07:08.960 --> 00:07:25.720 align:middle line:84% convert to g uniformly, then this means that then, 00:07:25.720 --> 00:07:29.780 align:middle line:84% therefore, g here is also continuous. 00:07:29.780 --> 00:07:35.840 align:middle line:90% 00:07:35.840 --> 00:07:38.480 align:middle line:84% So we proved this already, that if you 00:07:38.480 --> 00:07:41.080 align:middle line:84% have a sequence of continuous function 00:07:41.080 --> 00:07:47.320 align:middle line:84% that converges uniformly, then the limit is also continuous. 00:07:47.320 --> 00:07:58.960 align:middle line:84% And so then we can look at-- so we can therefore 00:07:58.960 --> 00:08:17.920 align:middle line:84% define f as a function on this interval here by that f of x 00:08:17.920 --> 00:08:23.480 align:middle line:84% is equal to the integral from x0. 00:08:23.480 --> 00:08:27.440 align:middle line:84% So I'm defining it, sorry, to be the c 00:08:27.440 --> 00:08:38.159 align:middle line:84% constant, c plus the integral from 0 to x of g of x, g of s 00:08:38.159 --> 00:08:40.880 align:middle line:90% ds. 00:08:40.880 --> 00:08:43.320 align:middle line:84% So I define this function like this. 00:08:43.320 --> 00:08:48.040 align:middle line:84% g is continuous, so this is a nice continuous function. 00:08:48.040 --> 00:09:05.340 align:middle line:84% And in fact, by the fundamental theorem of calculus, 00:09:05.340 --> 00:09:19.670 align:middle line:84% f here is differentiable with the derivative equal to g. 00:09:19.670 --> 00:09:22.680 align:middle line:90% 00:09:22.680 --> 00:09:39.240 align:middle line:84% We also have that we also have that 00:09:39.240 --> 00:09:46.520 align:middle line:84% if we're looking at f of x0, fn of x0, 00:09:46.520 --> 00:09:50.040 align:middle line:84% then we know already that these converts to c. 00:09:50.040 --> 00:09:53.720 align:middle line:84% This is the assumption one up there. 00:09:53.720 --> 00:09:59.000 align:middle line:84% But this here is the value of f at x0. 00:09:59.000 --> 00:09:59.940 align:middle line:90% So we have that. 00:09:59.940 --> 00:10:03.080 align:middle line:90% 00:10:03.080 --> 00:10:08.480 align:middle line:84% I want to now prove that not just at this point 00:10:08.480 --> 00:10:13.600 align:middle line:84% but that this sequence converges to f uniformly. 00:10:13.600 --> 00:10:16.780 align:middle line:90% b we get for free. 00:10:16.780 --> 00:10:31.420 align:middle line:84% So b follows from that this fn prime converges to g. 00:10:31.420 --> 00:10:35.960 align:middle line:84% And this here is the derivative of f. 00:10:35.960 --> 00:10:44.480 align:middle line:84% And we have this uniformly by assumption two. 00:10:44.480 --> 00:10:47.750 align:middle line:84% So all we need to show is-- so we just 00:10:47.750 --> 00:11:00.490 align:middle line:84% need to show that each fn converges to f uniformly. 00:11:00.490 --> 00:11:03.270 align:middle line:90% 00:11:03.270 --> 00:11:10.110 align:middle line:84% We just need to show that the property a-- 00:11:10.110 --> 00:11:14.970 align:middle line:90% and so we're looking at fn of x. 00:11:14.970 --> 00:11:17.990 align:middle line:84% We have to look at this difference. 00:11:17.990 --> 00:11:23.390 align:middle line:84% And for a given epsilon we have to find at little n. 00:11:23.390 --> 00:11:26.110 align:middle line:84% So we have to find a capital N so 00:11:26.110 --> 00:11:28.950 align:middle line:84% that if little n is bigger than this capital N, 00:11:28.950 --> 00:11:34.030 align:middle line:84% then this difference here is smaller than epsilon for all x. 00:11:34.030 --> 00:11:36.750 align:middle line:84% But let's try to estimate this here first. 00:11:36.750 --> 00:11:44.010 align:middle line:90% So this difference is f. 00:11:44.010 --> 00:11:47.250 align:middle line:90% 00:11:47.250 --> 00:11:51.550 align:middle line:84% So we want to estimate this difference. 00:11:51.550 --> 00:11:54.390 align:middle line:84% And we can use the fundamental theorem of calculus 00:11:54.390 --> 00:12:20.670 align:middle line:84% or use the fundamental theorem of calculus to write f of x. 00:12:20.670 --> 00:12:25.150 align:middle line:84% We can write this as the value at x0 00:12:25.150 --> 00:12:30.320 align:middle line:84% plus the integral of the derivative from this point. 00:12:30.320 --> 00:12:33.670 align:middle line:90% 00:12:33.670 --> 00:12:37.210 align:middle line:84% And then this should be less like this. 00:12:37.210 --> 00:12:39.870 align:middle line:84% We can use the fundamental theorem of calculus 00:12:39.870 --> 00:12:46.070 align:middle line:90% to just rewrite fn like that. 00:12:46.070 --> 00:12:49.150 align:middle line:84% And so now I want to look at this difference here. 00:12:49.150 --> 00:12:54.350 align:middle line:84% And so to look at this difference 00:12:54.350 --> 00:13:03.390 align:middle line:84% fn of dx minus f of x, now I can fill in this form of it. 00:13:03.390 --> 00:13:11.710 align:middle line:84% So I have that this is f of x0 plus the integral from x0 00:13:11.710 --> 00:13:16.950 align:middle line:90% to f prime s ds. 00:13:16.950 --> 00:13:21.590 align:middle line:84% And then I have to subtract f of x. 00:13:21.590 --> 00:13:27.150 align:middle line:84% But f of x was given by this formula here, 00:13:27.150 --> 00:13:38.910 align:middle line:84% and so minus c plus this integral here, g of s ds, 00:13:38.910 --> 00:13:39.690 align:middle line:90% like that. 00:13:39.690 --> 00:13:42.750 align:middle line:90% 00:13:42.750 --> 00:13:45.290 align:middle line:84% And so now I'm just rearranging terms. 00:13:45.290 --> 00:13:51.430 align:middle line:84% So I just think about this thing here as fn of x0 minus c. 00:13:51.430 --> 00:13:59.150 align:middle line:84% I take these two first, and then plus the integral of x0 00:13:59.150 --> 00:14:11.270 align:middle line:84% to x f prime n s ds minus this integral here, like that. 00:14:11.270 --> 00:14:15.390 align:middle line:84% And then I use the triangle inequality 00:14:15.390 --> 00:14:21.150 align:middle line:84% that this thing here is less than fn of x minus c-- 00:14:21.150 --> 00:14:27.470 align:middle line:84% sorry fn of x0 minus c, absolute value, like this. 00:14:27.470 --> 00:14:30.990 align:middle line:84% And then I have the second integral here. 00:14:30.990 --> 00:14:33.630 align:middle line:84% So the absolute value or the difference 00:14:33.630 --> 00:14:38.110 align:middle line:84% here, if you do it slowly like this, 00:14:38.110 --> 00:14:48.110 align:middle line:84% so you have this minus this in here of g of s ds, like that. 00:14:48.110 --> 00:14:53.050 align:middle line:84% Now I can think about this thing here as I can find the integral, 00:14:53.050 --> 00:14:55.470 align:middle line:84% so I can write this thing here as-- 00:14:55.470 --> 00:15:00.390 align:middle line:84% so I'm just keeping this first term here. 00:15:00.390 --> 00:15:03.790 align:middle line:84% And then the second here I can think 00:15:03.790 --> 00:15:08.630 align:middle line:84% about this as integral of the difference of this function. 00:15:08.630 --> 00:15:20.050 align:middle line:84% And this here was an n here, like that. 00:15:20.050 --> 00:15:36.910 align:middle line:90% 00:15:36.910 --> 00:15:44.190 align:middle line:84% So now, here, I can just take the absolute value sign. 00:15:44.190 --> 00:15:46.830 align:middle line:84% If I take the value I and under it, then 00:15:46.830 --> 00:15:49.910 align:middle line:84% I get something potentially larger. 00:15:49.910 --> 00:15:56.310 align:middle line:84% So this thing here is less than or equal to f 00:15:56.310 --> 00:15:59.550 align:middle line:90% of x minus 0 minus c. 00:15:59.550 --> 00:16:00.950 align:middle line:90% That's the first term. 00:16:00.950 --> 00:16:03.750 align:middle line:84% And now I'm replacing it by something 00:16:03.750 --> 00:16:09.030 align:middle line:84% that is bigger or equal to this, so where I'm just integrating 00:16:09.030 --> 00:16:15.710 align:middle line:84% from x0 to x, and then of this difference here, 00:16:15.710 --> 00:16:21.580 align:middle line:90% f prime s minus g of s ds. 00:16:21.580 --> 00:16:24.230 align:middle line:90% 00:16:24.230 --> 00:16:28.970 align:middle line:90% And now I'm using that the fn-- 00:16:28.970 --> 00:16:41.330 align:middle line:90% 00:16:41.330 --> 00:16:46.650 align:middle line:84% so actually, let me just write this thing here. 00:16:46.650 --> 00:16:48.430 align:middle line:90% So I can write this thing here. 00:16:48.430 --> 00:16:51.390 align:middle line:90% This one here, I'm keeping. 00:16:51.390 --> 00:16:52.930 align:middle line:90% This one here, I'm keeping. 00:16:52.930 --> 00:16:55.950 align:middle line:84% So I'm writing this as less than or equal to keeping 00:16:55.950 --> 00:16:56.640 align:middle line:90% the first term. 00:16:56.640 --> 00:17:00.190 align:middle line:90% 00:17:00.190 --> 00:17:02.830 align:middle line:84% And the second term, you see, this thing 00:17:02.830 --> 00:17:08.230 align:middle line:84% here is this difference between these two. 00:17:08.230 --> 00:17:13.710 align:middle line:84% Well, the difference is at some s is smaller 00:17:13.710 --> 00:17:17.470 align:middle line:90% than the maximum for all s. 00:17:17.470 --> 00:17:19.829 align:middle line:84% But the maximum for all s, that's 00:17:19.829 --> 00:17:25.990 align:middle line:90% what we call d fn prime g. 00:17:25.990 --> 00:17:28.470 align:middle line:90% That was the maximum over all s. 00:17:28.470 --> 00:17:32.110 align:middle line:84% And so now we just have-- so this function in here 00:17:32.110 --> 00:17:33.590 align:middle line:90% is less than this. 00:17:33.590 --> 00:17:35.590 align:middle line:84% And now we are using-- this is where 00:17:35.590 --> 00:17:38.390 align:middle line:84% I want to use that the interval is bounded. 00:17:38.390 --> 00:17:40.750 align:middle line:90% So I just get plus b minus a. 00:17:40.750 --> 00:17:45.470 align:middle line:84% This is the length of the interval, like that. 00:17:45.470 --> 00:17:50.110 align:middle line:90% And so now you see that s-- 00:17:50.110 --> 00:17:53.970 align:middle line:84% and now there's no reference to x on this side. 00:17:53.970 --> 00:17:58.550 align:middle line:84% And so you see that, as n goes to infinity, 00:17:58.550 --> 00:18:02.390 align:middle line:84% well, this thing here goes to 0 and this thing here 00:18:02.390 --> 00:18:14.680 align:middle line:84% goes to 0, so as this upper bound goes to 0. 00:18:14.680 --> 00:18:18.790 align:middle line:90% 00:18:18.790 --> 00:18:23.050 align:middle line:84% And again, note that there's no reference to x. 00:18:23.050 --> 00:18:24.730 align:middle line:90% There's a reference to x0. 00:18:24.730 --> 00:18:26.490 align:middle line:84% But that's just some fixed point. 00:18:26.490 --> 00:18:29.990 align:middle line:90% So this is uniform in x. 00:18:29.990 --> 00:18:42.530 align:middle line:84% So this implies uniform in x converges. 00:18:42.530 --> 00:18:46.710 align:middle line:90% 00:18:46.710 --> 00:18:55.750 align:middle line:84% And so I want to draw some consequences of this. 00:18:55.750 --> 00:18:59.550 align:middle line:84% First, let me look at what will be, 00:18:59.550 --> 00:19:07.430 align:middle line:84% shortly, a particular case of a more general phenomenon. 00:19:07.430 --> 00:19:17.590 align:middle line:84% So this example we looked at several times last time. 00:19:17.590 --> 00:19:21.010 align:middle line:84% But it's now applied in this case here. 00:19:21.010 --> 00:19:26.310 align:middle line:90% 00:19:26.310 --> 00:19:33.310 align:middle line:84% So suppose you're looking at the functions En of x. 00:19:33.310 --> 00:19:37.190 align:middle line:84% The En is where you're summing from k 00:19:37.190 --> 00:19:44.050 align:middle line:84% equal to 0 to n of x to the power k divided by k factorial. 00:19:44.050 --> 00:19:46.790 align:middle line:84% I looked at those last time, these functions. 00:19:46.790 --> 00:19:50.230 align:middle line:90% And we call them En. 00:19:50.230 --> 00:19:56.670 align:middle line:84% And then we also looked at the exponential function 00:19:56.670 --> 00:20:01.550 align:middle line:84% and the exponential function in terms of power series. 00:20:01.550 --> 00:20:03.710 align:middle line:90% That was just this power series. 00:20:03.710 --> 00:20:07.630 align:middle line:90% 00:20:07.630 --> 00:20:18.150 align:middle line:84% And we have that we have that En here so that for all-- 00:20:18.150 --> 00:20:21.350 align:middle line:90% we used Weinstrass M-test to-- 00:20:21.350 --> 00:20:37.150 align:middle line:84% so by Weinstrass M-test for all L-- 00:20:37.150 --> 00:20:40.140 align:middle line:90% L is a positive real number-- 00:20:40.140 --> 00:20:44.150 align:middle line:90% 00:20:44.150 --> 00:20:53.710 align:middle line:84% we had that these functions converge to E uniformly 00:20:53.710 --> 00:20:57.430 align:middle line:90% on an interval like this. 00:20:57.430 --> 00:20:59.990 align:middle line:84% They don't necessarily-- in fact, they don't. 00:20:59.990 --> 00:21:05.870 align:middle line:84% But the claim here about uniform convergence 00:21:05.870 --> 00:21:08.990 align:middle line:84% is just on every bounded interval. 00:21:08.990 --> 00:21:11.830 align:middle line:84% So you fix a bounded interval like that interval, 00:21:11.830 --> 00:21:13.610 align:middle line:84% and then you have uniform convergence. 00:21:13.610 --> 00:21:15.110 align:middle line:84% It's not like uniform convergence 00:21:15.110 --> 00:21:19.070 align:middle line:90% of all of R at the same time. 00:21:19.070 --> 00:21:26.850 align:middle line:84% So we have that, and now if we are looking at-- 00:21:26.850 --> 00:21:32.530 align:middle line:84% so now if we're looking at the function-- 00:21:32.530 --> 00:21:38.230 align:middle line:90% 00:21:38.230 --> 00:21:42.510 align:middle line:84% if we take now the derivative of the En's-- 00:21:42.510 --> 00:21:44.600 align:middle line:84% so what is the derivative of the En's? 00:21:44.600 --> 00:21:50.470 align:middle line:90% 00:21:50.470 --> 00:21:53.110 align:middle line:90% So the derivative of the En-- 00:21:53.110 --> 00:21:55.470 align:middle line:90% but the En is just a polynomial. 00:21:55.470 --> 00:22:00.790 align:middle line:84% So you can just differentiate each term on its own. 00:22:00.790 --> 00:22:15.190 align:middle line:84% So this here is the sum here of x to the power k minus 1 00:22:15.190 --> 00:22:19.030 align:middle line:90% divided by k factorial. 00:22:19.030 --> 00:22:24.310 align:middle line:84% And here, you would be summing from k equal to 0 to n. 00:22:24.310 --> 00:22:28.870 align:middle line:90% 00:22:28.870 --> 00:22:34.310 align:middle line:90% And now I can write this also. 00:22:34.310 --> 00:22:39.230 align:middle line:84% You see, because when k is equal to 0, 00:22:39.230 --> 00:22:41.270 align:middle line:84% the first term doesn't really count. 00:22:41.270 --> 00:22:44.710 align:middle line:84% So this thing here, I can write as-- 00:22:44.710 --> 00:22:47.710 align:middle line:84% I mean, it doesn't count because it's 0. 00:22:47.710 --> 00:22:50.270 align:middle line:84% So you don't really need to write that term. 00:22:50.270 --> 00:22:54.030 align:middle line:84% So you can just write it from k equal to 1 to n. 00:22:54.030 --> 00:23:00.030 align:middle line:84% And then you can write it again as k times x to the power 00:23:00.030 --> 00:23:04.510 align:middle line:90% k minus 1 over k factorial. 00:23:04.510 --> 00:23:11.350 align:middle line:90% But k factorial is just-- 00:23:11.350 --> 00:23:19.830 align:middle line:84% it's the same as k times k minus 1 factorial. 00:23:19.830 --> 00:23:25.350 align:middle line:84% So this means that k factorial over k 00:23:25.350 --> 00:23:29.440 align:middle line:84% is just the same as k minus 1 factorial. 00:23:29.440 --> 00:23:32.030 align:middle line:90% 00:23:32.030 --> 00:23:36.710 align:middle line:84% Or, if you will, if I take the inverse of this, 00:23:36.710 --> 00:23:45.210 align:middle line:84% k over k factorial is the same as 1 over k minus 1 factorial. 00:23:45.210 --> 00:23:50.350 align:middle line:84% It's just flipping the denominator and the denominator. 00:23:50.350 --> 00:23:57.170 align:middle line:84% So this finite sum I can write as k equal to 1 to n 00:23:57.170 --> 00:24:00.990 align:middle line:84% because you see, this thing here is exactly this. 00:24:00.990 --> 00:24:02.810 align:middle line:90% So we can just replace it. 00:24:02.810 --> 00:24:07.870 align:middle line:90% 00:24:07.870 --> 00:24:09.090 align:middle line:90% You can write it like that. 00:24:09.090 --> 00:24:14.390 align:middle line:90% 00:24:14.390 --> 00:24:21.670 align:middle line:84% But now this thing here, I'm summing from 1 to n. 00:24:21.670 --> 00:24:26.270 align:middle line:84% The first term, I get that is just this 00:24:26.270 --> 00:24:30.150 align:middle line:84% here by definition when k is equal to 1. 00:24:30.150 --> 00:24:31.390 align:middle line:90% That's 0 factorial. 00:24:31.390 --> 00:24:35.350 align:middle line:84% 0 factorial, by definition, is 1. 00:24:35.350 --> 00:24:38.070 align:middle line:84% So this thing here and this thing here, 00:24:38.070 --> 00:24:40.910 align:middle line:90% by definition, is also 1. 00:24:40.910 --> 00:24:42.330 align:middle line:90% So this is just 1. 00:24:42.330 --> 00:24:48.710 align:middle line:84% So starting off at 1, when k is equal to 2, then this becomes x. 00:24:48.710 --> 00:24:50.670 align:middle line:90% This will become 1 factorial. 00:24:50.670 --> 00:24:52.790 align:middle line:90% 1 factorial is 1. 00:24:52.790 --> 00:24:54.190 align:middle line:90% So this here is x. 00:24:54.190 --> 00:24:57.470 align:middle line:90% So this here is like this x. 00:24:57.470 --> 00:25:04.190 align:middle line:84% The next one, when k is equal to 3, then you get x squared. 00:25:04.190 --> 00:25:05.690 align:middle line:90% So you get x squared. 00:25:05.690 --> 00:25:09.570 align:middle line:84% And down here, you get 2 factorial. 00:25:09.570 --> 00:25:11.770 align:middle line:90% And so you continue like this. 00:25:11.770 --> 00:25:17.470 align:middle line:84% And the last one you get is x to the power n minus 1 over n 00:25:17.470 --> 00:25:20.050 align:middle line:90% minus 1 factorial. 00:25:20.050 --> 00:25:22.830 align:middle line:90% 00:25:22.830 --> 00:25:25.210 align:middle line:90% Let me just say-- 00:25:25.210 --> 00:25:32.134 align:middle line:84% so you see that this thing here is nothing but En minus 1 of x. 00:25:32.134 --> 00:25:35.950 align:middle line:90% 00:25:35.950 --> 00:25:45.030 align:middle line:84% So the derivative of the En is the previous element En minus 1. 00:25:45.030 --> 00:25:45.896 align:middle line:90% Yeah? 00:25:45.896 --> 00:25:46.670 align:middle line:84% AUDIENCE: I think that cleared it up. 00:25:46.670 --> 00:25:47.462 align:middle line:90% TOBIAS COLDING: OK. 00:25:47.462 --> 00:25:54.060 align:middle line:90% 00:25:54.060 --> 00:26:15.710 align:middle line:84% So now we have that these En, if I'm looking at the En, is 0. 00:26:15.710 --> 00:26:18.310 align:middle line:90% 00:26:18.310 --> 00:26:21.670 align:middle line:90% The value here of these guys-- 00:26:21.670 --> 00:26:27.230 align:middle line:84% the only term that matters is the first term, and that's 1. 00:26:27.230 --> 00:26:28.770 align:middle line:90% So this here for each n. 00:26:28.770 --> 00:26:37.830 align:middle line:90% 00:26:37.830 --> 00:26:39.830 align:middle line:90% So now what you have is-- 00:26:39.830 --> 00:26:43.430 align:middle line:84% so now we have that you have that if you're 00:26:43.430 --> 00:26:47.150 align:middle line:84% looking at this sequence, you have this. 00:26:47.150 --> 00:26:54.830 align:middle line:84% We have that the En prime is equal to En minus 1. 00:26:54.830 --> 00:26:57.090 align:middle line:84% And we already proved that these here-- 00:26:57.090 --> 00:26:58.590 align:middle line:84% and it doesn't matter whether you're 00:26:58.590 --> 00:27:00.150 align:middle line:90% looking at the previous one-- 00:27:00.150 --> 00:27:03.910 align:middle line:84% that they converge to E uniformly. 00:27:03.910 --> 00:27:14.850 align:middle line:84% This is a uniform convergence on this interval here. 00:27:14.850 --> 00:27:18.550 align:middle line:90% 00:27:18.550 --> 00:27:21.230 align:middle line:90% And then obviously, 3-- 00:27:21.230 --> 00:27:24.430 align:middle line:84% obviously, the En prime here, they 00:27:24.430 --> 00:27:32.976 align:middle line:90% are polynomial, so continuous. 00:27:32.976 --> 00:27:47.150 align:middle line:90% 00:27:47.150 --> 00:28:01.030 align:middle line:84% So therefore, by the theorem, there 00:28:01.030 --> 00:28:05.110 align:middle line:84% exists some function f So that these are 00:28:05.110 --> 00:28:07.830 align:middle line:90% converged to this uniformly. 00:28:07.830 --> 00:28:08.980 align:middle line:90% That's no great surprise. 00:28:08.980 --> 00:28:13.190 align:middle line:90% 00:28:13.190 --> 00:28:17.950 align:middle line:84% And we also have that this thing here, 00:28:17.950 --> 00:28:22.150 align:middle line:84% the derivative converges to the derivative of this 00:28:22.150 --> 00:28:26.350 align:middle line:90% here uniformly. 00:28:26.350 --> 00:28:29.100 align:middle line:84% And again, this is on integrals of this form. 00:28:29.100 --> 00:28:32.310 align:middle line:90% 00:28:32.310 --> 00:28:39.910 align:middle line:84% Well, we already know that it converges uniformly. 00:28:39.910 --> 00:28:42.670 align:middle line:90% This is just by-- 00:28:42.670 --> 00:28:44.930 align:middle line:84% I mean, it can only converge to one function 00:28:44.930 --> 00:28:46.790 align:middle line:84% because uniform convergence means 00:28:46.790 --> 00:28:50.030 align:middle line:84% that it's converging pointwise at each x 00:28:50.030 --> 00:28:55.270 align:middle line:84% and pointwise convergence, then the limit is just one thing. 00:28:55.270 --> 00:28:59.990 align:middle line:84% So we already know that this here is E of x. 00:28:59.990 --> 00:29:02.390 align:middle line:84% So you see that the conclusion here we 00:29:02.390 --> 00:29:15.670 align:middle line:84% get is now that this and this here is, as we said, En minus 1. 00:29:15.670 --> 00:29:20.310 align:middle line:84% So the conclusion that we get is that the E-- 00:29:20.310 --> 00:29:24.510 align:middle line:84% so this is the conclusion from the theorem up there-- 00:29:24.510 --> 00:29:33.302 align:middle line:84% that these En minus 1 converge uniformly to the derivative 00:29:33.302 --> 00:29:34.010 align:middle line:90% of this function. 00:29:34.010 --> 00:29:39.870 align:middle line:90% 00:29:39.870 --> 00:29:46.750 align:middle line:84% The theorem said that if you have these three conditions, 00:29:46.750 --> 00:29:52.470 align:middle line:84% then there exists a function f that is differentiable 00:29:52.470 --> 00:29:55.350 align:middle line:84% and the derivative of it is continuous. 00:29:55.350 --> 00:29:57.870 align:middle line:84% And you have these two properties. 00:29:57.870 --> 00:30:01.470 align:middle line:84% But we already know that these converge to this. 00:30:01.470 --> 00:30:04.750 align:middle line:84% So we've already identified what this function is. 00:30:04.750 --> 00:30:08.310 align:middle line:84% So the theorem just gave us that this function 00:30:08.310 --> 00:30:14.350 align:middle line:84% must be differentiable and the derivative continuous. 00:30:14.350 --> 00:30:18.350 align:middle line:84% And now we try to figure out what that derivative is. 00:30:18.350 --> 00:30:21.070 align:middle line:84% But the derivative of these functions 00:30:21.070 --> 00:30:25.030 align:middle line:84% are the same functions we started with, but just shifted 00:30:25.030 --> 00:30:26.550 align:middle line:90% one backwards. 00:30:26.550 --> 00:30:33.610 align:middle line:84% So these here, also, they converge to this. 00:30:33.610 --> 00:30:35.270 align:middle line:90% That's what the theorem say. 00:30:35.270 --> 00:30:45.510 align:middle line:84% But the En minus 1, they also converge to E. 00:30:45.510 --> 00:30:50.910 align:middle line:84% So this means that these two functions here, this here, 00:30:50.910 --> 00:30:54.670 align:middle line:90% and this here must be the same. 00:30:54.670 --> 00:30:59.350 align:middle line:84% So this means that E is differentiable 00:30:59.350 --> 00:31:02.550 align:middle line:90% and the derivative is itself. 00:31:02.550 --> 00:31:04.030 align:middle line:90% Yeah? 00:31:04.030 --> 00:31:07.910 align:middle line:84% AUDIENCE: So we defined this for all n greater than sum-- 00:31:07.910 --> 00:31:10.770 align:middle line:84% TOBIAS COLDING: This here was for all-- 00:31:10.770 --> 00:31:15.190 align:middle line:84% this here was defined for any nonnegative integer. 00:31:15.190 --> 00:31:26.020 align:middle line:90% 00:31:26.020 --> 00:31:31.320 align:middle line:84% And so I mean, when I'm talking about down here, 00:31:31.320 --> 00:31:33.420 align:middle line:84% then I'm thinking about n bigger or equal to 1 00:31:33.420 --> 00:31:36.440 align:middle line:84% because I didn't strictly speak define E minus 1. 00:31:36.440 --> 00:31:42.900 align:middle line:90% 00:31:42.900 --> 00:31:48.060 align:middle line:84% So now I want to do this more generally. 00:31:48.060 --> 00:31:53.110 align:middle line:90% So I take a power series. 00:31:53.110 --> 00:32:01.020 align:middle line:90% 00:32:01.020 --> 00:32:03.100 align:middle line:90% Also, by the way, the plan-- 00:32:03.100 --> 00:32:07.580 align:middle line:84% so after these lectures, there's two more lectures, Thursday 00:32:07.580 --> 00:32:08.960 align:middle line:90% and Tuesday next week. 00:32:08.960 --> 00:32:14.260 align:middle line:90% 00:32:14.260 --> 00:32:22.480 align:middle line:84% And I already updated the course plan to reflect this. 00:32:22.480 --> 00:32:25.300 align:middle line:84% So my plan is next time, I will prove 00:32:25.300 --> 00:32:29.100 align:middle line:84% existence of uniqueness of some ODE 00:32:29.100 --> 00:32:32.940 align:middle line:84% using things we have learned here about metric spaces. 00:32:32.940 --> 00:32:36.600 align:middle line:84% And then, Tuesday next week, we'll do a review of the class. 00:32:36.600 --> 00:32:38.870 align:middle line:84% So there won't be any new material on Tuesday. 00:32:38.870 --> 00:33:01.350 align:middle line:90% 00:33:01.350 --> 00:33:06.140 align:middle line:84% Also, for whatever it's worth, I decided in the end with a friend 00:33:06.140 --> 00:33:10.700 align:middle line:84% to turn in the lecture notes, to turn it into a book, 00:33:10.700 --> 00:33:14.280 align:middle line:84% just because the TPBB that was like the main reference, 00:33:14.280 --> 00:33:15.880 align:middle line:90% I didn't find that great anyway. 00:33:15.880 --> 00:33:22.460 align:middle line:84% It was also 1,000 pages, who would want to read 1,000 pages. 00:33:22.460 --> 00:33:26.160 align:middle line:84% And so the course notes here is shorter, 00:33:26.160 --> 00:33:28.340 align:middle line:84% as you probably have seen, and trying 00:33:28.340 --> 00:33:33.380 align:middle line:90% to focus on the key points. 00:33:33.380 --> 00:33:37.520 align:middle line:84% So eventually, there will be a book. 00:33:37.520 --> 00:33:59.860 align:middle line:90% 00:33:59.860 --> 00:34:03.960 align:middle line:84% So now I want to do this more generally. 00:34:03.960 --> 00:34:06.120 align:middle line:90% So we take a power series. 00:34:06.120 --> 00:34:10.540 align:middle line:84% So we have an as a sequence, and we're looking at a power series 00:34:10.540 --> 00:34:11.980 align:middle line:90% like this. 00:34:11.980 --> 00:34:13.270 align:middle line:90% So this is a power series. 00:34:13.270 --> 00:34:21.620 align:middle line:90% 00:34:21.620 --> 00:34:31.300 align:middle line:84% And so just to remind you, the radius of convergence 00:34:31.300 --> 00:34:37.340 align:middle line:90% is given as follows. 00:34:37.340 --> 00:34:44.940 align:middle line:84% So we take M, and M is defined to be the lim 00:34:44.940 --> 00:34:52.860 align:middle line:84% sup as n goes to infinity of the n-th root 00:34:52.860 --> 00:34:57.500 align:middle line:90% of absolute value of an. 00:34:57.500 --> 00:35:01.420 align:middle line:84% And then the radius of convergence, so R here, 00:35:01.420 --> 00:35:07.140 align:middle line:84% is then 1 over M with this usual convention 00:35:07.140 --> 00:35:10.380 align:middle line:84% that if M is infinity, this is 0. 00:35:10.380 --> 00:35:13.320 align:middle line:90% If M is 0, this is infinity. 00:35:13.320 --> 00:35:18.480 align:middle line:84% And so then we have, again, that we have that-- 00:35:18.480 --> 00:35:23.780 align:middle line:84% so we have that the series here, so if you're 00:35:23.780 --> 00:35:31.460 align:middle line:84% looking at this k equal to 0 up to n of an xn, 00:35:31.460 --> 00:35:49.420 align:middle line:84% this here converts to this limit here uniformly 00:35:49.420 --> 00:35:54.100 align:middle line:84% on each interval of this form here, 00:35:54.100 --> 00:36:01.580 align:middle line:84% where L here is strictly less than 1. 00:36:01.580 --> 00:36:07.101 align:middle line:84% And this is, again, is Weinstrass M-test. 00:36:07.101 --> 00:36:19.632 align:middle line:90% 00:36:19.632 --> 00:36:24.060 align:middle line:84% Now what I'm interested in is I'm interested in. 00:36:24.060 --> 00:36:27.140 align:middle line:84% So I want to be able to calculate 00:36:27.140 --> 00:36:28.600 align:middle line:90% the derivative of this. 00:36:28.600 --> 00:36:32.820 align:middle line:90% 00:36:32.820 --> 00:36:41.740 align:middle line:84% So we define a function f by that f of x is equal to the sum 00:36:41.740 --> 00:36:44.200 align:middle line:84% here from k equal to 0 to infinity. 00:36:44.200 --> 00:36:52.860 align:middle line:84% So this is the power series x so k, like that. 00:36:52.860 --> 00:37:05.220 align:middle line:84% And we want a formula for the derivative. 00:37:05.220 --> 00:37:11.060 align:middle line:84% And in fact, I want the formula for any derivative of-- 00:37:11.060 --> 00:37:14.310 align:middle line:84% so say the M derivative of this function. 00:37:14.310 --> 00:37:22.420 align:middle line:90% 00:37:22.420 --> 00:37:25.900 align:middle line:90% So now I look at-- 00:37:25.900 --> 00:37:28.380 align:middle line:84% so we had take this power series, 00:37:28.380 --> 00:37:33.030 align:middle line:84% and I'm looking at the power series that is given by-- 00:37:33.030 --> 00:37:38.320 align:middle line:90% 00:37:38.320 --> 00:37:42.140 align:middle line:84% so suppose you're looking at the power series 00:37:42.140 --> 00:37:47.820 align:middle line:84% where for each term, so you have in the original power series. 00:37:47.820 --> 00:37:51.580 align:middle line:90% It's an times x to the n. 00:37:51.580 --> 00:38:03.140 align:middle line:84% Then I'm looking at n an times x minus 1 like this. 00:38:03.140 --> 00:38:12.440 align:middle line:90% And here, I'm summing from 1-- 00:38:12.440 --> 00:38:15.180 align:middle line:90% 00:38:15.180 --> 00:38:19.980 align:middle line:84% I mean, it doesn't matter-- sum from 0, so 0 to infinity, 00:38:19.980 --> 00:38:22.980 align:middle line:90% like that. 00:38:22.980 --> 00:38:25.940 align:middle line:84% Now the radius of convergence for this-- so 00:38:25.940 --> 00:38:27.920 align:middle line:84% what is the radius of convergence? 00:38:27.920 --> 00:38:37.140 align:middle line:90% 00:38:37.140 --> 00:38:42.420 align:middle line:84% The radius of convergence for this 00:38:42.420 --> 00:38:50.920 align:middle line:84% is the n-th root of n times a to the n, like this. 00:38:50.920 --> 00:38:54.660 align:middle line:90% 00:38:54.660 --> 00:39:00.700 align:middle line:84% And this thing here is the same as n to the power 1 over n times 00:39:00.700 --> 00:39:06.580 align:middle line:90% the n-th root of an, like this. 00:39:06.580 --> 00:39:12.220 align:middle line:84% Now this thing here, if you're looking at n like this, 00:39:12.220 --> 00:39:14.180 align:middle line:90% this here is-- 00:39:14.180 --> 00:39:21.500 align:middle line:84% you can write this thing here as E, and then it's log n over n. 00:39:21.500 --> 00:39:26.460 align:middle line:84% This here, as n goes to infinity, converges to 0. 00:39:26.460 --> 00:39:32.660 align:middle line:84% So this thing here converges to 1 by E to the 1. 00:39:32.660 --> 00:39:35.780 align:middle line:84% And so you have that lim sup here. 00:39:35.780 --> 00:39:46.220 align:middle line:84% So the lim sup of the n-th root of n times an, like this-- 00:39:46.220 --> 00:39:49.820 align:middle line:84% this thing here is because this here, you 00:39:49.820 --> 00:39:54.200 align:middle line:84% can write this as the same as lim sup. 00:39:54.200 --> 00:39:58.140 align:middle line:84% n goes to infinity. n goes to infinity. 00:39:58.140 --> 00:40:02.420 align:middle line:84% It's the same as the n-th root of this thing here times 00:40:02.420 --> 00:40:04.220 align:middle line:90% the n-th root of-- 00:40:04.220 --> 00:40:05.120 align:middle line:90% like this. 00:40:05.120 --> 00:40:06.940 align:middle line:90% This here converts to 1. 00:40:06.940 --> 00:40:12.380 align:middle line:84% So it really just converges-- so it's equal to the lim sup 00:40:12.380 --> 00:40:19.380 align:middle line:84% as n goes to infinity of the n-th root of n. 00:40:19.380 --> 00:40:24.660 align:middle line:84% And that was what we call M originally. 00:40:24.660 --> 00:40:30.740 align:middle line:84% So this lim sup is the same as this lim sup. 00:40:30.740 --> 00:40:33.940 align:middle line:84% So this means that M for this new sequence 00:40:33.940 --> 00:40:36.700 align:middle line:84% is the same as M for the old sequence. 00:40:36.700 --> 00:40:42.260 align:middle line:84% So this means that the radius of convergence for the new sequence 00:40:42.260 --> 00:40:44.500 align:middle line:84% is equal to the radius of convergence 00:40:44.500 --> 00:40:45.846 align:middle line:90% from the old sequence. 00:40:45.846 --> 00:40:49.500 align:middle line:90% 00:40:49.500 --> 00:40:54.460 align:middle line:84% So if you take the derivative term by term 00:40:54.460 --> 00:40:56.900 align:middle line:84% and you form this new power series, 00:40:56.900 --> 00:40:59.900 align:middle line:84% then it has the same radius of convergence 00:40:59.900 --> 00:41:01.230 align:middle line:90% as the original passes. 00:41:01.230 --> 00:41:05.340 align:middle line:90% 00:41:05.340 --> 00:41:08.680 align:middle line:90% And so now you can iterate this. 00:41:08.680 --> 00:41:12.300 align:middle line:90% 00:41:12.300 --> 00:41:13.760 align:middle line:90% So you can iterate this. 00:41:13.760 --> 00:41:21.860 align:middle line:90% 00:41:21.860 --> 00:41:23.940 align:middle line:90% So you can take the-- 00:41:23.940 --> 00:41:26.700 align:middle line:84% so this is just taking the derivative term 00:41:26.700 --> 00:41:29.180 align:middle line:90% by term one time. 00:41:29.180 --> 00:41:33.260 align:middle line:84% If you now take the derivative, suppose 00:41:33.260 --> 00:41:36.500 align:middle line:84% you're starting with this sequence here. 00:41:36.500 --> 00:41:40.460 align:middle line:84% So you're starting with this power series. 00:41:40.460 --> 00:41:45.220 align:middle line:84% And you're taking the derivative term by term. 00:41:45.220 --> 00:41:48.460 align:middle line:84% Then you get as we did up there, you're 00:41:48.460 --> 00:41:50.620 align:middle line:90% getting this power series. 00:41:50.620 --> 00:41:55.490 align:middle line:84% So now you take a derivative again term by term. 00:41:55.490 --> 00:42:02.100 align:middle line:90% 00:42:02.100 --> 00:42:04.640 align:middle line:90% So you're getting this right. 00:42:04.640 --> 00:42:09.340 align:middle line:84% And the thing is that each time the radius of convergence 00:42:09.340 --> 00:42:22.820 align:middle line:84% for this same radius of convergence, these two here-- 00:42:22.820 --> 00:42:26.300 align:middle line:90% same radius of convergence-- 00:42:26.300 --> 00:42:35.540 align:middle line:84% it's because of the same principle, same radius, 00:42:35.540 --> 00:42:37.080 align:middle line:90% and so on. 00:42:37.080 --> 00:42:38.150 align:middle line:90% And so you can continue. 00:42:38.150 --> 00:42:40.860 align:middle line:90% 00:42:40.860 --> 00:42:46.940 align:middle line:84% And if you do this, if you do this M time, 00:42:46.940 --> 00:42:48.760 align:middle line:90% then you're getting n-- 00:42:48.760 --> 00:42:54.080 align:middle line:90% 00:42:54.080 --> 00:43:00.380 align:middle line:84% so if you do this M times, then you 00:43:00.380 --> 00:43:10.260 align:middle line:84% get that this thing here is x minus M, and you're getting an. 00:43:10.260 --> 00:43:23.100 align:middle line:84% And here, you're getting n minus 1, n minus 2, up to n 00:43:23.100 --> 00:43:25.720 align:middle line:90% minus M plus 1. 00:43:25.720 --> 00:43:28.980 align:middle line:90% 00:43:28.980 --> 00:43:32.260 align:middle line:90% So you're getting that. 00:43:32.260 --> 00:43:33.860 align:middle line:90% So that is after-- 00:43:33.860 --> 00:43:35.600 align:middle line:90% this is when you do it n times. 00:43:35.600 --> 00:43:37.420 align:middle line:84% And so again, these here will have 00:43:37.420 --> 00:43:39.040 align:middle line:90% the same radius of convergence. 00:43:39.040 --> 00:43:48.820 align:middle line:90% 00:43:48.820 --> 00:43:53.660 align:middle line:84% Now this thing here, if I want to write this a little bit 00:43:53.660 --> 00:43:57.740 align:middle line:84% more compactly, if I thinking about this thing here, n 00:43:57.740 --> 00:44:04.580 align:middle line:84% minus 2 up to n minus M plus 1, I 00:44:04.580 --> 00:44:08.760 align:middle line:84% can write this thing here as n factorial. 00:44:08.760 --> 00:44:12.020 align:middle line:90% 00:44:12.020 --> 00:44:14.440 align:middle line:90% But then I'm getting all these-- 00:44:14.440 --> 00:44:21.040 align:middle line:84% so if I wrote it as n factorial, then I would also have-- 00:44:21.040 --> 00:44:22.680 align:middle line:90% it would keep going like this. 00:44:22.680 --> 00:44:30.120 align:middle line:84% And the next one here would be n minus M. 00:44:30.120 --> 00:44:33.180 align:middle line:84% So if I write it-- so I can write this thing here times 00:44:33.180 --> 00:44:36.400 align:middle line:90% this factorial is equal to that. 00:44:36.400 --> 00:44:36.900 align:middle line:90% Yeah? 00:44:36.900 --> 00:44:38.740 align:middle line:90% Sorry. 00:44:38.740 --> 00:44:41.340 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:44:41.340 --> 00:44:50.300 align:middle line:90% 00:44:50.300 --> 00:44:53.340 align:middle line:84% TOBIAS COLDING: So I think you were complaining about one thing 00:44:53.340 --> 00:44:57.780 align:middle line:84% that I sort of pushed under the rug. 00:44:57.780 --> 00:45:09.060 align:middle line:84% And that is the following, that if you take the way we defined-- 00:45:09.060 --> 00:45:13.780 align:middle line:84% so if you take a power series, again, like this, 00:45:13.780 --> 00:45:20.340 align:middle line:84% and you define the radius of convergence, 00:45:20.340 --> 00:45:24.380 align:middle line:84% then you define it from this n element. 00:45:24.380 --> 00:45:26.460 align:middle line:84% I think that's what you're complaining about. 00:45:26.460 --> 00:45:30.380 align:middle line:84% And so you take the coefficient to the n power. 00:45:30.380 --> 00:45:32.860 align:middle line:90% That's not exactly what I did. 00:45:32.860 --> 00:45:34.780 align:middle line:84% Is that what you're complaining about? 00:45:34.780 --> 00:45:36.560 align:middle line:90% And that doesn't matter. 00:45:36.560 --> 00:45:39.980 align:middle line:84% And the thing is so when you define 00:45:39.980 --> 00:45:46.220 align:middle line:84% the radius of convergence, you took this n power, like this. 00:45:46.220 --> 00:45:52.380 align:middle line:84% But there was a coefficient to where this here was to the n. 00:45:52.380 --> 00:45:55.300 align:middle line:84% When I'm doing it over here, there's 00:45:55.300 --> 00:45:56.960 align:middle line:90% two ways of dealing with it. 00:45:56.960 --> 00:46:00.380 align:middle line:84% One is the one you suggest, and the other one-- and that 00:46:00.380 --> 00:46:04.060 align:middle line:84% was why I put it on slipped it under the rock. 00:46:04.060 --> 00:46:09.260 align:middle line:84% Is that that's the way the radius 00:46:09.260 --> 00:46:11.500 align:middle line:84% this with the radius of convergence, 00:46:11.500 --> 00:46:15.060 align:middle line:84% it comes from comparing it with the geometric series. 00:46:15.060 --> 00:46:16.900 align:middle line:84% And then it really doesn't matter 00:46:16.900 --> 00:46:19.820 align:middle line:84% for this term in terms of x whether you have shifted it 00:46:19.820 --> 00:46:20.440 align:middle line:90% or not. 00:46:20.440 --> 00:46:26.040 align:middle line:84% But if you do it, if you do it as you're suggesting, 00:46:26.040 --> 00:46:32.860 align:middle line:84% which would be totally fine, so then this thing here, what 00:46:32.860 --> 00:46:36.420 align:middle line:84% you really should have done is that we should have thought 00:46:36.420 --> 00:46:40.460 align:middle line:84% about this n, an, and then we should 00:46:40.460 --> 00:46:44.680 align:middle line:84% take this here, and instead of the n root, 00:46:44.680 --> 00:46:46.600 align:middle line:84% we should take the n minus 1 root. 00:46:46.600 --> 00:46:49.580 align:middle line:90% But that doesn't matter either. 00:46:49.580 --> 00:46:51.920 align:middle line:84% That's going to be the same also. 00:46:51.920 --> 00:46:54.860 align:middle line:84% So either way, you can check either of these two things. 00:46:54.860 --> 00:46:57.260 align:middle line:84% And I was thinking about the first 00:46:57.260 --> 00:46:58.890 align:middle line:90% and where it's coming from. 00:46:58.890 --> 00:47:12.300 align:middle line:90% 00:47:12.300 --> 00:47:14.720 align:middle line:84% Sorry, now I lost my thread here. 00:47:14.720 --> 00:47:21.780 align:middle line:84% So I was just observing that if you can write the n factorial 00:47:21.780 --> 00:47:26.740 align:middle line:84% as first taking all of these factors 00:47:26.740 --> 00:47:30.140 align:middle line:84% and then taking n minus 1, so this 00:47:30.140 --> 00:47:37.340 align:middle line:84% means that n factorial divided by n minus 1, 00:47:37.340 --> 00:47:48.500 align:middle line:84% so n minus m factorial, this is equal to n minus 1, n minus 2, 00:47:48.500 --> 00:47:53.320 align:middle line:84% down to n minus m plus 1, like that. 00:47:53.320 --> 00:47:56.780 align:middle line:90% 00:47:56.780 --> 00:48:00.020 align:middle line:84% But you see, that's exactly what we have done here. 00:48:00.020 --> 00:48:04.860 align:middle line:84% So this thing here, so what I conclude from this 00:48:04.860 --> 00:48:33.220 align:middle line:84% is that if I take a power series like this, 00:48:33.220 --> 00:48:42.020 align:middle line:84% and I take the derivative term by term, and I do this M times, 00:48:42.020 --> 00:48:50.420 align:middle line:84% then this is what I get is n factorial n minus M factorial 00:48:50.420 --> 00:49:05.800 align:middle line:90% and then an xn n minus M. 00:49:05.800 --> 00:49:10.180 align:middle line:84% So this is when we do it term by term, taking the derivative. 00:49:10.180 --> 00:49:13.940 align:middle line:84% And we already have that these here have 00:49:13.940 --> 00:49:15.720 align:middle line:90% the same radius of convergence. 00:49:15.720 --> 00:49:23.580 align:middle line:90% 00:49:23.580 --> 00:49:28.830 align:middle line:84% And so now we have the following theorem. 00:49:28.830 --> 00:49:32.660 align:middle line:90% 00:49:32.660 --> 00:49:35.580 align:middle line:84% Then if you take a power series, so you 00:49:35.580 --> 00:49:42.600 align:middle line:84% take a power series like this, R is the radius of convergence. 00:49:42.600 --> 00:49:54.260 align:middle line:90% 00:49:54.260 --> 00:49:58.700 align:middle line:84% And if L here is strictly less than some positive number 00:49:58.700 --> 00:50:02.740 align:middle line:84% but strictly less than the radius of convergence, 00:50:02.740 --> 00:50:13.000 align:middle line:84% then these guys here converge to-- 00:50:13.000 --> 00:50:16.780 align:middle line:84% so when you're summing, when you take a finite sum, 00:50:16.780 --> 00:50:27.130 align:middle line:84% these converge to this uniformly, but not only these. 00:50:27.130 --> 00:50:36.540 align:middle line:90% 00:50:36.540 --> 00:50:41.860 align:middle line:84% But not only that, but you also have that-- 00:50:41.860 --> 00:51:04.490 align:middle line:84% more interesting-- you have that this sequence here, k minus M-- 00:51:04.490 --> 00:51:21.010 align:middle line:84% k factorial divided by k minus M factorial ak xk minus M, 00:51:21.010 --> 00:51:24.250 align:middle line:84% that these sequence here converge 00:51:24.250 --> 00:51:29.130 align:middle line:90% uniformly to the derivative. 00:51:29.130 --> 00:51:40.090 align:middle line:84% This here is f of x to the M derivative of x of f. 00:51:40.090 --> 00:51:41.670 align:middle line:84% So we have that these functions-- 00:51:41.670 --> 00:51:45.170 align:middle line:84% but that we already proved that these functions converge, 00:51:45.170 --> 00:51:59.250 align:middle line:84% and their derivative here converge uniformly, like this, 00:51:59.250 --> 00:52:00.300 align:middle line:90% on the same interval. 00:52:00.300 --> 00:52:06.746 align:middle line:90% 00:52:06.746 --> 00:52:10.130 align:middle line:90% Now why is this the case? 00:52:10.130 --> 00:52:13.270 align:middle line:84% So this here is again it's following. 00:52:13.270 --> 00:52:15.280 align:middle line:90% So this follows the proof-- 00:52:15.280 --> 00:52:19.090 align:middle line:90% 00:52:19.090 --> 00:52:22.850 align:middle line:90% follows from the same theorem-- 00:52:22.850 --> 00:52:31.370 align:middle line:84% follows from the theorem we proved earlier, 00:52:31.370 --> 00:52:32.160 align:middle line:90% we showed earlier. 00:52:32.160 --> 00:52:36.170 align:middle line:90% 00:52:36.170 --> 00:52:39.570 align:middle line:90% And let's just go through that. 00:52:39.570 --> 00:52:41.250 align:middle line:90% So let me go through it. 00:52:41.250 --> 00:52:52.850 align:middle line:84% So let's assume, first, that M is equal to 1. 00:52:52.850 --> 00:52:56.698 align:middle line:90% So I just take one derivative-- 00:52:56.698 --> 00:53:02.610 align:middle line:84% well, then I'm looking at this sequence here, 00:53:02.610 --> 00:53:08.690 align:middle line:90% k xk k equal to 0 to n. 00:53:08.690 --> 00:53:15.170 align:middle line:84% I'm looking at these that converge to this infinite-- 00:53:15.170 --> 00:53:17.910 align:middle line:90% to converge to f. 00:53:17.910 --> 00:53:21.210 align:middle line:90% 00:53:21.210 --> 00:53:23.890 align:middle line:84% f, again, is this infinite power series. 00:53:23.890 --> 00:53:26.710 align:middle line:84% And I'm looking at the derivative of these guys. 00:53:26.710 --> 00:53:29.650 align:middle line:90% So these are these. 00:53:29.650 --> 00:53:38.450 align:middle line:84% And I know now that these here converge uniformly and these 00:53:38.450 --> 00:53:41.870 align:middle line:84% here converge uniformly the power series-- 00:53:41.870 --> 00:53:44.610 align:middle line:90% that is, this infinite series. 00:53:44.610 --> 00:53:47.170 align:middle line:84% This here was the finite part of it. 00:53:47.170 --> 00:53:50.910 align:middle line:84% And this here is the infinite part-- the whole thing. 00:53:50.910 --> 00:53:57.170 align:middle line:90% 00:53:57.170 --> 00:53:58.830 align:middle line:90% So we have that. 00:53:58.830 --> 00:54:03.690 align:middle line:90% 00:54:03.690 --> 00:54:05.830 align:middle line:90% This means, in particular-- 00:54:05.830 --> 00:54:11.690 align:middle line:84% so this thing here implies, of course, 00:54:11.690 --> 00:54:18.650 align:middle line:84% that when you're looking at 0, when you're looking at 0, if you 00:54:18.650 --> 00:54:23.450 align:middle line:84% evaluate this thing here at 0, then, in this one here, 00:54:23.450 --> 00:54:28.330 align:middle line:84% you get in this is the sequence of functions, 00:54:28.330 --> 00:54:34.830 align:middle line:84% f of x from before, these are the fn prime of x. 00:54:34.830 --> 00:54:38.010 align:middle line:90% 00:54:38.010 --> 00:54:40.670 align:middle line:90% And so now I claim that-- 00:54:40.670 --> 00:54:44.030 align:middle line:84% so the theorem that we proved earlier 00:54:44.030 --> 00:54:49.130 align:middle line:84% is that if you take a sequence of functions that 00:54:49.130 --> 00:54:53.590 align:middle line:84% are differentiable and the derivative is continuous-- 00:54:53.590 --> 00:54:56.930 align:middle line:84% these are all polynomials, so this is definitely the case. 00:54:56.930 --> 00:55:02.370 align:middle line:84% And we know that these values here fn-- 00:55:02.370 --> 00:55:08.530 align:middle line:84% in fact, at 0, they're equal to f of 0. 00:55:08.530 --> 00:55:10.890 align:middle line:90% So we had three conditions. 00:55:10.890 --> 00:55:14.690 align:middle line:84% We had that the fn in some particular 00:55:14.690 --> 00:55:19.730 align:middle line:84% point that we call x0, this is now 0, that this converged. 00:55:19.730 --> 00:55:24.210 align:middle line:84% We have that the f prime was converging uniformly. 00:55:24.210 --> 00:55:27.090 align:middle line:84% Well, we know that these converge uniformly. 00:55:27.090 --> 00:55:30.150 align:middle line:84% And then we had that the f prime should be continuous. 00:55:30.150 --> 00:55:31.830 align:middle line:84% But they were obviously continuous. 00:55:31.830 --> 00:55:35.090 align:middle line:90% They were polynomials. 00:55:35.090 --> 00:55:47.170 align:middle line:84% Then we had that the derivative converged uniformly and that 00:55:47.170 --> 00:55:50.890 align:middle line:84% the function that they converge to-- 00:55:50.890 --> 00:55:54.810 align:middle line:84% the derivative converge and also the original function 00:55:54.810 --> 00:56:02.490 align:middle line:84% converge and that the derivative converge 00:56:02.490 --> 00:56:06.090 align:middle line:84% uniformly to the derivative of the function 00:56:06.090 --> 00:56:08.530 align:middle line:90% that we converge to. 00:56:08.530 --> 00:56:12.330 align:middle line:84% So you see that this is just checking 00:56:12.330 --> 00:56:15.950 align:middle line:84% that we have the three property from the theorem before. 00:56:15.950 --> 00:56:18.090 align:middle line:84% And then we get the two conclusion. 00:56:18.090 --> 00:56:19.850 align:middle line:84% And the one conclusion that we're 00:56:19.850 --> 00:56:24.490 align:middle line:84% interested in is that this here is then the power 00:56:24.490 --> 00:56:27.130 align:middle line:90% series for the derivative of f. 00:56:27.130 --> 00:56:35.850 align:middle line:90% 00:56:35.850 --> 00:56:39.290 align:middle line:84% And then you can do the same thing with-- 00:56:39.290 --> 00:56:49.850 align:middle line:84% so let me just say that you can do the same thing if you 00:56:49.850 --> 00:56:51.970 align:middle line:90% are integrating. 00:56:51.970 --> 00:56:54.270 align:middle line:90% So here, we proved-- 00:56:54.270 --> 00:57:00.170 align:middle line:84% so suppose you take a sequence here that is converging to f. 00:57:00.170 --> 00:57:04.330 align:middle line:84% Well, suppose you take a power series here. 00:57:04.330 --> 00:57:14.730 align:middle line:84% Suppose you take this power series, ak xk k equal to 0. 00:57:14.730 --> 00:57:16.150 align:middle line:90% So this is the finite-- 00:57:16.150 --> 00:57:19.810 align:middle line:90% this is the power series, sorry. 00:57:19.810 --> 00:57:24.010 align:middle line:84% And suppose I'm interested in what is the integral from a 00:57:24.010 --> 00:57:33.770 align:middle line:84% to b of this power series f of x dx. 00:57:33.770 --> 00:57:44.930 align:middle line:84% And here, I'm assuming that a and b are in the interval, where 00:57:44.930 --> 00:57:48.970 align:middle line:90% the power series is convergent. 00:57:48.970 --> 00:57:52.610 align:middle line:84% So this means, of course, that if you think about it 00:57:52.610 --> 00:57:54.530 align:middle line:90% that you have here 0. 00:57:54.530 --> 00:57:59.710 align:middle line:84% You have R. You have here minus R. And then a and b is lying-- 00:57:59.710 --> 00:58:02.170 align:middle line:84% and so this means that this whole interval 00:58:02.170 --> 00:58:05.960 align:middle line:84% is lying comfortably inside some interval like this. 00:58:05.960 --> 00:58:11.750 align:middle line:90% 00:58:11.750 --> 00:58:18.570 align:middle line:84% And I'm asking what is the nice formula for this integral here? 00:58:18.570 --> 00:58:24.090 align:middle line:84% And so then, well, I can use not the theorem 00:58:24.090 --> 00:58:27.230 align:middle line:84% we proved earlier but the theorem we proved last time. 00:58:27.230 --> 00:58:29.170 align:middle line:90% And so what is that? 00:58:29.170 --> 00:58:36.290 align:middle line:84% So if I define fn of x to be equal to the sum here from k 00:58:36.290 --> 00:58:42.570 align:middle line:90% equals to 0 to n of ak xk-- 00:58:42.570 --> 00:58:45.850 align:middle line:90% sorry, x to the power k-- 00:58:45.850 --> 00:58:49.330 align:middle line:84% this thing here, these here converge 00:58:49.330 --> 00:58:55.850 align:middle line:84% uniformly to f of x on any interval 00:58:55.850 --> 00:59:00.490 align:middle line:84% like this when L here is strictly less than the radius 00:59:00.490 --> 00:59:01.990 align:middle line:90% of convergence. 00:59:01.990 --> 00:59:04.310 align:middle line:90% So they converge here uniformly. 00:59:04.310 --> 00:59:07.370 align:middle line:90% 00:59:07.370 --> 00:59:09.210 align:middle line:84% But the theorem we proved earlier 00:59:09.210 --> 00:59:14.490 align:middle line:84% was that if you take a uniform convergence like this 00:59:14.490 --> 00:59:17.370 align:middle line:84% and you take a and b in this interval, 00:59:17.370 --> 00:59:29.970 align:middle line:84% then the integral of these guys here converge to the integral 00:59:29.970 --> 00:59:30.930 align:middle line:90% like this. 00:59:30.930 --> 00:59:34.450 align:middle line:90% You have this convergence here. 00:59:34.450 --> 00:59:37.170 align:middle line:84% But these here are easy to integrate. 00:59:37.170 --> 00:59:42.050 align:middle line:90% 00:59:42.050 --> 00:59:50.170 align:middle line:84% But because they're polynomial, so this 00:59:50.170 --> 00:59:56.510 align:middle line:84% means that the fn s, the fn of s, 00:59:56.510 --> 01:00:00.530 align:middle line:84% when you integrate this from a to b ds, 01:00:00.530 --> 01:00:14.970 align:middle line:84% these guys are the sum here of ak sk k equal to 0 to n, 01:00:14.970 --> 01:00:21.930 align:middle line:84% and then the integral here, ds from a to b because it's just 01:00:21.930 --> 01:00:23.450 align:middle line:90% that polynomial. 01:00:23.450 --> 01:00:25.970 align:middle line:84% And so for polynomial, well, we can just 01:00:25.970 --> 01:00:35.210 align:middle line:84% use that it's a sum of these guys, ak sk ds k 01:00:35.210 --> 01:00:37.930 align:middle line:90% equals to 0 to n. 01:00:37.930 --> 01:00:46.770 align:middle line:84% And this thing here, when we integrate this, 01:00:46.770 --> 01:00:48.170 align:middle line:90% that's just the-- 01:00:48.170 --> 01:00:50.930 align:middle line:84% I mean, now it doesn't get much nicer unless you 01:00:50.930 --> 01:00:56.050 align:middle line:90% make one of them equal to 0. 01:00:56.050 --> 01:00:58.210 align:middle line:90% So you have this. 01:00:58.210 --> 01:01:08.050 align:middle line:84% So let me say that if a here was equal to 0, 01:01:08.050 --> 01:01:12.130 align:middle line:84% and B was equal to x, so if that's the case 01:01:12.130 --> 01:01:19.110 align:middle line:84% and you're looking at this integral from 0 to x of fn s ds, 01:01:19.110 --> 01:01:25.740 align:middle line:84% we know that this here converged to the integral of the power 01:01:25.740 --> 01:01:26.240 align:middle line:90% series. 01:01:26.240 --> 01:01:28.970 align:middle line:90% 01:01:28.970 --> 01:01:31.310 align:middle line:84% On the other hand, this integral here-- 01:01:31.310 --> 01:01:44.610 align:middle line:90% 01:01:44.610 --> 01:01:46.730 align:middle line:84% on the other hand, this integral here, 01:01:46.730 --> 01:01:56.430 align:middle line:84% if you're looking at the sum here and then sum, 01:01:56.430 --> 01:02:06.250 align:middle line:84% and now the a is 0, b is x, so then is this here ak xk, 01:02:06.250 --> 01:02:11.130 align:middle line:84% now when we integrate this term here, 01:02:11.130 --> 01:02:15.130 align:middle line:84% we're getting sum here k equal to 0 to n. 01:02:15.130 --> 01:02:29.750 align:middle line:84% And then this here becomes ak k plus 1 x to the power k plus 1. 01:02:29.750 --> 01:02:35.330 align:middle line:90% 01:02:35.330 --> 01:02:38.270 align:middle line:84% And so you see that if you're looking at-- 01:02:38.270 --> 01:02:45.450 align:middle line:84% so you see that these guys here now converge to the integral 01:02:45.450 --> 01:02:51.050 align:middle line:90% from 0 to x of f of s ds. 01:02:51.050 --> 01:02:53.290 align:middle line:84% That's what we get from the theorem. 01:02:53.290 --> 01:02:59.650 align:middle line:84% So the conclusion here is that if you take a power series 01:02:59.650 --> 01:03:02.810 align:middle line:84% and you're looking inside the radius of convergence, 01:03:02.810 --> 01:03:06.050 align:middle line:84% then the power series is differentiable as many times 01:03:06.050 --> 01:03:07.290 align:middle line:90% as you want. 01:03:07.290 --> 01:03:11.190 align:middle line:84% And the derivative is given by a power series. 01:03:11.190 --> 01:03:16.450 align:middle line:84% And it's just the derivative of the individual term by term. 01:03:16.450 --> 01:03:21.410 align:middle line:84% And likewise, if you want to integrate a power series, 01:03:21.410 --> 01:03:25.397 align:middle line:84% then the integral, at least when you're integrating because, 01:03:25.397 --> 01:03:27.230 align:middle line:84% otherwise, the formula is not quite as nice, 01:03:27.230 --> 01:03:31.930 align:middle line:84% but you do get a formula, but it just doesn't look so nice. 01:03:31.930 --> 01:03:34.910 align:middle line:84% But the integral of a power series, 01:03:34.910 --> 01:03:37.680 align:middle line:84% you can just integrate it term by term. 01:03:37.680 --> 01:03:41.130 align:middle line:90% 01:03:41.130 --> 01:03:44.610 align:middle line:84% So now the next thing we want to talk about-- 01:03:44.610 --> 01:03:50.050 align:middle line:84% and I'm just trying to set up for next lecture. 01:03:50.050 --> 01:03:54.050 align:middle line:84% And so we want to talk about differential equations 01:03:54.050 --> 01:03:58.090 align:middle line:84% and how we're using some of the stuff 01:03:58.090 --> 01:04:03.010 align:middle line:84% that we've proven in this class to solve differential equation. 01:04:03.010 --> 01:04:05.330 align:middle line:84% And it's an example where it may not 01:04:05.330 --> 01:04:13.890 align:middle line:84% be very obvious that you could use metric spaces to solve 01:04:13.890 --> 01:04:14.920 align:middle line:90% a differential equation. 01:04:14.920 --> 01:04:17.490 align:middle line:90% 01:04:17.490 --> 01:04:20.010 align:middle line:84% But this is nevertheless the case. 01:04:20.010 --> 01:04:23.920 align:middle line:84% So we're looking at a differential equation. 01:04:23.920 --> 01:04:31.250 align:middle line:90% 01:04:31.250 --> 01:04:39.588 align:middle line:84% So differential equation is an equation is an equation in-- 01:04:39.588 --> 01:04:44.170 align:middle line:90% 01:04:44.170 --> 01:04:57.070 align:middle line:84% an unknown in an unknown function and its derivative. 01:04:57.070 --> 01:05:02.570 align:middle line:90% 01:05:02.570 --> 01:05:09.290 align:middle line:84% So let me give you some examples of this that you take-- 01:05:09.290 --> 01:05:14.450 align:middle line:90% you're looking at a function f. 01:05:14.450 --> 01:05:18.930 align:middle line:84% You're trying to find a function f so that-- so this is 01:05:18.930 --> 01:05:22.330 align:middle line:90% like the first example. 01:05:22.330 --> 01:05:28.170 align:middle line:84% You're looking for a function f so that the derivative is x. 01:05:28.170 --> 01:05:30.250 align:middle line:84% Well, we already know that this here, 01:05:30.250 --> 01:05:43.250 align:middle line:84% the solutions to this f of x equal to 1/2 x squared plus some 01:05:43.250 --> 01:05:45.450 align:middle line:90% constant. 01:05:45.450 --> 01:05:48.370 align:middle line:84% The constant when you take derivative, you get 0. 01:05:48.370 --> 01:05:52.330 align:middle line:84% And the derivative of this thing here gives you x. 01:05:52.330 --> 01:05:56.850 align:middle line:84% And those are the only solutions. 01:05:56.850 --> 01:06:01.890 align:middle line:84% Another example here is where you're 01:06:01.890 --> 01:06:08.630 align:middle line:84% taking something that looks like f prime minus f squared, 01:06:08.630 --> 01:06:10.550 align:middle line:90% so f of x, like this. 01:06:10.550 --> 01:06:13.890 align:middle line:90% 01:06:13.890 --> 01:06:20.490 align:middle line:84% And if you're looking at the function here, f of x 01:06:20.490 --> 01:06:23.570 align:middle line:90% is equal to 1 minus 1 over x. 01:06:23.570 --> 01:06:29.130 align:middle line:84% If you take this function here and now you take the derivative, 01:06:29.130 --> 01:06:33.650 align:middle line:84% so now I'm using the chain rule, so I get 1 minus 1, 01:06:33.650 --> 01:06:34.870 align:middle line:90% like this, squared. 01:06:34.870 --> 01:06:37.210 align:middle line:84% And then I take the derivative of this. 01:06:37.210 --> 01:06:40.930 align:middle line:90% So that's minus 1. 01:06:40.930 --> 01:06:43.930 align:middle line:90% 01:06:43.930 --> 01:06:46.890 align:middle line:90% So that's minus 1 times minus 1. 01:06:46.890 --> 01:06:54.170 align:middle line:84% So this here is minus 1 minus x squared. 01:06:54.170 --> 01:06:57.970 align:middle line:84% And so you see that when you take this function here, 01:06:57.970 --> 01:06:58.740 align:middle line:90% the minus sign-- 01:06:58.740 --> 01:07:14.130 align:middle line:90% 01:07:14.130 --> 01:07:17.090 align:middle line:90% I think it should be a plus. 01:07:17.090 --> 01:07:19.730 align:middle line:90% So when you take-- 01:07:19.730 --> 01:07:21.230 align:middle line:90% no, no, no, it should be minus. 01:07:21.230 --> 01:07:24.290 align:middle line:84% No, no, that's right, sorry, yeah, thanks. 01:07:24.290 --> 01:07:25.650 align:middle line:90% This is wonderful. 01:07:25.650 --> 01:07:30.090 align:middle line:84% This should have been a minus like that. 01:07:30.090 --> 01:07:32.790 align:middle line:84% And so this here is now is indeed plus. 01:07:32.790 --> 01:07:34.140 align:middle line:90% And then it's this equation. 01:07:34.140 --> 01:07:43.906 align:middle line:90% 01:07:43.906 --> 01:07:50.850 align:middle line:84% The next example-- so again, a differential equation 01:07:50.850 --> 01:07:55.850 align:middle line:84% is something that involved an unknown function 01:07:55.850 --> 01:07:59.330 align:middle line:90% and its derivative. 01:07:59.330 --> 01:08:01.530 align:middle line:84% So suppose you take something that 01:08:01.530 --> 01:08:08.010 align:middle line:84% looks like f, f prime, f double prime equal to 0. 01:08:08.010 --> 01:08:13.690 align:middle line:84% If you're taking the function f equal to some constant, 01:08:13.690 --> 01:08:16.130 align:middle line:90% this is a solution. 01:08:16.130 --> 01:08:23.090 align:middle line:84% And if you're taking f equal to, say, x or some constant times x, 01:08:23.090 --> 01:08:25.370 align:middle line:90% this is also a solution. 01:08:25.370 --> 01:08:26.670 align:middle line:90% They're both solutions. 01:08:26.670 --> 01:08:30.970 align:middle line:90% 01:08:30.970 --> 01:08:33.069 align:middle line:84% And this is just because the second-- 01:08:33.069 --> 01:08:38.090 align:middle line:84% in this case, the second derivative is 0. 01:08:38.090 --> 01:08:41.250 align:middle line:84% So now what we will be interested in 01:08:41.250 --> 01:08:45.649 align:middle line:84% is we're going to be interested in what's 01:08:45.649 --> 01:08:47.710 align:middle line:84% called an ordinary differential equation. 01:08:47.710 --> 01:08:52.270 align:middle line:90% And we only looked at this. 01:08:52.270 --> 01:08:54.840 align:middle line:84% We only looked at function of just one variable. 01:08:54.840 --> 01:09:00.330 align:middle line:90% 01:09:00.330 --> 01:09:02.950 align:middle line:84% So this is what the ordinary referred to. 01:09:02.950 --> 01:09:05.569 align:middle line:90% 01:09:05.569 --> 01:09:13.200 align:middle line:84% So we want to look at ordinary differential equation. 01:09:13.200 --> 01:09:17.810 align:middle line:90% 01:09:17.810 --> 01:09:23.210 align:middle line:84% And an ordinary differential equation is abbreviated by ODE. 01:09:23.210 --> 01:09:26.250 align:middle line:90% 01:09:26.250 --> 01:09:28.250 align:middle line:84% And if it's not an ordinary, then it's 01:09:28.250 --> 01:09:32.630 align:middle line:84% usually a partial differential equation, 01:09:32.630 --> 01:09:38.350 align:middle line:84% which involve a function that is of more than one variable. 01:09:38.350 --> 01:09:42.050 align:middle line:84% So it will involve what's called the partial derivative 01:09:42.050 --> 01:09:42.890 align:middle line:90% of the functions. 01:09:42.890 --> 01:09:48.930 align:middle line:90% 01:09:48.930 --> 01:10:08.770 align:middle line:84% So the unknown function will be denoted by y. 01:10:08.770 --> 01:10:11.290 align:middle line:84% And the equation that we're looking at 01:10:11.290 --> 01:10:19.210 align:middle line:84% is that we want some particular value at, say, 0. 01:10:19.210 --> 01:10:25.610 align:middle line:84% And then we want the derivative satisfying an equation. 01:10:25.610 --> 01:10:30.670 align:middle line:84% And the equation that we're looking for is that we have-- 01:10:30.670 --> 01:10:34.730 align:middle line:90% 01:10:34.730 --> 01:10:38.750 align:middle line:90% so f here and g are given. 01:10:38.750 --> 01:10:43.430 align:middle line:84% So think about those as functions that are given. 01:10:43.430 --> 01:10:46.090 align:middle line:90% Those are two given functions. 01:10:46.090 --> 01:10:48.770 align:middle line:90% But they may not be super nice. 01:10:48.770 --> 01:10:53.210 align:middle line:84% But you know that f is differentiable 01:10:53.210 --> 01:11:00.590 align:middle line:84% and the derivative is continuous and g here is continuous. 01:11:00.590 --> 01:11:03.290 align:middle line:90% 01:11:03.290 --> 01:11:08.010 align:middle line:84% But in principle, you know they are given. 01:11:08.010 --> 01:11:09.930 align:middle line:90% And now you're looking for-- 01:11:09.930 --> 01:11:11.590 align:middle line:84% but it may be very, very complicated. 01:11:11.590 --> 01:11:16.760 align:middle line:84% Maybe f is sine or something like that. 01:11:16.760 --> 01:11:21.720 align:middle line:84% And so you're looking for an unknown function y 01:11:21.720 --> 01:11:24.240 align:middle line:84% so that the derivative-- so you think 01:11:24.240 --> 01:11:27.840 align:middle line:90% about y as a function of x. 01:11:27.840 --> 01:11:31.480 align:middle line:84% And the derivative of y is supposed 01:11:31.480 --> 01:11:36.960 align:middle line:84% to be equal to f but is now applied to this unknown function 01:11:36.960 --> 01:11:40.240 align:middle line:90% and then plus g of x. 01:11:40.240 --> 01:11:43.560 align:middle line:84% So this here is much more tame, but this is a bit complicated 01:11:43.560 --> 01:11:47.320 align:middle line:84% because it involves the unknown function. 01:11:47.320 --> 01:11:52.160 align:middle line:90% And so we want to see-- 01:11:52.160 --> 01:12:08.290 align:middle line:84% so we are asking the question, does this ODE have a solution? 01:12:08.290 --> 01:12:13.480 align:middle line:90% 01:12:13.480 --> 01:12:24.410 align:middle line:90% And is the solution unique? 01:12:24.410 --> 01:12:30.300 align:middle line:90% 01:12:30.300 --> 01:12:32.920 align:middle line:90% So we're looking for a function. 01:12:32.920 --> 01:12:35.580 align:middle line:84% So here's what we're looking for. 01:12:35.580 --> 01:12:40.160 align:middle line:90% 01:12:40.160 --> 01:12:46.600 align:middle line:84% We're looking for a function y defined on some interval into R. 01:12:46.600 --> 01:12:49.300 align:middle line:84% This function has to be differentiable. 01:12:49.300 --> 01:12:52.920 align:middle line:90% So this has to exist. 01:12:52.920 --> 01:13:02.080 align:middle line:84% A 0 has to be in that interval because a is just a constant. 01:13:02.080 --> 01:13:17.250 align:middle line:84% And then we want, obviously, y to satisfy this equation? 01:13:17.250 --> 01:13:22.560 align:middle line:90% 01:13:22.560 --> 01:13:24.560 align:middle line:90% That's what we're looking for. 01:13:24.560 --> 01:13:26.560 align:middle line:90% So this is existence. 01:13:26.560 --> 01:13:29.000 align:middle line:84% The thing is that what we will prove 01:13:29.000 --> 01:13:33.940 align:middle line:84% is that there exists some interval-- 01:13:33.940 --> 01:13:35.540 align:middle line:90% so we will prove the following. 01:13:35.540 --> 01:13:39.400 align:middle line:90% 01:13:39.400 --> 01:13:42.400 align:middle line:84% So we prove the following theorem, 01:13:42.400 --> 01:13:45.280 align:middle line:84% which has many different names, and it is 01:13:45.280 --> 01:13:46.730 align:middle line:90% attributed to various people. 01:13:46.730 --> 01:13:50.360 align:middle line:90% 01:13:50.360 --> 01:13:52.220 align:middle line:90% But it's I guess. 01:13:52.220 --> 01:13:57.200 align:middle line:90% 01:13:57.200 --> 01:14:08.080 align:middle line:84% it's most often named the Picard-Lindelöf theorem. 01:14:08.080 --> 01:14:18.920 align:middle line:84% And so this is that given f and g as above, 01:14:18.920 --> 01:14:23.840 align:middle line:84% so f has to be differentiable and the derivative continuous g 01:14:23.840 --> 01:14:25.300 align:middle line:90% just has to be continuous. 01:14:25.300 --> 01:14:28.160 align:middle line:90% 01:14:28.160 --> 01:14:43.880 align:middle line:84% Then for any a, there exists a delta greater than 0 such that-- 01:14:43.880 --> 01:14:47.360 align:middle line:90% 01:14:47.360 --> 01:14:54.480 align:middle line:84% and a solution to this differential equation 01:14:54.480 --> 01:15:05.320 align:middle line:84% is ODE on the interval from minus delta to delta. 01:15:05.320 --> 01:15:12.440 align:middle line:84% So we proved that there exists a solution near the origin. 01:15:12.440 --> 01:15:15.960 align:middle line:84% The thing is that a solution may blow up. 01:15:15.960 --> 01:15:19.580 align:middle line:84% So it may not exist on a very long interval. 01:15:19.580 --> 01:15:21.540 align:middle line:84% It may just exist on a small interval. 01:15:21.540 --> 01:15:25.200 align:middle line:90% It's all depending on f and g. 01:15:25.200 --> 01:15:28.880 align:middle line:84% But in particular, will depend on f. 01:15:28.880 --> 01:15:42.000 align:middle line:84% And that moreover, it is unique, meaning 01:15:42.000 --> 01:15:45.680 align:middle line:84% that if you take another solution 01:15:45.680 --> 01:15:50.720 align:middle line:84% defined on the same interval, then it must be that solution. 01:15:50.720 --> 01:16:00.080 align:middle line:84% And so now I just have a minute to-- but we will 01:16:00.080 --> 01:16:02.120 align:middle line:90% prove this next time anyway. 01:16:02.120 --> 01:16:07.780 align:middle line:84% But I just want to, as a teaser, if you will, 01:16:07.780 --> 01:16:19.480 align:middle line:84% I will just say that the proof of this 01:16:19.480 --> 01:16:25.260 align:middle line:84% uses the contraction mapping theorem. 01:16:25.260 --> 01:16:35.460 align:middle line:90% 01:16:35.460 --> 01:16:38.480 align:middle line:84% Maybe I will remind you what it is. 01:16:38.480 --> 01:16:53.160 align:middle line:84% So recall that the contraction mapping theorem 01:16:53.160 --> 01:16:54.671 align:middle line:90% is the following-- 01:16:54.671 --> 01:16:58.280 align:middle line:90% 01:16:58.280 --> 01:17:03.560 align:middle line:84% that suppose you take a metric space x and d-- 01:17:03.560 --> 01:17:07.800 align:middle line:84% so D is the metric, x is the space-- 01:17:07.800 --> 01:17:12.030 align:middle line:84% and suppose you take a map from x to itself. 01:17:12.030 --> 01:17:15.200 align:middle line:90% 01:17:15.200 --> 01:17:18.440 align:middle line:84% So then T is a map from x to itself. 01:17:18.440 --> 01:17:25.590 align:middle line:84% T is said to be a contracting map. 01:17:25.590 --> 01:17:32.520 align:middle line:90% 01:17:32.520 --> 01:17:37.560 align:middle line:84% If there exists some constant, that constant-- 01:17:37.560 --> 01:17:38.720 align:middle line:90% that's key. 01:17:38.720 --> 01:17:42.040 align:middle line:84% It has to be strictly less than 1 01:17:42.040 --> 01:17:47.800 align:middle line:84% so that if you measure distances between the images, 01:17:47.800 --> 01:17:52.560 align:middle line:84% so for any x and y in capital X-- 01:17:52.560 --> 01:17:56.440 align:middle line:84% this is for all x and y in capital 01:17:56.440 --> 01:18:01.440 align:middle line:84% X. If you look at the distance between the images of these two 01:18:01.440 --> 01:18:06.600 align:middle line:84% points, then it's less than or equal to this constant that 01:18:06.600 --> 01:18:12.200 align:middle line:84% is strictly less than 1, like this. 01:18:12.200 --> 01:18:15.760 align:middle line:90% So that's a contracting map. 01:18:15.760 --> 01:18:17.920 align:middle line:84% And the contracting mapping theorem 01:18:17.920 --> 01:18:21.200 align:middle line:84% that we prove, I'll remind you about it again next time, 01:18:21.200 --> 01:18:25.120 align:middle line:84% but just the contracting mapping theorem 01:18:25.120 --> 01:18:27.760 align:middle line:84% that we proved is that-- so we proved 01:18:27.760 --> 01:18:30.200 align:middle line:84% the following theorem, which is a contracting mapping 01:18:30.200 --> 01:18:43.520 align:middle line:84% theorem, that if x here is Cauchy complete, 01:18:43.520 --> 01:19:00.000 align:middle line:84% and T from x to x is a contracting map, 01:19:00.000 --> 01:19:05.770 align:middle line:90% then T has a unique. 01:19:05.770 --> 01:19:08.360 align:middle line:90% 01:19:08.360 --> 01:19:09.220 align:middle line:90% fixed point. 01:19:09.220 --> 01:19:14.940 align:middle line:90% 01:19:14.940 --> 01:19:18.980 align:middle line:84% So there exists a unique one-- often write like this-- 01:19:18.980 --> 01:19:31.040 align:middle line:84% there exists one and only one x0 so that this map takes x0 01:19:31.040 --> 01:19:31.820 align:middle line:90% to itself. 01:19:31.820 --> 01:19:34.520 align:middle line:84% So that's a contraction mapping theorem. 01:19:34.520 --> 01:19:39.800 align:middle line:84% And so it seems like what does ODE 01:19:39.800 --> 01:19:42.920 align:middle line:84% has to do with them, this abstract concept 01:19:42.920 --> 01:19:44.760 align:middle line:90% of a metric space? 01:19:44.760 --> 01:19:48.400 align:middle line:84% So it's tied together using this. 01:19:48.400 --> 01:19:52.320 align:middle line:90% And then the space x, it's not-- 01:19:52.320 --> 01:19:55.920 align:middle line:90% 01:19:55.920 --> 01:19:59.000 align:middle line:84% you may think that that's close, but it's not quite the case. 01:19:59.000 --> 01:20:01.960 align:middle line:84% You may think that it's like a continuous function 01:20:01.960 --> 01:20:04.280 align:middle line:90% on an interval like this. 01:20:04.280 --> 01:20:07.680 align:middle line:90% That's close. 01:20:07.680 --> 01:20:09.280 align:middle line:84% You may think that because that's 01:20:09.280 --> 01:20:11.640 align:middle line:84% a space we've talked about a lot, this 01:20:11.640 --> 01:20:13.320 align:middle line:90% here, with this metric d. 01:20:13.320 --> 01:20:15.920 align:middle line:84% We talked about with this maximum distance. 01:20:15.920 --> 01:20:17.860 align:middle line:84% It's not actually quite the case. 01:20:17.860 --> 01:20:22.200 align:middle line:84% It's a little bit more complicated than this. 01:20:22.200 --> 01:20:25.360 align:middle line:84% But x is related to this because x here 01:20:25.360 --> 01:20:33.680 align:middle line:84% will be a ball around the function that is constant, 01:20:33.680 --> 01:20:39.200 align:middle line:84% the function that is constant a in this space. 01:20:39.200 --> 01:20:44.080 align:middle line:84% And we have to choose this R carefully. 01:20:44.080 --> 01:20:46.600 align:middle line:90% And so that's a space x. 01:20:46.600 --> 01:20:49.940 align:middle line:84% And now I haven't defined the function the map g. 01:20:49.940 --> 01:20:51.500 align:middle line:90% But we'll do that next time. 01:20:51.500 --> 01:20:54.080 align:middle line:84% And I will remind you about this also. 01:20:54.080 --> 01:20:56.310 align:middle line:90% Any questions? 01:20:56.310 --> 01:21:02.000 align:middle line:90%