WEBVTT 00:00:00.000 --> 00:00:01.924 align:middle line:90% [SQUEAKING] 00:00:01.924 --> 00:00:03.367 align:middle line:90% [RUSTLING] 00:00:03.367 --> 00:00:04.810 align:middle line:90% [CLICKING] 00:00:04.810 --> 00:00:11.648 align:middle line:90% 00:00:11.648 --> 00:00:12.440 align:middle line:90% TOBIAS COLDING: OK. 00:00:12.440 --> 00:00:17.060 align:middle line:84% So first, let me look at a curve and plane. 00:00:17.060 --> 00:00:20.980 align:middle line:84% Talked a little bit about this last time at the very end. 00:00:20.980 --> 00:00:27.800 align:middle line:90% 00:00:27.800 --> 00:00:29.720 align:middle line:90% Oops. 00:00:29.720 --> 00:00:30.220 align:middle line:90% OK. 00:00:30.220 --> 00:00:34.720 align:middle line:84% So let's first talk about curves in the plane. 00:00:34.720 --> 00:00:42.880 align:middle line:84% So we have functions, f and g, on some interval, 00:00:42.880 --> 00:00:51.480 align:middle line:84% say, from a to b into R. And I'm assuming that f 00:00:51.480 --> 00:01:08.410 align:middle line:84% and g are differentiable and that the derivative 00:01:08.410 --> 00:01:09.470 align:middle line:90% are continuous. 00:01:09.470 --> 00:01:16.055 align:middle line:90% 00:01:16.055 --> 00:01:18.530 align:middle line:90% And then I'm thinking about-- 00:01:18.530 --> 00:01:20.430 align:middle line:84% so I'm thinking about the curves. 00:01:20.430 --> 00:01:23.230 align:middle line:90% So gamma here of s is a curve. 00:01:23.230 --> 00:01:26.675 align:middle line:90% 00:01:26.675 --> 00:01:32.910 align:middle line:84% And it's given by the gamma of s is f of s comma g of s. 00:01:32.910 --> 00:01:35.650 align:middle line:90% 00:01:35.650 --> 00:01:38.870 align:middle line:84% And I want to look at the length of this curve. 00:01:38.870 --> 00:01:42.790 align:middle line:84% So s here is running in the interval from a to b. 00:01:42.790 --> 00:01:45.730 align:middle line:90% 00:01:45.730 --> 00:01:47.490 align:middle line:90% And so the length-- 00:01:47.490 --> 00:01:50.310 align:middle line:84% and this is the definition of the length, 00:01:50.310 --> 00:01:52.610 align:middle line:90% but then we will justify it. 00:01:52.610 --> 00:01:59.740 align:middle line:90% So the length of gamma. 00:01:59.740 --> 00:02:03.070 align:middle line:90% 00:02:03.070 --> 00:02:08.470 align:middle line:84% So L is equal to the integral from a to b 00:02:08.470 --> 00:02:18.430 align:middle line:84% of square root of f prime of s squared plus g prime of s 00:02:18.430 --> 00:02:22.750 align:middle line:90% squared ds. 00:02:22.750 --> 00:02:25.230 align:middle line:90% Now, why is this the case? 00:02:25.230 --> 00:02:31.030 align:middle line:84% So why is this a sensible definition? 00:02:31.030 --> 00:02:38.750 align:middle line:84% So suppose that this thing here is f of a comma g of a. 00:02:38.750 --> 00:02:43.590 align:middle line:84% So this is the starting point of the curve. 00:02:43.590 --> 00:02:46.290 align:middle line:84% And suppose that the curve looks like this. 00:02:46.290 --> 00:02:48.660 align:middle line:90% So here is the endpoint. 00:02:48.660 --> 00:02:51.550 align:middle line:90% 00:02:51.550 --> 00:02:54.830 align:middle line:84% And now imagine that you take the interval 00:02:54.830 --> 00:03:01.320 align:middle line:84% from a to b and you partition it into a lot of small-- 00:03:01.320 --> 00:03:05.840 align:middle line:84% so you have this partition of this interval. 00:03:05.840 --> 00:03:09.140 align:middle line:90% And then you're looking at-- 00:03:09.140 --> 00:03:11.120 align:middle line:90% so we have the curve here. 00:03:11.120 --> 00:03:14.520 align:middle line:90% This is maybe xi minus 1. 00:03:14.520 --> 00:03:18.080 align:middle line:90% And here I have xi. 00:03:18.080 --> 00:03:24.640 align:middle line:84% And under the map gamma, this little interval 00:03:24.640 --> 00:03:29.410 align:middle line:84% here is mapped maybe into this piece of the curve. 00:03:29.410 --> 00:03:33.990 align:middle line:90% 00:03:33.990 --> 00:03:37.640 align:middle line:84% Now, this piece of the curve, I will then 00:03:37.640 --> 00:03:42.240 align:middle line:84% approximate it by the line segment from here to here. 00:03:42.240 --> 00:03:48.020 align:middle line:84% So if you do that, then it means that you are looking at-- 00:03:48.020 --> 00:03:51.660 align:middle line:84% so you have that you're looking-- 00:03:51.660 --> 00:03:53.360 align:middle line:90% so you have that-- 00:03:53.360 --> 00:03:57.020 align:middle line:90% so here you have xi minus 1. 00:03:57.020 --> 00:03:59.380 align:middle line:90% Here you have xi. 00:03:59.380 --> 00:04:06.500 align:middle line:84% And you're looking at the curve on this little interval. 00:04:06.500 --> 00:04:11.700 align:middle line:90% And there the curve goes from-- 00:04:11.700 --> 00:04:17.519 align:middle line:90% so the initial point is this. 00:04:17.519 --> 00:04:20.220 align:middle line:90% That's the image of that point. 00:04:20.220 --> 00:04:25.620 align:middle line:84% And the image of this is gamma of xi. 00:04:25.620 --> 00:04:27.060 align:middle line:90% Wow. 00:04:27.060 --> 00:04:31.670 align:middle line:84% And so the line segment from here to here-- 00:04:31.670 --> 00:04:34.660 align:middle line:90% 00:04:34.660 --> 00:04:37.300 align:middle line:84% so the length of this line segment 00:04:37.300 --> 00:04:40.380 align:middle line:90% if you have this point here-- 00:04:40.380 --> 00:04:47.860 align:middle line:84% so this is gamma of xi minus 1, and this is gamma of xi. 00:04:47.860 --> 00:04:54.410 align:middle line:84% Then the length here is the vector here. 00:04:54.410 --> 00:05:00.670 align:middle line:84% So gamma of xi minus gamma of xi minus 1. 00:05:00.670 --> 00:05:02.630 align:middle line:90% The length of this. 00:05:02.630 --> 00:05:06.490 align:middle line:90% But this thing here is-- 00:05:06.490 --> 00:05:10.110 align:middle line:90% this is the point. 00:05:10.110 --> 00:05:10.630 align:middle line:90% Sorry. 00:05:10.630 --> 00:05:11.130 align:middle line:90% Yeah. 00:05:11.130 --> 00:05:16.070 align:middle line:90% 00:05:16.070 --> 00:05:18.620 align:middle line:90% And the beginning point is this. 00:05:18.620 --> 00:05:28.150 align:middle line:90% 00:05:28.150 --> 00:05:31.910 align:middle line:84% So this here-- so it's the length of this. 00:05:31.910 --> 00:05:35.030 align:middle line:84% But the length of this by Pythagoras 00:05:35.030 --> 00:05:40.450 align:middle line:84% is the square of the coordinates. 00:05:40.450 --> 00:05:46.670 align:middle line:90% 00:05:46.670 --> 00:05:49.590 align:middle line:84% And then taking the square root of that. 00:05:49.590 --> 00:05:54.140 align:middle line:84% So this is what the length of the line segment here is. 00:05:54.140 --> 00:06:02.680 align:middle line:90% 00:06:02.680 --> 00:06:05.880 align:middle line:84% But now, the mean value theorem tells you 00:06:05.880 --> 00:06:10.200 align:middle line:84% that this thing here is essentially 00:06:10.200 --> 00:06:15.060 align:middle line:84% the derivative at some point in between these two. 00:06:15.060 --> 00:06:18.320 align:middle line:90% 00:06:18.320 --> 00:06:22.280 align:middle line:84% It actually is equal to, if you will, 00:06:22.280 --> 00:06:27.785 align:middle line:84% the derivative at some intermediate point 00:06:27.785 --> 00:06:32.440 align:middle line:84% times the length of that little interval. 00:06:32.440 --> 00:06:34.680 align:middle line:90% This is this difference here. 00:06:34.680 --> 00:06:41.240 align:middle line:84% And likewise-- so this is x, like this, 00:06:41.240 --> 00:06:42.920 align:middle line:90% some intermediate point. 00:06:42.920 --> 00:06:48.080 align:middle line:84% And likewise for this, there's some intermediate point 00:06:48.080 --> 00:06:51.860 align:middle line:90% times the length here. 00:06:51.860 --> 00:06:54.500 align:middle line:84% So this means that this little line segment 00:06:54.500 --> 00:07:06.020 align:middle line:84% here, the length of this little line segment is equal to-- 00:07:06.020 --> 00:07:09.700 align:middle line:84% and you see that when you take this thing here 00:07:09.700 --> 00:07:13.600 align:middle line:84% and you square it, then you can factor out. 00:07:13.600 --> 00:07:15.500 align:middle line:90% So let me just write it out. 00:07:15.500 --> 00:07:17.000 align:middle line:90% So you take this thing here. 00:07:17.000 --> 00:07:27.500 align:middle line:84% So it's f prime, like this, squared, delta xi squared plus, 00:07:27.500 --> 00:07:37.260 align:middle line:84% and then g prime squared delta xi squared. 00:07:37.260 --> 00:07:39.840 align:middle line:90% But now you can factor this out. 00:07:39.840 --> 00:07:48.260 align:middle line:84% So you can write this as square root of delta xi squared, 00:07:48.260 --> 00:07:55.510 align:middle line:84% and then f prime, like this, squared plus g 00:07:55.510 --> 00:08:01.030 align:middle line:84% prime in this intermediate point squared. 00:08:01.030 --> 00:08:03.470 align:middle line:84% And this here factors out both of them. 00:08:03.470 --> 00:08:06.890 align:middle line:84% And so now you can just-- when you take the square root, 00:08:06.890 --> 00:08:08.750 align:middle line:90% it actually goes out as-- 00:08:08.750 --> 00:08:10.830 align:middle line:90% this is anyway positive. 00:08:10.830 --> 00:08:12.720 align:middle line:90% So it becomes like this. 00:08:12.720 --> 00:08:17.510 align:middle line:90% 00:08:17.510 --> 00:08:27.430 align:middle line:84% And now the functions g prime and f prime are both continuous. 00:08:27.430 --> 00:08:30.790 align:middle line:84% So this thing here is roughly what 00:08:30.790 --> 00:08:33.390 align:middle line:90% it is at one of the endpoints. 00:08:33.390 --> 00:08:39.770 align:middle line:84% So this here is roughly equal to, like this. 00:08:39.770 --> 00:08:42.110 align:middle line:90% And then it's like one endpoint. 00:08:42.110 --> 00:08:43.930 align:middle line:90% Or you can even take the-- 00:08:43.930 --> 00:08:44.430 align:middle line:90% yeah. 00:08:44.430 --> 00:08:45.980 align:middle line:90% So it's like an endpoint. 00:08:45.980 --> 00:08:52.650 align:middle line:90% 00:08:52.650 --> 00:08:55.710 align:middle line:84% And so you see that this thing here now, when you sum up-- 00:08:55.710 --> 00:08:58.930 align:middle line:90% 00:08:58.930 --> 00:09:02.690 align:middle line:84% when you sum up all of these over this partition 00:09:02.690 --> 00:09:09.770 align:middle line:84% and you let the partition-- the delta xi goes to 0, 00:09:09.770 --> 00:09:13.890 align:middle line:84% then this here is going to converge to the integral 00:09:13.890 --> 00:09:24.010 align:middle line:84% from a to b of the square root of f prime of s squared plus g 00:09:24.010 --> 00:09:29.790 align:middle line:90% prime of s squared ds. 00:09:29.790 --> 00:09:31.390 align:middle line:90% So that's why. 00:09:31.390 --> 00:09:34.130 align:middle line:90% So you take this little curve. 00:09:34.130 --> 00:09:35.310 align:middle line:90% It's parameterized. 00:09:35.310 --> 00:09:37.930 align:middle line:84% So you take the curve parameterized by a to b. 00:09:37.930 --> 00:09:42.770 align:middle line:90% You're looking at a partition. 00:09:42.770 --> 00:09:47.420 align:middle line:84% And then, on each of these little intervals, 00:09:47.420 --> 00:09:53.420 align:middle line:84% you approximate the curve by the line segment. 00:09:53.420 --> 00:09:57.260 align:middle line:84% And then you let the-- as you let this refinement-- 00:09:57.260 --> 00:10:03.700 align:middle line:84% the length of these individual intervals goes to 0, 00:10:03.700 --> 00:10:08.280 align:middle line:84% then actually this approximation will have a limit. 00:10:08.280 --> 00:10:11.200 align:middle line:90% And that limit is this integral. 00:10:11.200 --> 00:10:16.820 align:middle line:84% So that's why we define it to be the arc length. 00:10:16.820 --> 00:10:18.020 align:middle line:90% OK. 00:10:18.020 --> 00:10:19.820 align:middle line:90% So that's arc length. 00:10:19.820 --> 00:10:22.330 align:middle line:84% And so let's look at an example of this. 00:10:22.330 --> 00:10:29.140 align:middle line:90% 00:10:29.140 --> 00:10:31.080 align:middle line:90% So we'll look at two examples. 00:10:31.080 --> 00:10:36.500 align:middle line:84% Let me first look at the case where f of s is equal to s 00:10:36.500 --> 00:10:44.940 align:middle line:84% and g of s is equal to s squared. 00:10:44.940 --> 00:10:46.380 align:middle line:90% Then I have that. 00:10:46.380 --> 00:10:48.440 align:middle line:90% So then the curve is-- 00:10:48.440 --> 00:10:54.160 align:middle line:84% so the curve gamma of s is s comma s squared. 00:10:54.160 --> 00:11:03.800 align:middle line:84% And f prime of s is equal to 1 and g prime of s is equal to 2s. 00:11:03.800 --> 00:11:06.520 align:middle line:90% And so this means that-- 00:11:06.520 --> 00:11:09.040 align:middle line:84% and let's say that it's parameterized, 00:11:09.040 --> 00:11:13.120 align:middle line:90% s here, from 0 to 1. 00:11:13.120 --> 00:11:15.920 align:middle line:84% So then the length here of this curve 00:11:15.920 --> 00:11:20.800 align:middle line:84% is the integral from 0 to 1 and then square root. 00:11:20.800 --> 00:11:23.780 align:middle line:84% And then you have to take this thing here squared. 00:11:23.780 --> 00:11:31.040 align:middle line:84% So that's 1 plus this here, 2s squared, ds. 00:11:31.040 --> 00:11:35.920 align:middle line:84% So this is-- and it's this integral here, 00:11:35.920 --> 00:11:39.640 align:middle line:90% 1 plus 4s squared. 00:11:39.640 --> 00:11:41.880 align:middle line:90% So that's the arc length. 00:11:41.880 --> 00:11:44.850 align:middle line:90% So it's given us this integral. 00:11:44.850 --> 00:11:51.650 align:middle line:84% The second example I want to look at for arc length 00:11:51.650 --> 00:11:56.670 align:middle line:90% is if you have the unit circle. 00:11:56.670 --> 00:11:58.760 align:middle line:84% So suppose that you have the unit circle. 00:11:58.760 --> 00:12:03.370 align:middle line:90% 00:12:03.370 --> 00:12:06.010 align:middle line:90% So here you have one. 00:12:06.010 --> 00:12:08.970 align:middle line:84% So this is supposed to be the unit circle. 00:12:08.970 --> 00:12:15.510 align:middle line:84% So it's x squared plus y squared equal to 1. 00:12:15.510 --> 00:12:18.020 align:middle line:84% It's a unit circle centered at the origin. 00:12:18.020 --> 00:12:20.610 align:middle line:90% 00:12:20.610 --> 00:12:26.170 align:middle line:84% Now, this means that if you take a point, 00:12:26.170 --> 00:12:31.450 align:middle line:84% then y here is-- so this here means 00:12:31.450 --> 00:12:36.090 align:middle line:84% that y is given in terms of x, where that y squared 00:12:36.090 --> 00:12:40.490 align:middle line:90% is equal to 1 minus x squared. 00:12:40.490 --> 00:12:44.940 align:middle line:84% And so if you assuming that you are in this upper, 00:12:44.940 --> 00:12:51.300 align:middle line:84% where y is positive-- so if y here is positive-- 00:12:51.300 --> 00:12:55.980 align:middle line:84% then you're looking at y equal to square root of 1 00:12:55.980 --> 00:12:58.420 align:middle line:90% minus x squared. 00:12:58.420 --> 00:13:06.500 align:middle line:84% So in this part here, where y is positive, you have the curve-- 00:13:06.500 --> 00:13:12.380 align:middle line:84% this part here of the unit circle. 00:13:12.380 --> 00:13:16.580 align:middle line:84% This one here is given by the parametrization 00:13:16.580 --> 00:13:23.140 align:middle line:84% gamma of s equal to s, and then square root of 1 00:13:23.140 --> 00:13:26.660 align:middle line:90% minus s squared. 00:13:26.660 --> 00:13:31.060 align:middle line:90% And s here is now running-- 00:13:31.060 --> 00:13:32.600 align:middle line:90% s is the first coordinate. 00:13:32.600 --> 00:13:35.460 align:middle line:90% So it's between minus 1 and 1. 00:13:35.460 --> 00:13:40.900 align:middle line:84% So s here is between minus 1 and 1. 00:13:40.900 --> 00:13:42.600 align:middle line:90% OK. 00:13:42.600 --> 00:13:48.920 align:middle line:84% Now suppose that you take some x here. 00:13:48.920 --> 00:13:51.760 align:middle line:84% And suppose that you want to look at, 00:13:51.760 --> 00:13:55.520 align:middle line:84% what is the arc length of this part of the circle? 00:13:55.520 --> 00:14:00.960 align:middle line:84% Well, then, in this case, if you're 00:14:00.960 --> 00:14:04.020 align:middle line:84% looking at the arc length of this part of the circle-- 00:14:04.020 --> 00:14:10.520 align:middle line:84% let's call that arc length theta. 00:14:10.520 --> 00:14:15.380 align:middle line:84% So theta here is equal to this part. 00:14:15.380 --> 00:14:20.040 align:middle line:84% It's not parameterized like this. 00:14:20.040 --> 00:14:21.620 align:middle line:90% But x here is-- 00:14:21.620 --> 00:14:23.560 align:middle line:90% so this here is-- 00:14:23.560 --> 00:14:28.680 align:middle line:90% so length of gamma. 00:14:28.680 --> 00:14:35.640 align:middle line:84% But now gamma-- by now, s here is running from x to 1. 00:14:35.640 --> 00:14:39.360 align:middle line:84% Because x is the first coordinate 00:14:39.360 --> 00:14:42.010 align:middle line:90% and it's running from-- 00:14:42.010 --> 00:14:45.970 align:middle line:84% the first coordinate is running from x, from this point, this 00:14:45.970 --> 00:14:47.955 align:middle line:90% here, to 1. 00:14:47.955 --> 00:14:51.810 align:middle line:90% 00:14:51.810 --> 00:14:56.690 align:middle line:84% And so now, let's try to calculate this length, 00:14:56.690 --> 00:14:59.850 align:middle line:84% or at least write it up as an integral. 00:14:59.850 --> 00:15:05.230 align:middle line:84% So if we write it up as an integral, we have that-- 00:15:05.230 --> 00:15:10.010 align:middle line:84% so in this case-- so gamma s is equal to s square root 00:15:10.010 --> 00:15:13.250 align:middle line:90% of 1 minus s squared. 00:15:13.250 --> 00:15:19.690 align:middle line:84% And so if we define f of s equal to s and g of s 00:15:19.690 --> 00:15:24.410 align:middle line:84% equal to square root of 1 minus s squared, 00:15:24.410 --> 00:15:27.530 align:middle line:90% then we take f prime-- 00:15:27.530 --> 00:15:30.610 align:middle line:90% well, that's equal to 1-- 00:15:30.610 --> 00:15:34.150 align:middle line:90% and g prime. 00:15:34.150 --> 00:15:36.530 align:middle line:90% So then we use the chain rule. 00:15:36.530 --> 00:15:41.910 align:middle line:84% So this is equal to 1 over 2 square root 00:15:41.910 --> 00:15:43.930 align:middle line:90% of 1 minus s squared. 00:15:43.930 --> 00:15:47.110 align:middle line:84% And then we have to take the derivative of this also. 00:15:47.110 --> 00:15:51.750 align:middle line:84% So this is times minus 2s, like that. 00:15:51.750 --> 00:15:55.270 align:middle line:90% 00:15:55.270 --> 00:15:58.470 align:middle line:84% And so this here is, of course, then equal to. 00:15:58.470 --> 00:15:59.990 align:middle line:90% The 2 cancels. 00:15:59.990 --> 00:16:03.350 align:middle line:84% And you're getting minus s over square root 00:16:03.350 --> 00:16:04.790 align:middle line:90% of 1 minus s squared. 00:16:04.790 --> 00:16:09.070 align:middle line:90% 00:16:09.070 --> 00:16:12.210 align:middle line:84% And so now, if you're looking at-- 00:16:12.210 --> 00:16:18.290 align:middle line:84% so f prime squared plus g prime squared, 00:16:18.290 --> 00:16:20.690 align:middle line:90% this is equal to f is 1. 00:16:20.690 --> 00:16:24.390 align:middle line:84% So this is 1 plus this thing here squared. 00:16:24.390 --> 00:16:26.590 align:middle line:90% So this is s squared. 00:16:26.590 --> 00:16:30.370 align:middle line:84% And then the denominator squared is just, of course, 00:16:30.370 --> 00:16:33.670 align:middle line:90% 1 minus s squared. 00:16:33.670 --> 00:16:40.840 align:middle line:84% And we can now put it with a common denominator. 00:16:40.840 --> 00:16:44.600 align:middle line:84% And so you see that this is 1 minus s squared. 00:16:44.600 --> 00:16:45.980 align:middle line:90% 1 minus s squared. 00:16:45.980 --> 00:16:48.160 align:middle line:90% That's how you write 1. 00:16:48.160 --> 00:16:52.420 align:middle line:84% And then plus s squared 1 minus s squared. 00:16:52.420 --> 00:16:56.280 align:middle line:84% And so you see that this whole thing is the s squared 00:16:56.280 --> 00:16:59.480 align:middle line:90% cancel in the denominator. 00:16:59.480 --> 00:17:03.779 align:middle line:84% And so you're getting 1 over 1 minus s squared. 00:17:03.779 --> 00:17:06.960 align:middle line:90% 00:17:06.960 --> 00:17:11.599 align:middle line:84% And so this means that the integral-- 00:17:11.599 --> 00:17:13.500 align:middle line:84% so this means that the integral-- 00:17:13.500 --> 00:17:17.720 align:middle line:90% 00:17:17.720 --> 00:17:19.180 align:middle line:90% so this length here-- 00:17:19.180 --> 00:17:22.280 align:middle line:90% 00:17:22.280 --> 00:17:25.579 align:middle line:90% and the length is-- 00:17:25.579 --> 00:17:29.720 align:middle line:84% so actually, we call the length theta. 00:17:29.720 --> 00:17:33.000 align:middle line:84% And it will be clear in a minute why I do this. 00:17:33.000 --> 00:17:39.170 align:middle line:84% So the length here is then the integral from x to 1. 00:17:39.170 --> 00:17:45.830 align:middle line:84% And then it is of the square root. 00:17:45.830 --> 00:17:49.290 align:middle line:84% So it's supposed to be the square root of this thing here. 00:17:49.290 --> 00:17:51.770 align:middle line:90% So it's the square root. 00:17:51.770 --> 00:17:58.160 align:middle line:84% So if you will, it's 1 square root of 1 minus s squared ds. 00:17:58.160 --> 00:18:02.050 align:middle line:90% 00:18:02.050 --> 00:18:04.930 align:middle line:90% So that's the length. 00:18:04.930 --> 00:18:09.050 align:middle line:84% But if you think about it-- so you have here the unit 00:18:09.050 --> 00:18:11.170 align:middle line:90% circle again. 00:18:11.170 --> 00:18:16.450 align:middle line:84% What we now calculated was that we take some x here, 00:18:16.450 --> 00:18:22.330 align:middle line:84% and then we calculate the length of this piece here. 00:18:22.330 --> 00:18:24.930 align:middle line:90% Well, that piece is-- 00:18:24.930 --> 00:18:27.750 align:middle line:90% this here is also-- 00:18:27.750 --> 00:18:30.050 align:middle line:90% this is how you define. 00:18:30.050 --> 00:18:31.820 align:middle line:90% If you will, this is the angle. 00:18:31.820 --> 00:18:34.790 align:middle line:90% 00:18:34.790 --> 00:18:38.310 align:middle line:90% So theta is the angle. 00:18:38.310 --> 00:18:41.790 align:middle line:90% And so you see that-- 00:18:41.790 --> 00:18:47.510 align:middle line:90% so theta here is the angle. 00:18:47.510 --> 00:18:52.790 align:middle line:90% And x here-- x here is-- 00:18:52.790 --> 00:18:54.190 align:middle line:90% and this point here-- 00:18:54.190 --> 00:19:02.270 align:middle line:84% sorry, this point here is cosine to the angle comma 00:19:02.270 --> 00:19:04.770 align:middle line:90% sine to the angle. 00:19:04.770 --> 00:19:06.310 align:middle line:90% That's this piece. 00:19:06.310 --> 00:19:12.940 align:middle line:84% So this means that x is actually cosine through the angle. 00:19:12.940 --> 00:19:20.990 align:middle line:90% 00:19:20.990 --> 00:19:26.550 align:middle line:90% So this means that theta here-- 00:19:26.550 --> 00:19:33.370 align:middle line:84% in terms of x, theta is the inverse function. 00:19:33.370 --> 00:19:40.280 align:middle line:84% So it's what's called arc cosine to x. 00:19:40.280 --> 00:19:45.580 align:middle line:84% So this arccosine, it means that it's the inverse function. 00:19:45.580 --> 00:19:51.960 align:middle line:84% If you take the inverse function on both sides of this then-- 00:19:51.960 --> 00:19:54.600 align:middle line:84% so if you take the inverse function-- 00:19:54.600 --> 00:19:57.840 align:middle line:90% so arccosine. 00:19:57.840 --> 00:20:00.680 align:middle line:84% So you get arccosine to x on this side. 00:20:00.680 --> 00:20:05.280 align:middle line:84% This side, you're getting arccosine and then 00:20:05.280 --> 00:20:08.600 align:middle line:90% cosine to theta. 00:20:08.600 --> 00:20:13.400 align:middle line:84% But these two functions together gives you the identity. 00:20:13.400 --> 00:20:16.140 align:middle line:90% So this is just theta. 00:20:16.140 --> 00:20:16.640 align:middle line:90% OK. 00:20:16.640 --> 00:20:20.160 align:middle line:90% 00:20:20.160 --> 00:20:20.820 align:middle line:90% OK. 00:20:20.820 --> 00:20:28.400 align:middle line:84% So in other words, what you have is that arccosine to x 00:20:28.400 --> 00:20:33.940 align:middle line:84% is given as the integral from x to 1 1 00:20:33.940 --> 00:20:39.000 align:middle line:84% over square root of 1 minus s squared ds. 00:20:39.000 --> 00:20:44.200 align:middle line:84% So you have this formula here, which is not so surprising. 00:20:44.200 --> 00:20:46.780 align:middle line:84% But anyway, you can get it from this. 00:20:46.780 --> 00:20:50.980 align:middle line:84% And so now you can get it-- so now, the fundamental theorem 00:20:50.980 --> 00:20:52.040 align:middle line:90% of calculus. 00:20:52.040 --> 00:21:09.920 align:middle line:84% So by the fundamental theorem of calculus, then you have that. 00:21:09.920 --> 00:21:12.380 align:middle line:84% So if you have a function that is 00:21:12.380 --> 00:21:15.340 align:middle line:84% given like an integral like this, 00:21:15.340 --> 00:21:20.700 align:middle line:84% then this thing here is the derivative. 00:21:20.700 --> 00:21:23.200 align:middle line:84% That's what the fundamental theorem of calculus say. 00:21:23.200 --> 00:21:25.700 align:middle line:84% Except you have to be a little bit careful because this 00:21:25.700 --> 00:21:28.620 align:middle line:90% is if the variable is up here. 00:21:28.620 --> 00:21:30.870 align:middle line:90% But now it's down here. 00:21:30.870 --> 00:21:36.290 align:middle line:84% Because it's down here, the derivative is minus this. 00:21:36.290 --> 00:21:36.790 align:middle line:90% OK. 00:21:36.790 --> 00:21:38.510 align:middle line:90% Let me explain that. 00:21:38.510 --> 00:21:43.390 align:middle line:84% So you get from the fundamental theorem of calculus, 00:21:43.390 --> 00:21:47.510 align:middle line:84% the derivative of this function, I claim, is minus this thing 00:21:47.510 --> 00:21:48.010 align:middle line:90% here. 00:21:48.010 --> 00:21:54.180 align:middle line:90% 00:21:54.180 --> 00:21:57.030 align:middle line:84% And then this should be evaluated at x. 00:21:57.030 --> 00:22:03.510 align:middle line:84% This is the arccosine at x is equal to that. 00:22:03.510 --> 00:22:05.530 align:middle line:90% So now let's just see that. 00:22:05.530 --> 00:22:07.150 align:middle line:90% So why is this the case? 00:22:07.150 --> 00:22:16.910 align:middle line:84% So you take-- so remember that if you 00:22:16.910 --> 00:22:20.950 align:middle line:84% have a function f of x and this is 00:22:20.950 --> 00:22:27.310 align:middle line:84% given as you're integrating from some a to some x, a is fixed. 00:22:27.310 --> 00:22:33.640 align:middle line:84% And then you're integrating a function like this. 00:22:33.640 --> 00:22:35.510 align:middle line:84% Assuming that the function is nice, 00:22:35.510 --> 00:22:40.200 align:middle line:84% et cetera, then we have that the fundamental theorem 00:22:40.200 --> 00:22:42.360 align:middle line:90% of calculus-- 00:22:42.360 --> 00:22:44.640 align:middle line:84% fundamental theorem of calculus-- 00:22:44.640 --> 00:22:51.600 align:middle line:84% the derivative of this function is little f. 00:22:51.600 --> 00:22:53.880 align:middle line:90% But suppose I now switch. 00:22:53.880 --> 00:22:57.880 align:middle line:84% So now suppose I now look-- so I don't look at this, 00:22:57.880 --> 00:23:00.500 align:middle line:90% but I look at-- 00:23:00.500 --> 00:23:09.760 align:middle line:84% suppose that g of x is the integral from x to a-- 00:23:09.760 --> 00:23:12.280 align:middle line:84% let's call it x to b, just because we usually 00:23:12.280 --> 00:23:20.200 align:middle line:84% have the endpoint as the last point, as b. 00:23:20.200 --> 00:23:23.480 align:middle line:84% So suppose that the function is like this. 00:23:23.480 --> 00:23:25.980 align:middle line:90% What is the derivative? 00:23:25.980 --> 00:23:30.080 align:middle line:84% But this is just the fundamental theorem. 00:23:30.080 --> 00:23:35.540 align:middle line:84% But one of the rules we had writing the integral 00:23:35.540 --> 00:23:38.280 align:middle line:90% is we can write this one here. 00:23:38.280 --> 00:23:41.940 align:middle line:84% So here I'm thinking about that a function little f 00:23:41.940 --> 00:23:44.640 align:middle line:84% is defined on some interval from a to b. 00:23:44.640 --> 00:23:48.220 align:middle line:90% 00:23:48.220 --> 00:23:51.940 align:middle line:84% And I can rewrite this thing here in this form 00:23:51.940 --> 00:23:55.460 align:middle line:84% here because I can just think about this thing 00:23:55.460 --> 00:24:04.300 align:middle line:84% here as the integral from a to b of f of s. 00:24:04.300 --> 00:24:05.120 align:middle line:90% So sorry. 00:24:05.120 --> 00:24:06.670 align:middle line:90% Let me write it over here. 00:24:06.670 --> 00:24:16.156 align:middle line:90% 00:24:16.156 --> 00:24:22.680 align:middle line:84% So I'm just making the trivial observation that we have-- 00:24:22.680 --> 00:24:28.590 align:middle line:84% so if we have a function from a to b into R, 00:24:28.590 --> 00:24:33.110 align:middle line:84% and we have some x that lie between a and b, 00:24:33.110 --> 00:24:36.270 align:middle line:84% then we proved that the integral from a 00:24:36.270 --> 00:24:43.550 align:middle line:84% to b of f, this thing here, is equal to the integral from a 00:24:43.550 --> 00:24:51.250 align:middle line:84% to x plus the integral from x to b. 00:24:51.250 --> 00:24:55.590 align:middle line:90% 00:24:55.590 --> 00:25:00.970 align:middle line:84% That was one of the rules, that if you take this-- 00:25:00.970 --> 00:25:03.910 align:middle line:84% if you have a dividing point in this interval, then 00:25:03.910 --> 00:25:06.430 align:middle line:84% you could integrate over the first half 00:25:06.430 --> 00:25:12.070 align:middle line:84% and then add the integral of the second half. 00:25:12.070 --> 00:25:17.510 align:middle line:84% But this thing here just tells you that this thing here you 00:25:17.510 --> 00:25:19.890 align:middle line:90% can rewrite as this minus that. 00:25:19.890 --> 00:25:27.090 align:middle line:84% So you have that the integral from x to b f of s ds 00:25:27.090 --> 00:25:32.730 align:middle line:84% is equal to the integral from a to b f of s ds 00:25:32.730 --> 00:25:44.250 align:middle line:84% minus the integral from a to x f of s ds. 00:25:44.250 --> 00:25:46.730 align:middle line:90% And so a and b are fixed. 00:25:46.730 --> 00:25:51.450 align:middle line:84% And when you think about this thing here as a function of x, 00:25:51.450 --> 00:25:53.010 align:middle line:90% then it's equal to-- 00:25:53.010 --> 00:25:54.970 align:middle line:90% this is just a constant. 00:25:54.970 --> 00:25:56.390 align:middle line:90% And then minus this. 00:25:56.390 --> 00:26:00.430 align:middle line:84% So when you take the derivative of this, this is a constant. 00:26:00.430 --> 00:26:02.650 align:middle line:90% So this derivative is 0. 00:26:02.650 --> 00:26:04.490 align:middle line:84% The derivative of this, well, that 00:26:04.490 --> 00:26:07.730 align:middle line:84% is straight the fundamental theorem of calculus. 00:26:07.730 --> 00:26:12.750 align:middle line:84% And because of this sign, you get minus f. 00:26:12.750 --> 00:26:14.650 align:middle line:84% And so that's what I applied over here, 00:26:14.650 --> 00:26:19.450 align:middle line:84% that if it is of this form here, then 00:26:19.450 --> 00:26:26.240 align:middle line:84% the derivative of this function is minus little f. 00:26:26.240 --> 00:26:26.740 align:middle line:90% OK? 00:26:26.740 --> 00:26:30.540 align:middle line:90% 00:26:30.540 --> 00:26:31.040 align:middle line:90% OK. 00:26:31.040 --> 00:26:34.080 align:middle line:90% So this was-- this is so-- 00:26:34.080 --> 00:26:38.660 align:middle line:84% so this is how the angle is defined, if you will. 00:26:38.660 --> 00:26:44.820 align:middle line:84% The angle is defined as the arc length of the curve. 00:26:44.820 --> 00:26:48.660 align:middle line:84% And we can calculate it using this formula here. 00:26:48.660 --> 00:26:50.690 align:middle line:90% And we're getting a nice-- 00:26:50.690 --> 00:26:54.740 align:middle line:90% 00:26:54.740 --> 00:26:59.240 align:middle line:84% I mean, we know what this arccosine-- 00:26:59.240 --> 00:27:01.660 align:middle line:84% we know actually what the derivative is just 00:27:01.660 --> 00:27:05.780 align:middle line:84% from this arc length and the fundamental theorem of calculus. 00:27:05.780 --> 00:27:07.320 align:middle line:90% OK. 00:27:07.320 --> 00:27:09.140 align:middle line:84% The next thing I want to talk about 00:27:09.140 --> 00:27:13.420 align:middle line:84% is I want to talk about pointwise and uniform 00:27:13.420 --> 00:27:16.620 align:middle line:90% convergence of functions. 00:27:16.620 --> 00:27:19.920 align:middle line:84% So you have a sequence of functions. 00:27:19.920 --> 00:27:26.600 align:middle line:90% 00:27:26.600 --> 00:27:29.290 align:middle line:84% You have a sequence of functions fn. 00:27:29.290 --> 00:27:32.920 align:middle line:90% 00:27:32.920 --> 00:27:38.120 align:middle line:84% So suppose you have some interval I, 00:27:38.120 --> 00:27:45.720 align:middle line:84% and you have a sequence of functions fn. 00:27:45.720 --> 00:27:51.530 align:middle line:84% And you have another function f also on this interval. 00:27:51.530 --> 00:27:54.520 align:middle line:90% 00:27:54.520 --> 00:27:57.220 align:middle line:84% Then we say-- so this is the definition. 00:27:57.220 --> 00:28:01.240 align:middle line:90% 00:28:01.240 --> 00:28:04.600 align:middle line:84% We say that the sequence here converges 00:28:04.600 --> 00:28:19.310 align:middle line:84% to f pointwise if, for all x fixed-- 00:28:19.310 --> 00:28:22.970 align:middle line:90% 00:28:22.970 --> 00:28:24.790 align:middle line:90% if you're looking at this-- 00:28:24.790 --> 00:28:28.930 align:middle line:84% so x here, think about x as being-- x could be anything, 00:28:28.930 --> 00:28:30.610 align:middle line:90% but it is fixed. 00:28:30.610 --> 00:28:33.790 align:middle line:84% And then you're looking at this sequence here. 00:28:33.790 --> 00:28:37.570 align:middle line:84% So this is a sequence of real numbers because x is fixed. 00:28:37.570 --> 00:28:42.590 align:middle line:84% Then you want this thing here to converge to f of x. 00:28:42.590 --> 00:28:43.090 align:middle line:90% Yeah? 00:28:43.090 --> 00:28:43.757 align:middle line:90% AUDIENCE: Sorry. 00:28:43.757 --> 00:28:45.830 align:middle line:84% I had a quick question about the last thing. 00:28:45.830 --> 00:28:51.330 align:middle line:84% So how do you know that the integral of a to b fs ds is 0? 00:28:51.330 --> 00:28:53.090 align:middle line:90% TOBIAS COLDING: So no. 00:28:53.090 --> 00:28:58.910 align:middle line:84% So the thing is that what we're looking at here-- 00:28:58.910 --> 00:28:59.890 align:middle line:90% so I was asking-- 00:28:59.890 --> 00:29:02.410 align:middle line:90% 00:29:02.410 --> 00:29:04.390 align:middle line:90% so this here is-- 00:29:04.390 --> 00:29:06.370 align:middle line:84% so if you have a function, capital F, 00:29:06.370 --> 00:29:09.650 align:middle line:84% that is given like this, the fundamental theorem of calculus 00:29:09.650 --> 00:29:11.970 align:middle line:84% gives you what the derivative is. 00:29:11.970 --> 00:29:15.370 align:middle line:84% So I'm just asking, suppose that the function-- 00:29:15.370 --> 00:29:20.940 align:middle line:84% that the variable was not up here but it was down here. 00:29:20.940 --> 00:29:22.700 align:middle line:90% What is the derivative? 00:29:22.700 --> 00:29:25.440 align:middle line:84% And so in order to figure out that derivative, 00:29:25.440 --> 00:29:28.100 align:middle line:90% I'm just using that-- 00:29:28.100 --> 00:29:31.820 align:middle line:84% so if I was integrating over the whole interval, 00:29:31.820 --> 00:29:33.540 align:middle line:90% a and b are fixed. 00:29:33.540 --> 00:29:38.040 align:middle line:84% So this is just some number, whatever it is. 00:29:38.040 --> 00:29:40.340 align:middle line:90% But it's probably not 0. 00:29:40.340 --> 00:29:43.620 align:middle line:84% I mean, it generally wouldn't be 0, but it's just some constant. 00:29:43.620 --> 00:29:49.180 align:middle line:84% And so for each x you get the same constant here. 00:29:49.180 --> 00:29:52.580 align:middle line:84% And then it's equal to this plus that. 00:29:52.580 --> 00:29:55.160 align:middle line:84% But this is what we're interested in. 00:29:55.160 --> 00:29:57.520 align:middle line:84% So I'm just moving this 1 over on the other side. 00:29:57.520 --> 00:29:59.020 align:middle line:90% So I get this thing here. 00:29:59.020 --> 00:30:01.580 align:middle line:84% But remember, this here is a constant. 00:30:01.580 --> 00:30:03.740 align:middle line:84% So when you take the derivative, it's 0. 00:30:03.740 --> 00:30:07.960 align:middle line:90% 00:30:07.960 --> 00:30:08.460 align:middle line:90% OK. 00:30:08.460 --> 00:30:09.260 align:middle line:90% So-- 00:30:09.260 --> 00:30:09.980 align:middle line:90% AUDIENCE: For this problem-- 00:30:09.980 --> 00:30:10.460 align:middle line:90% TOBIAS COLDING: Yeah? 00:30:10.460 --> 00:30:12.293 align:middle line:84% AUDIENCE: --if you were to just substitute-- 00:30:12.293 --> 00:30:14.660 align:middle line:84% set b equal to a, then wouldn't you 00:30:14.660 --> 00:30:18.160 align:middle line:84% just get that a equal integral from a to a goes to 0? 00:30:18.160 --> 00:30:19.320 align:middle line:90% And then you can get this-- 00:30:19.320 --> 00:30:21.208 align:middle line:84% TOBIAS COLDING: But that's-- but if-- 00:30:21.208 --> 00:30:21.708 align:middle line:90% right. 00:30:21.708 --> 00:30:24.520 align:middle line:90% 00:30:24.520 --> 00:30:26.140 align:middle line:90% If b is equal to a-- 00:30:26.140 --> 00:30:29.640 align:middle line:90% 00:30:29.640 --> 00:30:32.200 align:middle line:90% if b is equal to a, yeah. 00:30:32.200 --> 00:30:34.120 align:middle line:90% I mean, but it's not. 00:30:34.120 --> 00:30:38.453 align:middle line:84% I mean, so then this interval is just a trivial interval. 00:30:38.453 --> 00:30:39.120 align:middle line:90% AUDIENCE: Right. 00:30:39.120 --> 00:30:42.020 align:middle line:84% But doesn't it allow you to get to the same identity there? 00:30:42.020 --> 00:30:42.937 align:middle line:90% TOBIAS COLDING: Right. 00:30:42.937 --> 00:30:45.260 align:middle line:84% But the thing is-- so the reason why-- 00:30:45.260 --> 00:30:47.220 align:middle line:90% so why did I want to do it? 00:30:47.220 --> 00:30:47.760 align:middle line:90% Yes. 00:30:47.760 --> 00:30:53.540 align:middle line:84% But it's like-- but it's kind of using something implicitly. 00:30:53.540 --> 00:30:54.420 align:middle line:90% You see what I mean? 00:30:54.420 --> 00:30:57.680 align:middle line:84% I'm assuming that x is between a and b 00:30:57.680 --> 00:31:02.320 align:middle line:84% because when we took this interval, if x is not 00:31:02.320 --> 00:31:04.040 align:middle line:84% between a, b, we have to figure out, 00:31:04.040 --> 00:31:07.760 align:middle line:84% what the hell do we mean by this sort of integral anyway? 00:31:07.760 --> 00:31:13.560 align:middle line:84% And so, yes, it works, even if you 00:31:13.560 --> 00:31:16.000 align:middle line:84% have the ordering differently, but you just 00:31:16.000 --> 00:31:17.970 align:middle line:84% have to-- all I want to point out here 00:31:17.970 --> 00:31:21.690 align:middle line:84% is that if it is downstairs and not upstairs, 00:31:21.690 --> 00:31:26.270 align:middle line:84% you have to just remember the sign. 00:31:26.270 --> 00:31:26.770 align:middle line:90% OK. 00:31:26.770 --> 00:31:30.050 align:middle line:90% 00:31:30.050 --> 00:31:30.550 align:middle line:90% OK. 00:31:30.550 --> 00:31:33.690 align:middle line:84% So this here-- so I have a sequence 00:31:33.690 --> 00:31:35.970 align:middle line:90% of functions on the interval. 00:31:35.970 --> 00:31:39.610 align:middle line:84% I have another function on the same interval. 00:31:39.610 --> 00:31:45.450 align:middle line:84% And I will say that this sequence is converging pointwise 00:31:45.450 --> 00:31:51.490 align:middle line:84% to f if for all x fixed, when we're looking at this sequence, 00:31:51.490 --> 00:31:54.610 align:middle line:90% it converges to that. 00:31:54.610 --> 00:31:57.640 align:middle line:84% And so let me look at two examples of that. 00:31:57.640 --> 00:32:00.650 align:middle line:90% 00:32:00.650 --> 00:32:04.470 align:middle line:90% So the first one is-- 00:32:04.470 --> 00:32:15.010 align:middle line:90% 00:32:15.010 --> 00:32:20.110 align:middle line:84% so the first example I want to look at, 00:32:20.110 --> 00:32:27.950 align:middle line:84% this is where the functions fn of x is equal to x to the n. 00:32:27.950 --> 00:32:32.530 align:middle line:90% And x here is between 0 and 1. 00:32:32.530 --> 00:32:36.110 align:middle line:90% 00:32:36.110 --> 00:32:44.790 align:middle line:84% And so this is my fn and my function f, which is now, 00:32:44.790 --> 00:32:47.590 align:middle line:84% of course, also a function on this interval. 00:32:47.590 --> 00:32:56.790 align:middle line:84% This is equal to 0 if x is bigger or equal to 0 00:32:56.790 --> 00:32:59.230 align:middle line:90% but strictly less than 1. 00:32:59.230 --> 00:33:05.350 align:middle line:90% And it's 1 if x is equal to 1. 00:33:05.350 --> 00:33:07.030 align:middle line:90% OK. 00:33:07.030 --> 00:33:10.690 align:middle line:84% So of course, this function f is not continuous. 00:33:10.690 --> 00:33:12.250 align:middle line:90% The other ones are continuous. 00:33:12.250 --> 00:33:15.960 align:middle line:90% 00:33:15.960 --> 00:33:24.920 align:middle line:84% Now, I claim that this sequence-- so the sequence fn 00:33:24.920 --> 00:33:28.800 align:middle line:90% is converging pointwise to f. 00:33:28.800 --> 00:33:34.860 align:middle line:90% So fn converges to f pointwise. 00:33:34.860 --> 00:33:39.080 align:middle line:90% 00:33:39.080 --> 00:33:44.360 align:middle line:90% Because if you fix x-- 00:33:44.360 --> 00:33:56.680 align:middle line:84% let's say, fix x, and let's assume that it's between-- 00:33:56.680 --> 00:34:00.380 align:middle line:90% sorry, it's between 0-- 00:34:00.380 --> 00:34:02.120 align:middle line:90% it's strictly less than 1. 00:34:02.120 --> 00:34:05.000 align:middle line:90% So you're fixing x. 00:34:05.000 --> 00:34:07.800 align:middle line:84% And let's first deal with the case where that x 00:34:07.800 --> 00:34:10.080 align:middle line:90% is strictly less than 1. 00:34:10.080 --> 00:34:12.320 align:middle line:84% Well, if I'm looking at x to the n-- 00:34:12.320 --> 00:34:14.810 align:middle line:90% so it takes some small number. 00:34:14.810 --> 00:34:17.449 align:middle line:84% Small just means that it's strictly less than 1. 00:34:17.449 --> 00:34:20.530 align:middle line:84% And I multiply it by itself n time. 00:34:20.530 --> 00:34:22.593 align:middle line:90% Well, that certainly goes to 0. 00:34:22.593 --> 00:34:26.610 align:middle line:90% 00:34:26.610 --> 00:34:33.650 align:middle line:84% If x here is fixed but it's not 1, then x to the n. 00:34:33.650 --> 00:34:36.850 align:middle line:84% If x is 1, then this here is still 1. 00:34:36.850 --> 00:34:40.969 align:middle line:90% 00:34:40.969 --> 00:34:42.909 align:middle line:90% So you see that in this case-- 00:34:42.909 --> 00:34:46.870 align:middle line:90% so this here is f to the n of x. 00:34:46.870 --> 00:34:50.170 align:middle line:90% And this here is f of x. 00:34:50.170 --> 00:34:53.389 align:middle line:84% So these things here, I mean, they're actually equal to this. 00:34:53.389 --> 00:34:55.469 align:middle line:90% So in particular, they converge. 00:34:55.469 --> 00:34:59.810 align:middle line:90% And this here is what f of x is. 00:34:59.810 --> 00:35:05.090 align:middle line:84% And this here is, again, fn of x. 00:35:05.090 --> 00:35:09.050 align:middle line:84% So you have that this sequence of functions 00:35:09.050 --> 00:35:13.710 align:middle line:84% converge to this slightly crazy function. 00:35:13.710 --> 00:35:15.530 align:middle line:90% So that's the first example. 00:35:15.530 --> 00:35:16.850 align:middle line:90% They converge pointwise. 00:35:16.850 --> 00:35:19.870 align:middle line:90% 00:35:19.870 --> 00:35:24.800 align:middle line:84% The next example I want to look at is-- 00:35:24.800 --> 00:35:28.470 align:middle line:90% 00:35:28.470 --> 00:35:35.950 align:middle line:84% so the next example is where the functions are-- 00:35:35.950 --> 00:35:38.645 align:middle line:90% you're just summing here. 00:35:38.645 --> 00:35:39.270 align:middle line:90% You're summing. 00:35:39.270 --> 00:35:46.510 align:middle line:84% You just take the power series for the exponential function. 00:35:46.510 --> 00:35:51.610 align:middle line:84% So you're summing where k is equal to 1 to n. 00:35:51.610 --> 00:35:54.610 align:middle line:90% And that's my fn of x. 00:35:54.610 --> 00:35:57.190 align:middle line:90% 00:35:57.190 --> 00:36:03.510 align:middle line:84% And then my f of x is the infinite sum, which 00:36:03.510 --> 00:36:06.590 align:middle line:90% is the exponential function. 00:36:06.590 --> 00:36:10.920 align:middle line:84% And we have already proved that if you fix x-- 00:36:10.920 --> 00:36:12.340 align:middle line:90% so we fix x-- 00:36:12.340 --> 00:36:20.560 align:middle line:84% then this fn here of x converges to f of x for each fixed x. 00:36:20.560 --> 00:36:23.310 align:middle line:90% For each fixed x. 00:36:23.310 --> 00:36:30.960 align:middle line:90% 00:36:30.960 --> 00:36:34.880 align:middle line:84% So this sequence of functions pointwise converges 00:36:34.880 --> 00:36:38.760 align:middle line:90% to the exponential function. 00:36:38.760 --> 00:36:42.760 align:middle line:84% So this is one notion of convergence for functions. 00:36:42.760 --> 00:36:46.660 align:middle line:84% And there's another notion that is also useful. 00:36:46.660 --> 00:36:48.360 align:middle line:90% They're both useful. 00:36:48.360 --> 00:36:50.040 align:middle line:90% The other one is stronger. 00:36:50.040 --> 00:36:53.240 align:middle line:84% And so of course, if you have the stronger version, 00:36:53.240 --> 00:36:57.720 align:middle line:84% you often get something much better. 00:36:57.720 --> 00:36:59.100 align:middle line:84% Oh, I shouldn't have erased this. 00:36:59.100 --> 00:37:00.720 align:middle line:90% But anyway. 00:37:00.720 --> 00:37:05.480 align:middle line:90% And so the second notion is-- 00:37:05.480 --> 00:37:13.840 align:middle line:84% so this is the notion of uniform convergence. 00:37:13.840 --> 00:37:21.540 align:middle line:90% 00:37:21.540 --> 00:37:27.600 align:middle line:84% Uniform means that it's uniform in x. 00:37:27.600 --> 00:37:33.500 align:middle line:84% So what we have here is we have a sequence of functions 00:37:33.500 --> 00:37:37.780 align:middle line:84% and f defined on some interval into R. 00:37:37.780 --> 00:37:40.700 align:middle line:84% And then we say that this sequence converges 00:37:40.700 --> 00:37:56.460 align:middle line:84% to f uniformly if, for all epsilon greater than 0, 00:37:56.460 --> 00:38:00.900 align:middle line:84% there exists a capital N so that if little n is bigger 00:38:00.900 --> 00:38:08.910 align:middle line:84% than this capital N, then f of x minus fn of x 00:38:08.910 --> 00:38:13.050 align:middle line:84% is smaller than epsilon for all x. 00:38:13.050 --> 00:38:16.670 align:middle line:90% 00:38:16.670 --> 00:38:24.150 align:middle line:84% And again, the uniformness, it holds for all x. 00:38:24.150 --> 00:38:28.170 align:middle line:84% You can use the same capital N for all x. 00:38:28.170 --> 00:38:33.030 align:middle line:90% 00:38:33.030 --> 00:38:45.550 align:middle line:84% And so now, now what you have is that if a sequence converges 00:38:45.550 --> 00:38:50.590 align:middle line:84% uniformly, then it converges pointwise. 00:38:50.590 --> 00:38:54.950 align:middle line:84% So we have the following lemma that follows directly 00:38:54.950 --> 00:38:58.110 align:middle line:90% from the definition. 00:38:58.110 --> 00:38:58.660 align:middle line:90% So lemma. 00:38:58.660 --> 00:39:01.510 align:middle line:90% 00:39:01.510 --> 00:39:16.890 align:middle line:84% So a uniform convergence of fn to f 00:39:16.890 --> 00:39:37.730 align:middle line:84% implies pointwise convergence of fn to f. 00:39:37.730 --> 00:39:38.350 align:middle line:90% OK. 00:39:38.350 --> 00:39:43.330 align:middle line:84% So uniform convergence implies pointwise convergence. 00:39:43.330 --> 00:39:44.910 align:middle line:90% And that's clear. 00:39:44.910 --> 00:39:47.610 align:middle line:84% That's really just the definition 00:39:47.610 --> 00:39:51.090 align:middle line:84% if I had written out what it means for-- 00:39:51.090 --> 00:39:54.050 align:middle line:84% so this here, I've written out what it means 00:39:54.050 --> 00:39:57.330 align:middle line:90% to have uniform convergence. 00:39:57.330 --> 00:39:59.530 align:middle line:84% If it was pointwise convergence, then 00:39:59.530 --> 00:40:01.190 align:middle line:90% you have these functions again. 00:40:01.190 --> 00:40:04.420 align:middle line:84% And you have that this converges to that pointwise. 00:40:04.420 --> 00:40:07.360 align:middle line:90% It means that the sequence fn-- 00:40:07.360 --> 00:40:12.340 align:middle line:84% so once you fix x, then the sequence fn of x-- 00:40:12.340 --> 00:40:23.820 align:middle line:84% so once x is fixed, then these here converge to this. 00:40:23.820 --> 00:40:26.660 align:middle line:84% But saying that this sequence here converge 00:40:26.660 --> 00:40:29.380 align:middle line:84% means that for all epsilon greater than 0, 00:40:29.380 --> 00:40:35.820 align:middle line:84% there exists an N. But this N is now allowed to depend on x so 00:40:35.820 --> 00:40:39.340 align:middle line:84% that if little n is bigger than this capital N, 00:40:39.340 --> 00:40:46.380 align:middle line:84% then fn of x minus f of x is smaller than epsilon. 00:40:46.380 --> 00:40:52.220 align:middle line:84% So you see that clearly uniform convergence implies pointwise. 00:40:52.220 --> 00:40:55.820 align:middle line:84% And again, uniform just means that this capital N 00:40:55.820 --> 00:40:58.320 align:middle line:84% can be chosen to be the same for all x. 00:40:58.320 --> 00:41:01.780 align:middle line:90% 00:41:01.780 --> 00:41:04.850 align:middle line:84% So now let's look at the examples. 00:41:04.850 --> 00:41:09.070 align:middle line:84% Unfortunately, I erased the most important one. 00:41:09.070 --> 00:41:11.530 align:middle line:84% So I will just go through it again. 00:41:11.530 --> 00:41:15.670 align:middle line:84% Let me just talk about the example again. 00:41:15.670 --> 00:41:22.450 align:middle line:84% And then we'll see why this example here does not. 00:41:22.450 --> 00:41:25.550 align:middle line:84% So we have this example from before, 00:41:25.550 --> 00:41:32.550 align:middle line:84% where the sequence of functions was x to the n. 00:41:32.550 --> 00:41:36.270 align:middle line:90% And x here was between 0 and 1. 00:41:36.270 --> 00:41:39.870 align:middle line:84% And then we had this function f that 00:41:39.870 --> 00:41:43.270 align:middle line:84% was this function that was not continuous. 00:41:43.270 --> 00:41:47.150 align:middle line:84% But this function was given to be equal to 0 00:41:47.150 --> 00:41:53.950 align:middle line:84% if x here is strictly less than 1 00:41:53.950 --> 00:41:59.602 align:middle line:90% and it was 1 if x is equal to 1. 00:41:59.602 --> 00:42:03.116 align:middle line:90% That was this function f. 00:42:03.116 --> 00:42:07.410 align:middle line:84% And we know-- we've already shown that the fn converges 00:42:07.410 --> 00:42:09.010 align:middle line:90% to f pointwise. 00:42:09.010 --> 00:42:12.150 align:middle line:90% But we want to see now-- 00:42:12.150 --> 00:42:21.330 align:middle line:84% so I claim that fn does not converge. 00:42:21.330 --> 00:42:29.690 align:middle line:84% So this does not converge uniformly to f. 00:42:29.690 --> 00:42:33.150 align:middle line:84% We know already that it converges pointwise. 00:42:33.150 --> 00:42:39.050 align:middle line:84% And I claim that this sequence does not converge uniformly. 00:42:39.050 --> 00:42:42.850 align:middle line:90% So now, why is this the case? 00:42:42.850 --> 00:42:47.850 align:middle line:84% Well, this is because, you see, by the intermediate value 00:42:47.850 --> 00:42:54.330 align:middle line:84% theorem, the functions fn are continuous. 00:42:54.330 --> 00:42:57.490 align:middle line:90% At 0, they take the value 0. 00:42:57.490 --> 00:43:00.300 align:middle line:90% At 1, they take the value 1. 00:43:00.300 --> 00:43:04.360 align:middle line:84% So for each n, by the intermediate value theorem-- 00:43:04.360 --> 00:43:22.220 align:middle line:90% 00:43:22.220 --> 00:43:40.220 align:middle line:84% since each fn is continuous, then there exists an xn, 00:43:40.220 --> 00:43:44.140 align:middle line:90% but it really will depend on n. 00:43:44.140 --> 00:43:52.800 align:middle line:84% So that fn of xn will take the value one half. 00:43:52.800 --> 00:43:55.200 align:middle line:90% So the functions here-- 00:43:55.200 --> 00:43:56.820 align:middle line:90% this is the functions-- 00:43:56.820 --> 00:43:57.500 align:middle line:90% oops. 00:43:57.500 --> 00:44:00.320 align:middle line:90% This is the functions. 00:44:00.320 --> 00:44:04.760 align:middle line:90% This is the function fn. 00:44:04.760 --> 00:44:07.240 align:middle line:90% At 0, it takes a value of 0. 00:44:07.240 --> 00:44:10.620 align:middle line:90% At 1, it takes a value 1. 00:44:10.620 --> 00:44:16.100 align:middle line:84% So there is somewhere where a half is achieved. 00:44:16.100 --> 00:44:18.060 align:middle line:90% And this is at this-- it's here. 00:44:18.060 --> 00:44:21.700 align:middle line:90% 00:44:21.700 --> 00:44:23.100 align:middle line:90% So you have that. 00:44:23.100 --> 00:44:27.560 align:middle line:84% And again, this xn is depending on n. 00:44:27.560 --> 00:44:30.800 align:middle line:84% And the thing is that as n gets large, 00:44:30.800 --> 00:44:36.040 align:middle line:84% this point will have to move very close to 1. 00:44:36.040 --> 00:44:44.640 align:middle line:84% But if you're now looking at fn of x minus f of x-- 00:44:44.640 --> 00:44:51.440 align:middle line:90% 00:44:51.440 --> 00:44:52.800 align:middle line:90% so we want to-- 00:44:52.800 --> 00:44:55.240 align:middle line:84% if it was converging uniformly, it 00:44:55.240 --> 00:44:58.770 align:middle line:84% would mean that this difference here would be smaller 00:44:58.770 --> 00:45:01.810 align:middle line:90% than some epsilon for all x. 00:45:01.810 --> 00:45:09.170 align:middle line:84% But if you're just looking at this point here, 00:45:09.170 --> 00:45:12.210 align:middle line:84% this difference here is this point 00:45:12.210 --> 00:45:16.610 align:middle line:90% here, xn, will lie strictly-- 00:45:16.610 --> 00:45:22.210 align:middle line:84% xn here will lie-- this is where this value here is a half. 00:45:22.210 --> 00:45:23.590 align:middle line:90% It's not 1. 00:45:23.590 --> 00:45:28.770 align:middle line:84% So it's strictly between 0 and 1. 00:45:28.770 --> 00:45:37.210 align:middle line:84% So you see that for each n, this function f at xn, 00:45:37.210 --> 00:45:38.950 align:middle line:90% this will always be 0. 00:45:38.950 --> 00:45:43.110 align:middle line:84% This is because the function f is 0 except at 1. 00:45:43.110 --> 00:45:45.850 align:middle line:84% And this point is definitely not 1. 00:45:45.850 --> 00:45:47.450 align:middle line:90% So you have that. 00:45:47.450 --> 00:45:53.130 align:middle line:84% So you see that this thing here, this here is 0. 00:45:53.130 --> 00:45:56.560 align:middle line:84% So this is just the value of this, the absolute value. 00:45:56.560 --> 00:45:57.720 align:middle line:90% But that's actually a half. 00:45:57.720 --> 00:46:01.580 align:middle line:90% So this is always a half. 00:46:01.580 --> 00:46:06.020 align:middle line:84% So the function would converge uniformly as long as-- 00:46:06.020 --> 00:46:09.260 align:middle line:90% 00:46:09.260 --> 00:46:13.700 align:middle line:84% so uniform convergence would mean-- 00:46:13.700 --> 00:46:15.560 align:middle line:90% so uniform convergence. 00:46:15.560 --> 00:46:21.140 align:middle line:90% 00:46:21.140 --> 00:46:29.540 align:middle line:90% So now let's show. 00:46:29.540 --> 00:46:40.240 align:middle line:84% So let's prove that fn does not converge uniformly to f. 00:46:40.240 --> 00:46:45.300 align:middle line:90% 00:46:45.300 --> 00:46:51.500 align:middle line:90% And so I just pick my epsilon. 00:46:51.500 --> 00:46:57.810 align:middle line:84% So I pick an epsilon less than half. 00:46:57.810 --> 00:47:03.270 align:middle line:84% And then if it was converging-- and suppose not-- 00:47:03.270 --> 00:47:08.000 align:middle line:90% 00:47:08.000 --> 00:47:17.700 align:middle line:84% that is, suppose fn converges to f uniformly. 00:47:17.700 --> 00:47:21.440 align:middle line:90% 00:47:21.440 --> 00:47:23.180 align:middle line:90% This means that for all epsilon. 00:47:23.180 --> 00:47:25.140 align:middle line:84% But I just use this epsilon here. 00:47:25.140 --> 00:47:29.040 align:middle line:90% I just use this one, epsilon. 00:47:29.040 --> 00:47:37.440 align:middle line:84% But then I would have that there exists capital N so that-- 00:47:37.440 --> 00:47:42.040 align:middle line:84% I mean, I can even take it to be equal to a half. 00:47:42.040 --> 00:47:47.560 align:middle line:84% So there exists capital N such that 00:47:47.560 --> 00:47:53.930 align:middle line:84% if little n is bigger than capital N, then this thing here, 00:47:53.930 --> 00:48:01.850 align:middle line:84% minus f of x, would have to be strictly less than half 00:48:01.850 --> 00:48:02.910 align:middle line:90% for all x. 00:48:02.910 --> 00:48:06.730 align:middle line:90% 00:48:06.730 --> 00:48:09.970 align:middle line:84% If it was converging uniformly, then in particular, 00:48:09.970 --> 00:48:15.970 align:middle line:84% for this one choice of epsilon, you would have this for all x. 00:48:15.970 --> 00:48:19.310 align:middle line:84% But we already know that if we are looking at-- 00:48:19.310 --> 00:48:22.530 align:middle line:84% now depending on what this is, but that's fine. 00:48:22.530 --> 00:48:26.450 align:middle line:84% If you insert x equal to xn, then this 00:48:26.450 --> 00:48:28.990 align:middle line:90% is actually equal to a half. 00:48:28.990 --> 00:48:31.410 align:middle line:90% So this here does not hold. 00:48:31.410 --> 00:48:32.830 align:middle line:90% This is the contradiction. 00:48:32.830 --> 00:48:39.090 align:middle line:90% 00:48:39.090 --> 00:48:42.680 align:middle line:84% Now, in the other example that we looked at before-- 00:48:42.680 --> 00:48:50.810 align:middle line:90% 00:48:50.810 --> 00:48:54.840 align:middle line:84% in the other example we have that-- 00:48:54.840 --> 00:48:58.830 align:middle line:90% 00:48:58.830 --> 00:49:03.470 align:middle line:84% in the other example, which is this example here, 00:49:03.470 --> 00:49:06.410 align:middle line:84% this sequence here is converging. 00:49:06.410 --> 00:49:10.590 align:middle line:84% And I'll return to this in a minute. 00:49:10.590 --> 00:49:14.470 align:middle line:84% So in this case, these here converge 00:49:14.470 --> 00:49:26.350 align:middle line:84% to f uniformly on each interval-- 00:49:26.350 --> 00:49:27.850 align:middle line:90% on each bounded interval. 00:49:27.850 --> 00:49:35.550 align:middle line:84% So each interval of the form minus L 00:49:35.550 --> 00:49:38.870 align:middle line:84% to L. So on each bounded interval. 00:49:38.870 --> 00:49:39.670 align:middle line:90% OK. 00:49:39.670 --> 00:49:42.540 align:middle line:84% And we will talk about this more in just a second. 00:49:42.540 --> 00:49:45.910 align:middle line:90% 00:49:45.910 --> 00:49:48.820 align:middle line:90% OK. 00:49:48.820 --> 00:49:58.200 align:middle line:84% And because this is the content of what's 00:49:58.200 --> 00:50:01.480 align:middle line:90% called Weierstrass M-test. 00:50:01.480 --> 00:50:06.800 align:middle line:84% So this holds more generally that if you take a sequence-- 00:50:06.800 --> 00:50:07.560 align:middle line:90% so let's see. 00:50:07.560 --> 00:50:09.420 align:middle line:90% Weierstrass M-test. 00:50:09.420 --> 00:50:14.840 align:middle line:90% 00:50:14.840 --> 00:50:31.600 align:middle line:84% So in Weierstrass M-test, you have a sequence 00:50:31.600 --> 00:50:36.980 align:middle line:84% of functions, fn, and you have that these sequence-- 00:50:36.980 --> 00:50:45.640 align:middle line:84% so fn, sorry, is functions on some interval into R. 00:50:45.640 --> 00:50:58.770 align:middle line:84% And then you have that the fn are bounded by some constant Mn. 00:50:58.770 --> 00:51:02.810 align:middle line:90% And then you have that the Mn-- 00:51:02.810 --> 00:51:05.070 align:middle line:84% so these are clearly non-negative numbers. 00:51:05.070 --> 00:51:08.630 align:middle line:84% And you have that this series here-- 00:51:08.630 --> 00:51:12.010 align:middle line:90% 00:51:12.010 --> 00:51:16.190 align:middle line:90% this here is finite. 00:51:16.190 --> 00:51:19.470 align:middle line:84% So this is a series of non-negative numbers. 00:51:19.470 --> 00:51:23.290 align:middle line:84% So to say that this series is summable 00:51:23.290 --> 00:51:24.940 align:middle line:84% is the same as saying that it's finite. 00:51:24.940 --> 00:51:29.330 align:middle line:90% 00:51:29.330 --> 00:51:33.130 align:middle line:84% And now the claim is-- so then Weierstrass M-test is-- 00:51:33.130 --> 00:51:38.130 align:middle line:90% 00:51:38.130 --> 00:51:41.910 align:middle line:84% so this is that if this here is the case, 00:51:41.910 --> 00:51:49.570 align:middle line:84% then this sequence converges to some f uniformly. 00:51:49.570 --> 00:51:53.870 align:middle line:90% 00:51:53.870 --> 00:51:56.480 align:middle line:90% That's Weierstrass M-test. 00:51:56.480 --> 00:51:59.790 align:middle line:90% 00:51:59.790 --> 00:52:00.990 align:middle line:90% Yeah? 00:52:00.990 --> 00:52:02.510 align:middle line:84% AUDIENCE: What does this mean here? 00:52:02.510 --> 00:52:05.168 align:middle line:84% Is it the absolute value of the function? 00:52:05.168 --> 00:52:06.210 align:middle line:90% TOBIAS COLDING: Oh, yeah. 00:52:06.210 --> 00:52:09.870 align:middle line:84% So this here-- so this here means-- 00:52:09.870 --> 00:52:15.350 align:middle line:84% this here just means that fn of x is less than 00:52:15.350 --> 00:52:17.730 align:middle line:90% or equal to M for all x. 00:52:17.730 --> 00:52:25.070 align:middle line:90% 00:52:25.070 --> 00:52:26.030 align:middle line:90% Yeah. 00:52:26.030 --> 00:52:31.370 align:middle line:84% That's a kind of quite common used situation. 00:52:31.370 --> 00:52:34.030 align:middle line:90% 00:52:34.030 --> 00:52:35.918 align:middle line:90% OK. 00:52:35.918 --> 00:52:37.620 align:middle line:90% So let's try to prove that. 00:52:37.620 --> 00:52:45.270 align:middle line:90% 00:52:45.270 --> 00:52:47.410 align:middle line:90% So we are looking at-- 00:52:47.410 --> 00:52:53.000 align:middle line:90% 00:52:53.000 --> 00:52:54.900 align:middle line:90% so suppose we look at-- 00:52:54.900 --> 00:52:58.080 align:middle line:90% suppose we define Sn of x. 00:52:58.080 --> 00:53:03.880 align:middle line:84% So this is just when you're summing from k equal to 0 00:53:03.880 --> 00:53:08.160 align:middle line:90% to n of fk of x. 00:53:08.160 --> 00:53:10.240 align:middle line:90% Suppose you're looking at this. 00:53:10.240 --> 00:53:12.160 align:middle line:90% And you want to prove-- 00:53:12.160 --> 00:53:13.410 align:middle line:90% so I want to show-- 00:53:13.410 --> 00:53:20.320 align:middle line:90% 00:53:20.320 --> 00:53:21.940 align:middle line:90% yeah, sorry about that. 00:53:21.940 --> 00:53:23.460 align:middle line:84% This is not what we want to prove. 00:53:23.460 --> 00:53:25.360 align:middle line:90% Sorry. 00:53:25.360 --> 00:53:27.880 align:middle line:90% Sorry, I should have said. 00:53:27.880 --> 00:53:28.720 align:middle line:90% Yeah. 00:53:28.720 --> 00:53:31.220 align:middle line:84% It's not these functions that converge. 00:53:31.220 --> 00:53:33.560 align:middle line:90% It's the sum of them. 00:53:33.560 --> 00:53:36.000 align:middle line:84% So what I want to prove is that-- 00:53:36.000 --> 00:53:38.560 align:middle line:84% I want to prove that this thing here-- 00:53:38.560 --> 00:53:45.780 align:middle line:84% as k equal to 0 to n, that this function here 00:53:45.780 --> 00:53:50.270 align:middle line:90% converge uniformly. 00:53:50.270 --> 00:53:50.770 align:middle line:90% OK? 00:53:50.770 --> 00:53:53.740 align:middle line:90% 00:53:53.740 --> 00:53:56.140 align:middle line:90% So think about it. 00:53:56.140 --> 00:54:02.740 align:middle line:84% You want this to probably apply to some power series. 00:54:02.740 --> 00:54:09.700 align:middle line:84% And so the individual fk is like this polynomial. 00:54:09.700 --> 00:54:13.860 align:middle line:90% It's of some degree. 00:54:13.860 --> 00:54:16.100 align:middle line:90% OK. 00:54:16.100 --> 00:54:20.020 align:middle line:90% So we define this function Sn. 00:54:20.020 --> 00:54:26.170 align:middle line:84% And I want to prove this Sn and then converge. 00:54:26.170 --> 00:54:30.220 align:middle line:90% 00:54:30.220 --> 00:54:33.660 align:middle line:84% And if it's OK, I think it's better 00:54:33.660 --> 00:54:37.620 align:middle line:84% to call this something else than little f, to call it S, 00:54:37.620 --> 00:54:40.800 align:middle line:84% because it's not these guys that converge. 00:54:40.800 --> 00:54:42.270 align:middle line:84% It's the sum of them that converge. 00:54:42.270 --> 00:54:46.288 align:middle line:90% 00:54:46.288 --> 00:54:52.910 align:middle line:84% So we want to show that these guys here, the Sn converts 00:54:52.910 --> 00:54:54.640 align:middle line:90% to this S uniform. 00:54:54.640 --> 00:54:59.310 align:middle line:90% 00:54:59.310 --> 00:55:02.590 align:middle line:90% That's what we want to prove. 00:55:02.590 --> 00:55:06.830 align:middle line:84% Now, let's look at the difference 00:55:06.830 --> 00:55:14.150 align:middle line:90% between Sn of x and Sm of x. 00:55:14.150 --> 00:55:21.710 align:middle line:84% And so here I'm going to think about n as being larger than m. 00:55:21.710 --> 00:55:24.010 align:middle line:84% Suppose I'm thinking about this thing here. 00:55:24.010 --> 00:55:26.550 align:middle line:90% 00:55:26.550 --> 00:55:32.910 align:middle line:84% Well, this here is when you're summing the first-- 00:55:32.910 --> 00:55:35.870 align:middle line:84% I guess you're starting at k equal to 0. 00:55:35.870 --> 00:55:42.330 align:middle line:84% So then we're summing the first n plus 1 of those guys. 00:55:42.330 --> 00:55:43.783 align:middle line:90% Here, we're summing more. 00:55:43.783 --> 00:55:45.450 align:middle line:84% We're summing those guys, but then we're 00:55:45.450 --> 00:55:47.610 align:middle line:90% also summing some more. 00:55:47.610 --> 00:55:50.730 align:middle line:90% And so this thing here-- 00:55:50.730 --> 00:55:52.710 align:middle line:90% so let me just write it out. 00:55:52.710 --> 00:55:59.450 align:middle line:90% So k equal to 0 to n of fk of x. 00:55:59.450 --> 00:56:03.250 align:middle line:84% And the second one is when we're summing. 00:56:03.250 --> 00:56:04.550 align:middle line:90% n is bigger than m. 00:56:04.550 --> 00:56:06.360 align:middle line:84% So here, we are actually summing fewer. 00:56:06.360 --> 00:56:10.050 align:middle line:90% 00:56:10.050 --> 00:56:14.890 align:middle line:84% And so of course, the difference is between these two are, 00:56:14.890 --> 00:56:21.010 align:middle line:84% how many more of these guys are there than those guys? 00:56:21.010 --> 00:56:23.610 align:middle line:84% So this is summing absolute value, 00:56:23.610 --> 00:56:33.410 align:middle line:84% where we're summing from k equal to m plus 1 up to n of fk of x. 00:56:33.410 --> 00:56:35.870 align:middle line:90% So that's what it's equal to. 00:56:35.870 --> 00:56:37.950 align:middle line:84% And we didn't even have to have absolute value. 00:56:37.950 --> 00:56:41.460 align:middle line:84% But in a minute, it's useful to have absolute value. 00:56:41.460 --> 00:56:44.060 align:middle line:84% Because now, of course, this sum here 00:56:44.060 --> 00:56:49.730 align:middle line:84% is less than the sum of the absolute values k 00:56:49.730 --> 00:56:53.500 align:middle line:90% equal to m plus 1 to n. 00:56:53.500 --> 00:56:56.420 align:middle line:84% And then we take the absolute value 00:56:56.420 --> 00:57:00.510 align:middle line:84% in under the summation sign, like this. 00:57:00.510 --> 00:57:04.740 align:middle line:84% But now we can use our condition here. 00:57:04.740 --> 00:57:11.700 align:middle line:84% So this thing here is less than or equal to the sum here from k 00:57:11.700 --> 00:57:16.580 align:middle line:90% equal to m plus 1 up to n of Mk. 00:57:16.580 --> 00:57:22.260 align:middle line:90% 00:57:22.260 --> 00:57:29.880 align:middle line:84% But those guys, this is just a sequence of real numbers. 00:57:29.880 --> 00:57:30.880 align:middle line:90% This is not a function. 00:57:30.880 --> 00:57:32.380 align:middle line:90% It's just real numbers. 00:57:32.380 --> 00:57:33.900 align:middle line:90% And they converge. 00:57:33.900 --> 00:57:36.020 align:middle line:90% So this means that-- 00:57:36.020 --> 00:57:39.430 align:middle line:90% so this means that given-- 00:57:39.430 --> 00:57:45.670 align:middle line:84% this means that you have-- that you can make this thing here 00:57:45.670 --> 00:57:50.790 align:middle line:84% as small as you want, as long as m is sufficiently large. 00:57:50.790 --> 00:57:54.190 align:middle line:90% 00:57:54.190 --> 00:58:06.400 align:middle line:90% So given-- so given-- 00:58:06.400 --> 00:58:09.630 align:middle line:90% 00:58:09.630 --> 00:58:13.830 align:middle line:84% so given epsilon greater than 0, there 00:58:13.830 --> 00:58:19.510 align:middle line:84% exists a capital N such that if m here 00:58:19.510 --> 00:58:28.870 align:middle line:84% is bigger than capital N, then the sum here of the tail, 00:58:28.870 --> 00:58:37.230 align:middle line:84% this thing here, up to anything else of this Mk, 00:58:37.230 --> 00:58:38.890 align:middle line:90% that this is less than epsilon. 00:58:38.890 --> 00:58:41.410 align:middle line:90% 00:58:41.410 --> 00:58:44.470 align:middle line:84% And so you see that this is really saying-- 00:58:44.470 --> 00:58:50.450 align:middle line:84% so this implies that the Sn for each x-- 00:58:50.450 --> 00:58:55.410 align:middle line:84% for each x, you can think about this thing here as a sequence. 00:58:55.410 --> 00:58:59.490 align:middle line:84% And so you have that now this sequence 00:58:59.490 --> 00:59:02.670 align:middle line:90% here is a Cauchy sequence. 00:59:02.670 --> 00:59:05.300 align:middle line:84% For each x, this here is a Cauchy sequence. 00:59:05.300 --> 00:59:13.370 align:middle line:90% 00:59:13.370 --> 00:59:17.410 align:middle line:84% So this means that this Sn of x, they 00:59:17.410 --> 00:59:19.770 align:middle line:84% will actually converge to something. 00:59:19.770 --> 00:59:23.170 align:middle line:84% And that's something I call S of x. 00:59:23.170 --> 00:59:26.410 align:middle line:84% So that gives you a function S of x. 00:59:26.410 --> 00:59:30.350 align:middle line:84% But it gives you more because you see, 00:59:30.350 --> 00:59:33.050 align:middle line:84% it actually shows you something about-- 00:59:33.050 --> 00:59:39.380 align:middle line:84% because this inequality here, that this difference here 00:59:39.380 --> 00:59:40.740 align:middle line:90% is less than that. 00:59:40.740 --> 00:59:44.820 align:middle line:90% That holds for all x. 00:59:44.820 --> 00:59:48.300 align:middle line:90% So this means that-- 00:59:48.300 --> 00:59:51.860 align:middle line:84% this means that as you're sending-- 00:59:51.860 --> 00:59:55.630 align:middle line:90% so this means that you have-- 00:59:55.630 --> 00:59:58.740 align:middle line:90% 00:59:58.740 --> 01:00:01.700 align:middle line:90% so now let n go to infinity. 01:00:01.700 --> 01:00:04.320 align:middle line:84% If you take this inequality here-- 01:00:04.320 --> 01:00:07.420 align:middle line:84% so you we have this inequality, Sn 01:00:07.420 --> 01:00:15.820 align:middle line:84% of x minus Sm of x is less than or equal to the sum 01:00:15.820 --> 01:00:22.860 align:middle line:84% here, k equal to n plus 1 to n of Mk. 01:00:22.860 --> 01:00:24.680 align:middle line:90% So we have this inequality. 01:00:24.680 --> 01:00:28.900 align:middle line:84% And in fact, you can write it as if you're 01:00:28.900 --> 01:00:30.340 align:middle line:90% summing over the entirety. 01:00:30.340 --> 01:00:37.600 align:middle line:90% 01:00:37.600 --> 01:00:40.920 align:middle line:84% And so now this side of it, this side 01:00:40.920 --> 01:00:43.920 align:middle line:90% does not depend on little n. 01:00:43.920 --> 01:00:46.640 align:middle line:84% This here holds as long as all here we 01:00:46.640 --> 01:00:50.680 align:middle line:84% require was that this here was bigger than that. 01:00:50.680 --> 01:00:54.080 align:middle line:84% And so as n goes to infinity, you 01:00:54.080 --> 01:01:02.780 align:middle line:84% have that S of x minus Sm of x will have to be less than 01:01:02.780 --> 01:01:03.950 align:middle line:90% or equal to the sum. 01:01:03.950 --> 01:01:06.960 align:middle line:90% 01:01:06.960 --> 01:01:11.800 align:middle line:84% And the key point here for this Weierstrass M-test 01:01:11.800 --> 01:01:15.280 align:middle line:90% is that this here holds for all. 01:01:15.280 --> 01:01:19.160 align:middle line:90% This inequality holds for all x. 01:01:19.160 --> 01:01:20.980 align:middle line:90% You see that little n is gone. 01:01:20.980 --> 01:01:26.080 align:middle line:84% There's only this capital N. Sorry, the M. 01:01:26.080 --> 01:01:29.780 align:middle line:90% And so now we again-- 01:01:29.780 --> 01:01:35.630 align:middle line:84% so as long as you choose some N insufficiently large, 01:01:35.630 --> 01:01:39.530 align:middle line:84% then if little m is bigger or equal to this N, 01:01:39.530 --> 01:01:43.930 align:middle line:84% then you can make this sum here that have no reference to x. 01:01:43.930 --> 01:01:45.916 align:middle line:84% You can make this less than epsilon. 01:01:45.916 --> 01:01:48.730 align:middle line:90% 01:01:48.730 --> 01:01:51.370 align:middle line:84% So this here is less than epsilon. 01:01:51.370 --> 01:01:53.930 align:middle line:84% And you have this inequality for all x. 01:01:53.930 --> 01:01:57.710 align:middle line:84% So you have this one here less than this for all x. 01:01:57.710 --> 01:01:59.830 align:middle line:84% And that's exactly uniform convergence. 01:01:59.830 --> 01:02:00.530 align:middle line:90% Yeah? 01:02:00.530 --> 01:02:03.090 align:middle line:84% AUDIENCE: Where did we get the first-- 01:02:03.090 --> 01:02:05.890 align:middle line:84% when we said that the sum of these Mk's are 01:02:05.890 --> 01:02:08.590 align:middle line:84% less than epsilon, where did that come from? 01:02:08.590 --> 01:02:13.050 align:middle line:90% 01:02:13.050 --> 01:02:16.970 align:middle line:90% TOBIAS COLDING: So let me just-- 01:02:16.970 --> 01:02:19.750 align:middle line:84% I'll get to this in just a second. 01:02:19.750 --> 01:02:23.690 align:middle line:84% But let me just say what is the strategy here. 01:02:23.690 --> 01:02:27.016 align:middle line:90% So we have this sequence. 01:02:27.016 --> 01:02:30.170 align:middle line:90% 01:02:30.170 --> 01:02:32.220 align:middle line:84% Maybe I erased the statement, unfortunately. 01:02:32.220 --> 01:02:38.660 align:middle line:84% So we have this sequence of functions on an interval. 01:02:38.660 --> 01:02:45.380 align:middle line:84% And you had that this function here was bounded like this. 01:02:45.380 --> 01:02:49.900 align:middle line:84% And then you had that these are just real numbers. 01:02:49.900 --> 01:02:56.900 align:middle line:84% And you had that these real numbers here were summable. 01:02:56.900 --> 01:03:01.164 align:middle line:84% So these are non-negative, real numbers, of course. 01:03:01.164 --> 01:03:05.020 align:middle line:90% And so then-- and I claim-- 01:03:05.020 --> 01:03:10.820 align:middle line:84% so the claim is that the sum here of these guys, these 01:03:10.820 --> 01:03:17.620 align:middle line:84% here, I claim that these guys here converge uniformly to some 01:03:17.620 --> 01:03:20.340 align:middle line:90% function S of x. 01:03:20.340 --> 01:03:25.460 align:middle line:84% The first step is to say that they actually-- 01:03:25.460 --> 01:03:28.700 align:middle line:84% that there exists some function S, 01:03:28.700 --> 01:03:32.480 align:middle line:84% and they converge to that function pointwise. 01:03:32.480 --> 01:03:40.240 align:middle line:84% In order to prove that, I proved that these sums here, 01:03:40.240 --> 01:03:43.700 align:middle line:84% for each fixed x, is a Cauchy sequence. 01:03:43.700 --> 01:03:47.320 align:middle line:84% Once there is a Cauchy sequence for each fixed x, 01:03:47.320 --> 01:03:49.800 align:middle line:90% then they will be convergent. 01:03:49.800 --> 01:03:55.000 align:middle line:84% And the limits are now defined to be S of x. 01:03:55.000 --> 01:03:57.200 align:middle line:90% So that gives me a function. 01:03:57.200 --> 01:03:58.840 align:middle line:84% And then the thing is that I have 01:03:58.840 --> 01:04:01.240 align:middle line:84% to upgrade this convergence, which 01:04:01.240 --> 01:04:06.220 align:middle line:84% initially is pointwise to uniform convergence. 01:04:06.220 --> 01:04:08.580 align:middle line:90% And so now you are asking-- 01:04:08.580 --> 01:04:10.200 align:middle line:90% so now-- 01:04:10.200 --> 01:04:14.440 align:middle line:84% AUDIENCE: Why did-- so we got the sum of Mk's But then where 01:04:14.440 --> 01:04:14.960 align:middle line:90% did we-- 01:04:14.960 --> 01:04:16.292 align:middle line:90% how did we just [INAUDIBLE]? 01:04:16.292 --> 01:04:17.500 align:middle line:90% TOBIAS COLDING: That's right. 01:04:17.500 --> 01:04:21.600 align:middle line:84% And so now-- so this here was just very general. 01:04:21.600 --> 01:04:25.180 align:middle line:84% And it just used this inequality. 01:04:25.180 --> 01:04:27.360 align:middle line:90% That's all it used. 01:04:27.360 --> 01:04:31.370 align:middle line:84% Now I'm using that these guys are summable. 01:04:31.370 --> 01:04:34.410 align:middle line:84% So this means that if I'm starting this summation 01:04:34.410 --> 01:04:37.370 align:middle line:84% sufficiently far out, then I can make 01:04:37.370 --> 01:04:39.676 align:middle line:90% it less than a given epsilon. 01:04:39.676 --> 01:04:40.490 align:middle line:90% AUDIENCE: Oh, OK. 01:04:40.490 --> 01:04:41.810 align:middle line:90% Because it's technically-- 01:04:41.810 --> 01:04:43.643 align:middle line:84% TOBIAS COLDING: Because it's just like that. 01:04:43.643 --> 01:04:45.330 align:middle line:90% These are non-negative number. 01:04:45.330 --> 01:04:48.010 align:middle line:84% And that they are summable just means 01:04:48.010 --> 01:04:52.470 align:middle line:84% that if I go out really, really far, then from there on out, 01:04:52.470 --> 01:04:55.890 align:middle line:84% it makes very little contribution to the total sum. 01:04:55.890 --> 01:04:58.570 align:middle line:90% OK. 01:04:58.570 --> 01:04:59.070 align:middle line:90% OK. 01:04:59.070 --> 01:05:01.330 align:middle line:90% So right. 01:05:01.330 --> 01:05:04.950 align:middle line:90% So right. 01:05:04.950 --> 01:05:11.370 align:middle line:84% So you see the first thing is, for each fixed x, 01:05:11.370 --> 01:05:17.210 align:middle line:84% I prove that these Sn of x is a Cauchy sequence that then allow 01:05:17.210 --> 01:05:21.170 align:middle line:90% me to define a function S of x. 01:05:21.170 --> 01:05:25.190 align:middle line:84% And I get that the sequence Sn of x is converging. 01:05:25.190 --> 01:05:27.810 align:middle line:84% The sequence function is converging pointwise. 01:05:27.810 --> 01:05:30.090 align:middle line:84% And then again, once I have that, 01:05:30.090 --> 01:05:33.450 align:middle line:84% then I need to upgrade it to be uniform convergence. 01:05:33.450 --> 01:05:35.290 align:middle line:90% And that's what we did here. 01:05:35.290 --> 01:05:35.790 align:middle line:90% OK? 01:05:35.790 --> 01:05:38.670 align:middle line:90% 01:05:38.670 --> 01:05:39.750 align:middle line:90% OK. 01:05:39.750 --> 01:05:44.710 align:middle line:90% So now let's try to apply this. 01:05:44.710 --> 01:05:45.870 align:middle line:90% Yeah? 01:05:45.870 --> 01:05:52.030 align:middle line:84% AUDIENCE: So what if the Mk's are kind of converging sequence 01:05:52.030 --> 01:05:58.070 align:middle line:84% but in the small side, the 0 side, is on a low index? 01:05:58.070 --> 01:06:05.270 align:middle line:84% So then, as I go further out, it's getting bigger. 01:06:05.270 --> 01:06:07.090 align:middle line:84% TOBIAS COLDING: Can we look at an example 01:06:07.090 --> 01:06:09.630 align:middle line:90% and see if it becomes clear? 01:06:09.630 --> 01:06:13.275 align:middle line:90% So let's look at a sort of-- 01:06:13.275 --> 01:06:16.670 align:middle line:90% let's look at an example. 01:06:16.670 --> 01:06:21.830 align:middle line:84% Let's look at a very-- yeah, let's look at an example. 01:06:21.830 --> 01:06:25.550 align:middle line:84% So in fact, let's look at the function, the example we just 01:06:25.550 --> 01:06:27.760 align:middle line:84% looked at, where we said that it followed 01:06:27.760 --> 01:06:30.820 align:middle line:90% from Weierstrass M-test. 01:06:30.820 --> 01:06:34.600 align:middle line:84% So what was the example I just looked at? 01:06:34.600 --> 01:06:39.480 align:middle line:90% So I take a function fn of x. 01:06:39.480 --> 01:06:46.360 align:middle line:84% This is now x to the n divided by n factorial. 01:06:46.360 --> 01:06:51.400 align:middle line:90% Now I define Sn of x. 01:06:51.400 --> 01:06:55.000 align:middle line:84% So this is now the sum from k equal to 0 01:06:55.000 --> 01:06:57.960 align:middle line:90% to n of this fn of x-- 01:06:57.960 --> 01:06:59.680 align:middle line:90% fk of x, sorry. 01:06:59.680 --> 01:07:02.080 align:middle line:84% So this means that this thing here 01:07:02.080 --> 01:07:10.440 align:middle line:84% is the sum from k equal to 0 to n of xk over k factorial. 01:07:10.440 --> 01:07:11.600 align:middle line:90% OK? 01:07:11.600 --> 01:07:14.720 align:middle line:90% And I want to show that this-- 01:07:14.720 --> 01:07:23.480 align:middle line:84% so I want to show that this Sn of x 01:07:23.480 --> 01:07:34.930 align:middle line:84% converges to the exponential function uniformly 01:07:34.930 --> 01:07:39.370 align:middle line:90% on an interval of this form. 01:07:39.370 --> 01:07:42.010 align:middle line:84% So L is some fixed number, and I'm 01:07:42.010 --> 01:07:45.950 align:middle line:84% looking between minus L to L. And I claim that they converge. 01:07:45.950 --> 01:07:49.230 align:middle line:84% Now, let's try to apply Weierstrass M-test. 01:07:49.230 --> 01:07:53.210 align:middle line:90% And so what are the Mn? 01:07:53.210 --> 01:07:56.170 align:middle line:84% The Mn, in this case, the natural one 01:07:56.170 --> 01:08:01.670 align:middle line:84% is L to the power n times n factorial. 01:08:01.670 --> 01:08:04.622 align:middle line:90% 01:08:04.622 --> 01:08:10.730 align:middle line:84% If I'm looking at those numbers, these are non-negative numbers. 01:08:10.730 --> 01:08:16.490 align:middle line:84% And we already know that this sequence of non-negative numbers 01:08:16.490 --> 01:08:20.569 align:middle line:84% is summable, that these non-negative numbers, 01:08:20.569 --> 01:08:22.970 align:middle line:84% like this, that they are summable. 01:08:22.970 --> 01:08:29.670 align:middle line:90% 01:08:29.670 --> 01:08:32.109 align:middle line:90% And suppose now I have-- 01:08:32.109 --> 01:08:36.950 align:middle line:90% suppose now I take this fn of x. 01:08:36.950 --> 01:08:41.870 align:middle line:84% So this is this function x to the power n over n factorial. 01:08:41.870 --> 01:08:46.830 align:middle line:84% x here, remember, is in this bounded interval. 01:08:46.830 --> 01:08:50.950 align:middle line:90% Well, let's look at fn of x. 01:08:50.950 --> 01:08:56.470 align:middle line:90% So this is xn over n factorial. 01:08:56.470 --> 01:08:58.470 align:middle line:84% But this thing here-- so this is just 01:08:58.470 --> 01:09:02.710 align:middle line:90% x to the power n n factorial. 01:09:02.710 --> 01:09:05.130 align:middle line:90% This is why it's lying in here. 01:09:05.130 --> 01:09:08.710 align:middle line:84% So the absolute value is bounded by L. 01:09:08.710 --> 01:09:13.250 align:middle line:90% So this is L n over n factorial. 01:09:13.250 --> 01:09:20.470 align:middle line:84% So I have that this sequence is less than this Mn. 01:09:20.470 --> 01:09:22.939 align:middle line:90% And so now, Weierstrass. 01:09:22.939 --> 01:09:27.399 align:middle line:84% But we restrict to this interval here. 01:09:27.399 --> 01:09:34.840 align:middle line:84% And so now Weierstrass M-test tells you that the sum here, 01:09:34.840 --> 01:09:41.920 align:middle line:84% when you're summing from k to 0 to n xk over k factorial, 01:09:41.920 --> 01:09:52.220 align:middle line:84% that these actually converge to the limit uniformly on-- 01:09:52.220 --> 01:09:56.160 align:middle line:84% But remember, we are restricting. 01:09:56.160 --> 01:10:01.320 align:middle line:84% And the limit here, we already have defined the limit 01:10:01.320 --> 01:10:03.030 align:middle line:90% to be the exponential function. 01:10:03.030 --> 01:10:08.290 align:middle line:90% 01:10:08.290 --> 01:10:10.200 align:middle line:90% Is that clear or not? 01:10:10.200 --> 01:10:13.520 align:middle line:90% Does that answer, or not really? 01:10:13.520 --> 01:10:14.880 align:middle line:90% AUDIENCE: I'm not sure. 01:10:14.880 --> 01:10:16.140 align:middle line:90% TOBIAS COLDING: OK. 01:10:16.140 --> 01:10:19.070 align:middle line:84% AUDIENCE: So I guess my question was, 01:10:19.070 --> 01:10:23.465 align:middle line:84% is it that Mn must be a convergent series? 01:10:23.465 --> 01:10:24.340 align:middle line:90% TOBIAS COLDING: Yeah. 01:10:24.340 --> 01:10:26.060 align:middle line:90% So Mn-- that's right. 01:10:26.060 --> 01:10:28.620 align:middle line:90% So Mn. 01:10:28.620 --> 01:10:34.385 align:middle line:84% Mn, in this example or in the general example, it's-- 01:10:34.385 --> 01:10:37.180 align:middle line:90% 01:10:37.180 --> 01:10:40.440 align:middle line:84% so why is the Mn so much easier to determine? 01:10:40.440 --> 01:10:42.200 align:middle line:90% Because they are just numbers. 01:10:42.200 --> 01:10:44.280 align:middle line:84% They are just non-negative numbers. 01:10:44.280 --> 01:10:48.980 align:middle line:84% And you're just asking if the series of non-negative numbers-- 01:10:48.980 --> 01:10:51.340 align:middle line:84% it's not function, it's non-negative numbers-- 01:10:51.340 --> 01:10:53.160 align:middle line:90% if that's convergent or not. 01:10:53.160 --> 01:10:56.740 align:middle line:84% And so since they're non-negative, 01:10:56.740 --> 01:11:01.145 align:middle line:84% it just means that the sum is finite. 01:11:01.145 --> 01:11:07.920 align:middle line:84% And so that's much easier to determine. 01:11:07.920 --> 01:11:08.420 align:middle line:90% OK. 01:11:08.420 --> 01:11:11.680 align:middle line:90% 01:11:11.680 --> 01:11:17.660 align:middle line:84% So now we can do the same argument more generally. 01:11:17.660 --> 01:11:26.790 align:middle line:84% So this leads us to the following, that suppose 01:11:26.790 --> 01:11:28.080 align:middle line:90% you take a power series. 01:11:28.080 --> 01:11:33.750 align:middle line:90% 01:11:33.750 --> 01:11:36.030 align:middle line:84% So suppose you take a power series. 01:11:36.030 --> 01:11:42.210 align:middle line:84% So you're summing here n equal to 0 to infinity a n xn. 01:11:42.210 --> 01:11:44.310 align:middle line:90% This is a power series. 01:11:44.310 --> 01:11:45.370 align:middle line:90% So power series. 01:11:45.370 --> 01:11:48.110 align:middle line:84% Remember that a power series is where 01:11:48.110 --> 01:11:53.070 align:middle line:84% you're taking a sequence of numbers, real numbers, 01:11:53.070 --> 01:11:56.630 align:middle line:84% and you think about these as coefficients in this polynomial. 01:11:56.630 --> 01:11:58.870 align:middle line:84% And then you're summing over them. 01:11:58.870 --> 01:12:07.350 align:middle line:84% But it's like-- a n is the coefficient to x to the power n. 01:12:07.350 --> 01:12:08.690 align:middle line:90% So this is a power series. 01:12:08.690 --> 01:12:10.410 align:middle line:90% And a power series again means-- 01:12:10.410 --> 01:12:14.390 align:middle line:84% so you think about it as a series because for each x, 01:12:14.390 --> 01:12:16.970 align:middle line:90% it's like a regular series. 01:12:16.970 --> 01:12:23.090 align:middle line:84% But then it's like the value that you get depends on x. 01:12:23.090 --> 01:12:26.990 align:middle line:84% And so for this, we defined the radius of convergence. 01:12:26.990 --> 01:12:34.610 align:middle line:90% 01:12:34.610 --> 01:12:38.930 align:middle line:84% And let me remind you how this radius of convergence 01:12:38.930 --> 01:12:41.090 align:middle line:90% was defined. 01:12:41.090 --> 01:12:47.710 align:middle line:84% You take the a n, and you take the absolute value of it, 01:12:47.710 --> 01:12:50.650 align:middle line:84% and then you take the n-th root of that. 01:12:50.650 --> 01:12:56.150 align:middle line:84% And then you're looking at limsup as n goes to infinity. 01:12:56.150 --> 01:13:00.418 align:middle line:90% 01:13:00.418 --> 01:13:02.890 align:middle line:90% That gives you some number. 01:13:02.890 --> 01:13:06.490 align:middle line:84% That number potentially could be infinity. 01:13:06.490 --> 01:13:09.990 align:middle line:84% It's always non-negative, and it will always give you a number. 01:13:09.990 --> 01:13:12.250 align:middle line:84% It could be 0, it could be infinity, 01:13:12.250 --> 01:13:15.010 align:middle line:84% and it could be anything in between. 01:13:15.010 --> 01:13:17.560 align:middle line:84% And then the radius of convergence-- 01:13:17.560 --> 01:13:21.980 align:middle line:90% 01:13:21.980 --> 01:13:33.770 align:middle line:84% of convergence of the power series is-- 01:13:33.770 --> 01:13:36.660 align:middle line:90% 01:13:36.660 --> 01:13:43.260 align:middle line:84% so this is right by R. And this is 1 over M. 01:13:43.260 --> 01:13:47.473 align:middle line:84% And where this convention in this case-- usually, 01:13:47.473 --> 01:13:51.240 align:middle line:84% you don't divide by 0 or infinity. 01:13:51.240 --> 01:13:54.560 align:middle line:84% But in this case, if M here is 0, 01:13:54.560 --> 01:13:58.180 align:middle line:84% this means that the radius of convergence is infinite. 01:13:58.180 --> 01:14:04.300 align:middle line:84% If M is infinite, then it means that the radius of convergence 01:14:04.300 --> 01:14:04.940 align:middle line:90% is 0. 01:14:04.940 --> 01:14:10.000 align:middle line:84% And now the slightly surprising fact. 01:14:10.000 --> 01:14:11.260 align:middle line:90% But we proved this. 01:14:11.260 --> 01:14:13.860 align:middle line:84% But I'm just reminding you about it, 01:14:13.860 --> 01:14:22.710 align:middle line:84% that the thing is that the power series is convergent. 01:14:22.710 --> 01:14:24.230 align:middle line:90% So here you have 0. 01:14:24.230 --> 01:14:27.090 align:middle line:90% It's convergent inside. 01:14:27.090 --> 01:14:33.030 align:middle line:84% If this here is R, it's convergent inside here, 01:14:33.030 --> 01:14:35.670 align:middle line:90% and it's divergent outside. 01:14:35.670 --> 01:14:40.782 align:middle line:84% But it's not completely clear what happened on the boundary. 01:14:40.782 --> 01:14:47.010 align:middle line:84% So you have to check the boundary separately. 01:14:47.010 --> 01:14:48.930 align:middle line:90% And that depends on the power. 01:14:48.930 --> 01:14:52.110 align:middle line:84% There's no general formula for that. 01:14:52.110 --> 01:14:58.510 align:middle line:84% So now the theorem here is that-- 01:14:58.510 --> 01:15:00.330 align:middle line:90% so the theorem is like this. 01:15:00.330 --> 01:15:04.430 align:middle line:84% And this is just generalizing this example that we just did. 01:15:04.430 --> 01:15:07.990 align:middle line:84% And the theorem is the following, 01:15:07.990 --> 01:15:10.000 align:middle line:84% that if you take a power series-- 01:15:10.000 --> 01:15:13.430 align:middle line:90% 01:15:13.430 --> 01:15:24.810 align:middle line:84% so if you have a power series and R 01:15:24.810 --> 01:15:40.070 align:middle line:84% is the radius of convergence and L here, 01:15:40.070 --> 01:15:45.210 align:middle line:84% it has to be strictly less than the radius of convergence, 01:15:45.210 --> 01:15:55.530 align:middle line:84% then these functions here, k equal to 0 to n a n-- 01:15:55.530 --> 01:16:01.250 align:middle line:84% ak sorry, xk, that these functions here 01:16:01.250 --> 01:16:27.400 align:middle line:84% converge to the limit here uniformly on this interval here, 01:16:27.400 --> 01:16:29.260 align:middle line:90% from minus L to L. 01:16:29.260 --> 01:16:32.460 align:middle line:84% Remember that L had to be strictly less 01:16:32.460 --> 01:16:36.836 align:middle line:90% than the radius of convergence. 01:16:36.836 --> 01:16:39.660 align:middle line:90% And so this is extremely useful. 01:16:39.660 --> 01:16:45.940 align:middle line:84% If you have-- and it will be clear next time why it's useful 01:16:45.940 --> 01:16:48.940 align:middle line:84% because if you have uniform convergence, 01:16:48.940 --> 01:16:51.580 align:middle line:90% then it becomes very easy-- 01:16:51.580 --> 01:16:53.660 align:middle line:84% if you have uniform convergence, then you 01:16:53.660 --> 01:16:56.620 align:middle line:84% can argue about if you want to take 01:16:56.620 --> 01:17:00.220 align:middle line:84% the derivative of integrating this thing over here, 01:17:00.220 --> 01:17:03.540 align:middle line:84% if you have uniform convergence, then 01:17:03.540 --> 01:17:08.100 align:middle line:84% it will say that then a lot of operations 01:17:08.100 --> 01:17:10.480 align:middle line:84% will be more precise about that next time. 01:17:10.480 --> 01:17:14.880 align:middle line:84% But a lot of operations, like integration and differentiation, 01:17:14.880 --> 01:17:19.580 align:middle line:84% you can do that over here and then go to the limit. 01:17:19.580 --> 01:17:24.520 align:middle line:84% So instead of if you were to integrate this thing here. 01:17:24.520 --> 01:17:29.120 align:middle line:84% So just say that-- but this seems maybe very, very hard. 01:17:29.120 --> 01:17:31.720 align:middle line:84% But the thing is, if you can integrate here, 01:17:31.720 --> 01:17:34.360 align:middle line:84% then you can just use the usual sum, 01:17:34.360 --> 01:17:36.720 align:middle line:90% the usual formula for the sum. 01:17:36.720 --> 01:17:38.760 align:middle line:84% It's just the sum of the integral. 01:17:38.760 --> 01:17:42.680 align:middle line:84% And likewise for differentiation. 01:17:42.680 --> 01:17:45.510 align:middle line:90% And so, OK, maybe I will just-- 01:17:45.510 --> 01:17:48.420 align:middle line:90% 01:17:48.420 --> 01:17:48.920 align:middle line:90% yeah. 01:17:48.920 --> 01:17:50.760 align:middle line:90% Maybe I'll just stop here. 01:17:50.760 --> 01:17:55.480 align:middle line:84% I wrote out the proof in the notes. 01:17:55.480 --> 01:17:59.080 align:middle line:84% And I already posted the notes, the lecture notes. 01:17:59.080 --> 01:18:00.720 align:middle line:90% Are there any questions? 01:18:00.720 --> 01:18:05.560 align:middle line:90% 01:18:05.560 --> 01:18:06.320 align:middle line:90% OK. 01:18:06.320 --> 01:18:07.870 align:middle line:90% Great. 01:18:07.870 --> 01:18:11.000 align:middle line:90%