WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.920 align:middle line:90% [RUSTLING] 00:00:03.920 --> 00:00:06.370 align:middle line:90% [CLICKING] 00:00:06.370 --> 00:00:11.100 align:middle line:90% 00:00:11.100 --> 00:00:16.640 align:middle line:84% TOBIAS COLDING: OK, so assume that again that we have-- 00:00:16.640 --> 00:00:17.810 align:middle line:90% so this is a sequence. 00:00:17.810 --> 00:00:21.760 align:middle line:90% 00:00:21.760 --> 00:00:26.700 align:middle line:84% Then from a sequence, we can form another sequence. 00:00:26.700 --> 00:00:46.280 align:middle line:84% So from a-n we can form another sequence, at least. 00:00:46.280 --> 00:00:49.490 align:middle line:90% If a-n is bounded from above. 00:00:49.490 --> 00:00:58.080 align:middle line:90% 00:00:58.080 --> 00:01:03.530 align:middle line:84% So we can form another sequence, and that's what we take. 00:01:03.530 --> 00:01:12.150 align:middle line:84% B-n is going to be the soup of the set of a, starting at n. 00:01:12.150 --> 00:01:16.946 align:middle line:90% 00:01:16.946 --> 00:01:21.400 align:middle line:84% So this here is the sequence b-n, OK? 00:01:21.400 --> 00:01:24.036 align:middle line:90% 00:01:24.036 --> 00:01:27.110 align:middle line:84% But again, implicitly here we are assuming. 00:01:27.110 --> 00:01:46.570 align:middle line:84% So assume that there exists some c, so that for all n a-n 00:01:46.570 --> 00:01:47.470 align:middle line:90% is less than c. 00:01:47.470 --> 00:01:51.250 align:middle line:84% In other words, the c are bounded from above, right? 00:01:51.250 --> 00:01:56.290 align:middle line:84% So if the sequence a-n is bounded from above, 00:01:56.290 --> 00:02:02.430 align:middle line:84% then this here gives us a real number, 00:02:02.430 --> 00:02:11.190 align:middle line:84% and we see that if you're looking at b-n plus 1, 00:02:11.190 --> 00:02:20.590 align:middle line:84% this is now sup of a-n plus 1, a-n plus 2. 00:02:20.590 --> 00:02:25.750 align:middle line:84% And this here is sup over a smaller set, 00:02:25.750 --> 00:02:31.750 align:middle line:84% so this is less than the sup where 00:02:31.750 --> 00:02:33.500 align:middle line:90% we take sup over a larger set. 00:02:33.500 --> 00:02:36.230 align:middle line:90% 00:02:36.230 --> 00:02:37.860 align:middle line:90% And that here is b-n. 00:02:37.860 --> 00:02:40.830 align:middle line:90% 00:02:40.830 --> 00:02:45.410 align:middle line:84% So we have that if you take any sequence, 00:02:45.410 --> 00:02:48.610 align:middle line:84% and let's say that that sequence is bounded from above, 00:02:48.610 --> 00:02:52.950 align:middle line:84% then you can form another sequence b-n, 00:02:52.950 --> 00:02:56.740 align:middle line:84% and b-n here is monotone decreasing. 00:02:56.740 --> 00:03:05.160 align:middle line:90% 00:03:05.160 --> 00:03:22.400 align:middle line:84% If the sequence b-n is bounded from below, 00:03:22.400 --> 00:03:25.040 align:middle line:84% then the b-n here in the sequence 00:03:25.040 --> 00:03:30.680 align:middle line:84% here converge, and it converges to the infimum, right? 00:03:30.680 --> 00:03:34.160 align:middle line:84% This is this with monotone converging. 00:03:34.160 --> 00:03:37.640 align:middle line:84% Sequences, it's converging to the infimum 00:03:37.640 --> 00:03:44.050 align:middle line:90% of the b-n like that. 00:03:44.050 --> 00:03:44.550 align:middle line:90% OK. 00:03:44.550 --> 00:03:55.460 align:middle line:90% 00:03:55.460 --> 00:03:58.060 align:middle line:84% So likewise, although we won't really 00:03:58.060 --> 00:04:07.920 align:middle line:84% use this here, but likewise, again, 00:04:07.920 --> 00:04:13.380 align:middle line:84% if a-n is any sequence, any sequence, 00:04:13.380 --> 00:04:15.910 align:middle line:84% this time I want it to be bounded from below. 00:04:15.910 --> 00:04:22.700 align:middle line:90% 00:04:22.700 --> 00:04:25.540 align:middle line:90% Then you can find a sequence. 00:04:25.540 --> 00:04:31.140 align:middle line:84% From a-n, you can form a sequence, c-n, and c-n 00:04:31.140 --> 00:04:35.210 align:middle line:84% is just where you're taking the infimum, starting at a-n. 00:04:35.210 --> 00:04:40.240 align:middle line:90% 00:04:40.240 --> 00:04:44.980 align:middle line:84% And you see that if you're looking at c-n plus 1, then 00:04:44.980 --> 00:04:48.310 align:middle line:84% you take the infimum over a smaller set. 00:04:48.310 --> 00:04:55.750 align:middle line:90% 00:04:55.750 --> 00:04:58.630 align:middle line:90% So this is a smaller set. 00:04:58.630 --> 00:05:04.550 align:middle line:84% And so this here is bigger than where you're taking the infimum 00:05:04.550 --> 00:05:11.710 align:middle line:90% from a-n, and that's c-n. 00:05:11.710 --> 00:05:17.550 align:middle line:84% So the sequence c-n is therefore monotone increasing. 00:05:17.550 --> 00:05:19.690 align:middle line:90% And if it is bounded-- 00:05:19.690 --> 00:05:38.710 align:middle line:84% so if c-n Is bounded from above, then the sequence c-n converge 00:05:38.710 --> 00:05:48.230 align:middle line:84% to some c, and that c is just a sup of the c;s, right? 00:05:48.230 --> 00:05:55.490 align:middle line:84% And we talked a little bit about this earlier, 00:05:55.490 --> 00:05:59.650 align:middle line:84% but now we will use it in just a second. 00:05:59.650 --> 00:06:04.490 align:middle line:84% The thing is, though, that in the first case here, 00:06:04.490 --> 00:06:13.340 align:middle line:84% we call b here, B is written as the limsup of the a-n. 00:06:13.340 --> 00:06:15.850 align:middle line:90% 00:06:15.850 --> 00:06:25.580 align:middle line:84% And in the second case, it's called the liminf of the a-n. 00:06:25.580 --> 00:06:26.080 align:middle line:90% OK. 00:06:26.080 --> 00:06:33.010 align:middle line:90% 00:06:33.010 --> 00:06:35.610 align:middle line:84% So why is this relevant here for us? 00:06:35.610 --> 00:06:40.170 align:middle line:84% So what we were looking at last time-- 00:06:40.170 --> 00:06:43.050 align:middle line:84% and I'm only going to use the limsup. 00:06:43.050 --> 00:06:46.630 align:middle line:90% So last time, we had a sequence. 00:06:46.630 --> 00:06:57.060 align:middle line:84% So a-n was a sequence, and we defined 00:06:57.060 --> 00:06:59.500 align:middle line:90% what we call a power series. 00:06:59.500 --> 00:07:02.860 align:middle line:84% And again, power series, think about that 00:07:02.860 --> 00:07:06.260 align:middle line:84% x as being just some fixed number, 00:07:06.260 --> 00:07:07.900 align:middle line:90% and you have the sequence a-n. 00:07:07.900 --> 00:07:14.740 align:middle line:84% X is some fixed number, and then you form the sequence a-n x-n, 00:07:14.740 --> 00:07:16.980 align:middle line:90% right? 00:07:16.980 --> 00:07:23.180 align:middle line:84% And so for each x fixed, this gives you some number. 00:07:23.180 --> 00:07:24.860 align:middle line:90% Maybe, if it's converged. 00:07:24.860 --> 00:07:26.360 align:middle line:90% I mean, you can ask for each x. 00:07:26.360 --> 00:07:29.420 align:middle line:84% You can ask, does this sequence converge or not? 00:07:29.420 --> 00:07:32.260 align:middle line:84% And this here is called a power series. 00:07:32.260 --> 00:07:34.760 align:middle line:84% So you're summing from 0 to whatever. 00:07:34.760 --> 00:07:36.900 align:middle line:84% Or you're starting from one, or whatever you're 00:07:36.900 --> 00:07:38.840 align:middle line:90% starting with up to infinity. 00:07:38.840 --> 00:07:41.700 align:middle line:90% So this is a power series. 00:07:41.700 --> 00:07:48.840 align:middle line:84% And so now for power series, the question 00:07:48.840 --> 00:07:54.880 align:middle line:84% is, how serious is the question is, 00:07:54.880 --> 00:08:01.510 align:middle line:84% for which, for which x does it converge? 00:08:01.510 --> 00:08:08.520 align:middle line:90% 00:08:08.520 --> 00:08:10.800 align:middle line:84% So that's what we're interested in,. 00:08:10.800 --> 00:08:13.963 align:middle line:90% 00:08:13.963 --> 00:08:15.880 align:middle line:84% So just let me remind you, because we're going 00:08:15.880 --> 00:08:18.580 align:middle line:90% to use that anyway in a minute. 00:08:18.580 --> 00:08:24.280 align:middle line:90% So example from last time. 00:08:24.280 --> 00:08:29.760 align:middle line:90% 00:08:29.760 --> 00:08:34.600 align:middle line:84% The example we looked at last time was we're looking at x-n. 00:08:34.600 --> 00:08:36.840 align:middle line:84% And we looked at some other examples, too, 00:08:36.840 --> 00:08:39.880 align:middle line:84% but this was the one we're focused on. 00:08:39.880 --> 00:08:42.020 align:middle line:90% So this here is a power series. 00:08:42.020 --> 00:08:47.750 align:middle line:84% The a-n in this case here are 1 over n factorial. 00:08:47.750 --> 00:08:53.810 align:middle line:90% 00:08:53.810 --> 00:09:00.250 align:middle line:84% And for each fixed x, so for x fixed 00:09:00.250 --> 00:09:03.770 align:middle line:84% we looked at whether or not this sequence converges. 00:09:03.770 --> 00:09:07.730 align:middle line:84% And so we think about this thing here. 00:09:07.730 --> 00:09:08.950 align:middle line:90% We thought this here. 00:09:08.950 --> 00:09:11.410 align:middle line:84% We thought about it as just a sequence 00:09:11.410 --> 00:09:16.270 align:middle line:90% like this, sum of a sequence. 00:09:16.270 --> 00:09:21.890 align:middle line:84% So this is as a series like this, where the d-n here is now 00:09:21.890 --> 00:09:24.790 align:middle line:90% x to the n over n factorial. 00:09:24.790 --> 00:09:27.360 align:middle line:90% 00:09:27.360 --> 00:09:32.290 align:middle line:84% And so in order to determine for which x it converges-- so just 00:09:32.290 --> 00:09:33.870 align:middle line:90% think about x as being fixed. 00:09:33.870 --> 00:09:35.300 align:middle line:90% We use the ratio test. 00:09:35.300 --> 00:09:41.290 align:middle line:90% 00:09:41.290 --> 00:09:46.670 align:middle line:84% So we use the ratio test, and the ratio test 00:09:46.670 --> 00:09:52.270 align:middle line:84% said that if you're looking at d-n plus 1 over d-n, 00:09:52.270 --> 00:09:58.470 align:middle line:84% this here is where d-n is just x to the n plus 1, 00:09:58.470 --> 00:10:07.430 align:middle line:84% over n plus 1 factorial, divided by x to the n over n factorial. 00:10:07.430 --> 00:10:09.350 align:middle line:90% And that's the same. 00:10:09.350 --> 00:10:13.290 align:middle line:84% So when you divide by a fraction, you just switch. 00:10:13.290 --> 00:10:17.230 align:middle line:84% So this is the same as x to the n plus 1, divided 00:10:17.230 --> 00:10:25.310 align:middle line:84% by n plus 1 factorial, times n factorial, times x to the n. 00:10:25.310 --> 00:10:27.710 align:middle line:84% And you just think about this, the top 00:10:27.710 --> 00:10:32.710 align:middle line:84% here, as x to the n times x, times n factorial. 00:10:32.710 --> 00:10:37.110 align:middle line:84% And the denominator you think about as n plus 1 times n 00:10:37.110 --> 00:10:39.310 align:middle line:90% factorial, times x to the n. 00:10:39.310 --> 00:10:44.070 align:middle line:84% And you see that this here and this here cancel with those two. 00:10:44.070 --> 00:10:50.160 align:middle line:90% And so you get x over n plus 1. 00:10:50.160 --> 00:10:55.920 align:middle line:84% And this here converges to 0, so therefore the power series 00:10:55.920 --> 00:10:58.160 align:middle line:90% is convergent for all x. 00:10:58.160 --> 00:11:00.780 align:middle line:90% 00:11:00.780 --> 00:11:06.400 align:middle line:84% So that was a particular example of a power series, 00:11:06.400 --> 00:11:11.720 align:middle line:84% and it worked well with the ratio test. 00:11:11.720 --> 00:11:13.980 align:middle line:84% But the word test actually is better. 00:11:13.980 --> 00:11:18.940 align:middle line:84% And this was related to a point somebody raised last time. 00:11:18.940 --> 00:11:23.740 align:middle line:84% And that point was for the ratio test, 00:11:23.740 --> 00:11:28.280 align:middle line:84% it was implicitly somehow in the ratio test, was something 00:11:28.280 --> 00:11:32.160 align:middle line:84% about that from some stage on, the next element 00:11:32.160 --> 00:11:34.380 align:middle line:84% is smaller than the previous one. 00:11:34.380 --> 00:11:38.520 align:middle line:84% And you don't really need to have that. 00:11:38.520 --> 00:11:43.440 align:middle line:84% I mean, you certainly don't need it for a series to converge, 00:11:43.440 --> 00:11:49.260 align:middle line:84% but it's not even necessary if you apply the right test. 00:11:49.260 --> 00:11:56.100 align:middle line:84% And so this brings us to what's called 00:11:56.100 --> 00:12:00.460 align:middle line:90% the radius of convergence. 00:12:00.460 --> 00:12:08.620 align:middle line:84% So this is a crucial concept, and by this crucial concept, 00:12:08.620 --> 00:12:12.140 align:middle line:84% because you see a priori what you could imagine-- 00:12:12.140 --> 00:12:18.200 align:middle line:84% so suppose you just write down a power series like this. 00:12:18.200 --> 00:12:22.180 align:middle line:90% 00:12:22.180 --> 00:12:28.060 align:middle line:84% Maybe it's convergent for when x is equal to 5, 00:12:28.060 --> 00:12:30.980 align:middle line:84% it's divergent when x is equal to 10, 00:12:30.980 --> 00:12:35.440 align:middle line:84% and maybe it's converging again if x is equal to 20. 00:12:35.440 --> 00:12:38.500 align:middle line:84% That's not actually going to be the case. 00:12:38.500 --> 00:12:41.320 align:middle line:84% So there's something called radius of convergence. 00:12:41.320 --> 00:12:50.990 align:middle line:90% 00:12:50.990 --> 00:12:53.190 align:middle line:84% And the thing is that what you get out 00:12:53.190 --> 00:12:57.750 align:middle line:84% of this radius of convergence is that a power series like this 00:12:57.750 --> 00:13:02.830 align:middle line:84% is convergent on some interval, and then outside this interval 00:13:02.830 --> 00:13:04.670 align:middle line:90% is divergent. 00:13:04.670 --> 00:13:08.950 align:middle line:84% So it's not like you have some that is convergent here, 00:13:08.950 --> 00:13:12.550 align:middle line:84% it's divergent there, and then it's converging again. 00:13:12.550 --> 00:13:14.090 align:middle line:84% That's not what's going to happen. 00:13:14.090 --> 00:13:16.590 align:middle line:90% OK? 00:13:16.590 --> 00:13:20.990 align:middle line:84% Part of this we already saw in the different tests 00:13:20.990 --> 00:13:27.830 align:middle line:84% that at the boundary of this interval, 00:13:27.830 --> 00:13:29.750 align:middle line:84% the test is inconclusive, so you have 00:13:29.750 --> 00:13:33.950 align:middle line:84% to check the boundaries separately. 00:13:33.950 --> 00:13:38.790 align:middle line:84% And so the idea is that you check-- 00:13:38.790 --> 00:13:45.370 align:middle line:84% so you have a power series like this, 00:13:45.370 --> 00:13:48.810 align:middle line:84% and you think about-- again, you're thinking about d-n 00:13:48.810 --> 00:13:51.022 align:middle line:90% as x to the n. 00:13:51.022 --> 00:13:54.490 align:middle line:84% X, again, is thought of being fixed, 00:13:54.490 --> 00:13:56.750 align:middle line:90% and now you apply the root test. 00:13:56.750 --> 00:13:58.530 align:middle line:84% So apply the root test, but I'm going 00:13:58.530 --> 00:14:03.000 align:middle line:84% to apply it in a better form than what we did last time. 00:14:03.000 --> 00:14:07.050 align:middle line:90% 00:14:07.050 --> 00:14:10.300 align:middle line:84% Maybe I'll just state first the better form of the root test. 00:14:10.300 --> 00:14:16.970 align:middle line:90% 00:14:16.970 --> 00:14:21.530 align:middle line:84% Here's the real version of the root test. 00:14:21.530 --> 00:14:25.050 align:middle line:90% So suppose you take a series. 00:14:25.050 --> 00:14:26.990 align:middle line:84% Suppose you have some series here. 00:14:26.990 --> 00:14:30.364 align:middle line:90% 00:14:30.364 --> 00:14:34.330 align:middle line:90% Let's just call it d. 00:14:34.330 --> 00:14:36.040 align:middle line:84% Suppose you take a series like this. 00:14:36.040 --> 00:14:41.900 align:middle line:90% 00:14:41.900 --> 00:14:45.900 align:middle line:84% So now the root test, you took d-n, you 00:14:45.900 --> 00:14:49.270 align:middle line:84% took the absolute value, and you took the n-th root. 00:14:49.270 --> 00:14:52.440 align:middle line:90% 00:14:52.440 --> 00:14:55.500 align:middle line:84% When I looked at the root test last time, 00:14:55.500 --> 00:14:59.700 align:middle line:84% we looked at whether this is convergent or not. 00:14:59.700 --> 00:15:03.010 align:middle line:84% So we said if it was converging to something, right? 00:15:03.010 --> 00:15:06.740 align:middle line:84% but you could form this thing here. 00:15:06.740 --> 00:15:09.480 align:middle line:84% So you could form from this here. 00:15:09.480 --> 00:15:10.520 align:middle line:90% You could form this. 00:15:10.520 --> 00:15:14.560 align:middle line:84% This is now a sequence, and you could look at limsup of this. 00:15:14.560 --> 00:15:18.980 align:middle line:90% 00:15:18.980 --> 00:15:25.340 align:middle line:84% These here are all non-negative numbers, 00:15:25.340 --> 00:15:30.740 align:middle line:84% and this limsup here is now going to be either infinite 00:15:30.740 --> 00:15:34.260 align:middle line:84% or it's going to be some real non-negative number. 00:15:34.260 --> 00:15:37.740 align:middle line:84% This here is going to be infinite. 00:15:37.740 --> 00:15:40.440 align:middle line:84% Usually, we don't talk about that it's converging if it's 00:15:40.440 --> 00:15:43.850 align:middle line:90% infinite or it's some number. 00:15:43.850 --> 00:15:46.600 align:middle line:90% 00:15:46.600 --> 00:15:50.040 align:middle line:90% Let's call it d, I guess. 00:15:50.040 --> 00:15:52.800 align:middle line:90% OK? 00:15:52.800 --> 00:16:01.260 align:middle line:84% And the thing is that if you said r equal to 1 over d-- 00:16:01.260 --> 00:16:04.270 align:middle line:90% 00:16:04.270 --> 00:16:04.770 align:middle line:90% sorry. 00:16:04.770 --> 00:16:08.860 align:middle line:90% 00:16:08.860 --> 00:16:12.400 align:middle line:84% This is just a root test, the adult version of the root test, 00:16:12.400 --> 00:16:15.540 align:middle line:84% because this applies to all sequences. 00:16:15.540 --> 00:16:18.040 align:middle line:90% For any sequence, you have this. 00:16:18.040 --> 00:16:22.420 align:middle line:84% For any sequence d-n, if you're looking at this series, 00:16:22.420 --> 00:16:27.840 align:middle line:84% you can look at this, and it's either infinity or d. 00:16:27.840 --> 00:16:32.240 align:middle line:84% And then if d here is less than 1, 00:16:32.240 --> 00:16:34.180 align:middle line:90% then the series is convergent. 00:16:34.180 --> 00:16:40.690 align:middle line:90% 00:16:40.690 --> 00:16:45.390 align:middle line:84% If d is bigger than 1, the series is divergent. 00:16:45.390 --> 00:16:51.970 align:middle line:90% 00:16:51.970 --> 00:16:55.510 align:middle line:84% And if d is equal to 1, it's inconclusive. 00:16:55.510 --> 00:17:02.610 align:middle line:90% 00:17:02.610 --> 00:17:07.210 align:middle line:84% The difference between this and what we talked about last time 00:17:07.210 --> 00:17:12.810 align:middle line:84% was that last time we assumed that this thing here converged. 00:17:12.810 --> 00:17:17.170 align:middle line:84% Here, I'm not assuming it, but then I'm looking at the limsup 00:17:17.170 --> 00:17:18.690 align:middle line:90% instead. 00:17:18.690 --> 00:17:24.810 align:middle line:84% So this does give us a number, and it's just 00:17:24.810 --> 00:17:29.530 align:middle line:84% a matter of whether this limsup is smaller than 1. 00:17:29.530 --> 00:17:31.990 align:middle line:84% And then it's convergent or bigger than 1, 00:17:31.990 --> 00:17:33.110 align:middle line:90% then it's divergent. 00:17:33.110 --> 00:17:36.830 align:middle line:90% 00:17:36.830 --> 00:17:39.430 align:middle line:84% Now I'll come back to the proof, but the proof 00:17:39.430 --> 00:17:41.970 align:middle line:84% is really kind of the same as the one we gave. 00:17:41.970 --> 00:17:45.790 align:middle line:84% The one we gave, really, if you mimic 00:17:45.790 --> 00:17:50.430 align:middle line:84% what we did in the special case last time, 00:17:50.430 --> 00:17:54.030 align:middle line:84% then the proof would still apply to this case 00:17:54.030 --> 00:18:01.110 align:middle line:84% here and give this result. But let's 00:18:01.110 --> 00:18:06.130 align:middle line:84% take this more fancy version of the root test. 00:18:06.130 --> 00:18:07.740 align:middle line:90% Let's take that for granted. 00:18:07.740 --> 00:18:15.870 align:middle line:90% 00:18:15.870 --> 00:18:20.870 align:middle line:84% So now I have a power series like this. 00:18:20.870 --> 00:18:31.750 align:middle line:84% Just for clarity, I define d-n to be just a-n, times x-n. 00:18:31.750 --> 00:18:42.880 align:middle line:84% And I'm interested in is for what x, for what x 00:18:42.880 --> 00:18:44.380 align:middle line:90% does the power series converge? 00:18:44.380 --> 00:19:00.720 align:middle line:90% 00:19:00.720 --> 00:19:07.560 align:middle line:84% So you are looking at limsup of the n-th root of d-n 00:19:07.560 --> 00:19:16.155 align:middle line:84% is a-n times x-n, and you take the absolute value of that. 00:19:16.155 --> 00:19:18.640 align:middle line:84% That's what you're looking at, and you're 00:19:18.640 --> 00:19:23.120 align:middle line:84% asking is this smaller than 1 or bigger than 1? 00:19:23.120 --> 00:19:27.997 align:middle line:84% But this thing here, right, this is the same as limsup, right? 00:19:27.997 --> 00:19:29.580 align:middle line:84% If you're looking at this thing here-- 00:19:29.580 --> 00:19:32.100 align:middle line:84% let's do a little side calculation-- if you're 00:19:32.100 --> 00:19:38.860 align:middle line:84% looking at the n-th root of a-n times x-n, 00:19:38.860 --> 00:19:44.780 align:middle line:84% this thing here is the same as the n-th root of this product 00:19:44.780 --> 00:19:45.280 align:middle line:90% here. 00:19:45.280 --> 00:19:49.220 align:middle line:84% Absolute value is the same as this times 00:19:49.220 --> 00:19:53.500 align:middle line:90% x absolute value to n. 00:19:53.500 --> 00:19:55.820 align:middle line:84% And when you take the n-th root, you 00:19:55.820 --> 00:19:59.820 align:middle line:84% may as well take the n-th root of each of those two factors 00:19:59.820 --> 00:20:01.460 align:middle line:90% individually. 00:20:01.460 --> 00:20:02.920 align:middle line:90% So this is what you get. 00:20:02.920 --> 00:20:08.020 align:middle line:90% 00:20:08.020 --> 00:20:11.500 align:middle line:84% But this thing here, you see, this thing here, 00:20:11.500 --> 00:20:15.380 align:middle line:84% the n-th root-- this is whatever it is. 00:20:15.380 --> 00:20:18.020 align:middle line:90% Can't do much about that. 00:20:18.020 --> 00:20:20.740 align:middle line:84% But this thing here, that's just the norm 00:20:20.740 --> 00:20:24.540 align:middle line:84% of x, and the norm of x has the property that when you raise it 00:20:24.540 --> 00:20:26.580 align:middle line:84% to the n-th power, you're getting this. 00:20:26.580 --> 00:20:29.670 align:middle line:90% So this is the n. 00:20:29.670 --> 00:20:32.310 align:middle line:84% So this means that this thing here 00:20:32.310 --> 00:20:41.920 align:middle line:84% is the same as the norm of x limsup of the n-th root of a-n. 00:20:41.920 --> 00:20:46.240 align:middle line:90% 00:20:46.240 --> 00:20:47.570 align:middle line:90% Often you do it like this. 00:20:47.570 --> 00:20:49.130 align:middle line:90% You don't have to do this. 00:20:49.130 --> 00:20:52.850 align:middle line:84% It's usually clear anyway, but often you write it like that. 00:20:52.850 --> 00:20:55.456 align:middle line:90% 00:20:55.456 --> 00:21:00.630 align:middle line:84% So now what you do is you are asking 00:21:00.630 --> 00:21:03.890 align:middle line:84% when is this number here smaller than 1? 00:21:03.890 --> 00:21:05.250 align:middle line:90% When is it bigger than 1? 00:21:05.250 --> 00:21:08.490 align:middle line:90% 00:21:08.490 --> 00:21:10.750 align:middle line:84% You see, now you define something 00:21:10.750 --> 00:21:14.750 align:middle line:84% that's called the radius of convergence. 00:21:14.750 --> 00:21:17.150 align:middle line:90% And so this is the following. 00:21:17.150 --> 00:21:24.790 align:middle line:90% So the radius of convergence-- 00:21:24.790 --> 00:21:30.850 align:middle line:84% So this is just radius of convergence. 00:21:30.850 --> 00:21:37.170 align:middle line:84% The radius of convergence is just a number, 00:21:37.170 --> 00:21:41.650 align:middle line:84% and that number is radius of convergence. 00:21:41.650 --> 00:21:44.430 align:middle line:84% And we have to just interpret it a little bit. 00:21:44.430 --> 00:21:51.800 align:middle line:84% So it's 1 over limsup of the n-th root. 00:21:51.800 --> 00:21:56.330 align:middle line:90% 00:21:56.330 --> 00:22:00.410 align:middle line:84% This thing here could, of course, be infinite. 00:22:00.410 --> 00:22:05.690 align:middle line:84% If this is infinite, then the radius, by definition, this here 00:22:05.690 --> 00:22:08.370 align:middle line:90% would be 0. 00:22:08.370 --> 00:22:21.290 align:middle line:84% Where this here is equal to 0 if this limsup is infinite. 00:22:21.290 --> 00:22:25.190 align:middle line:90% And otherwise, it's always a-n. 00:22:25.190 --> 00:22:28.460 align:middle line:84% Otherwise, there's one other case you have to define, 00:22:28.460 --> 00:22:32.820 align:middle line:84% that if it's infinite, if the limsup-- 00:22:32.820 --> 00:22:35.780 align:middle line:90% 00:22:35.780 --> 00:22:42.340 align:middle line:84% sorry, this should be absolute value if this limsup is zero. 00:22:42.340 --> 00:22:44.980 align:middle line:90% otherwise. 00:22:44.980 --> 00:22:46.180 align:middle line:90% Otherwise, there's nothing. 00:22:46.180 --> 00:22:49.060 align:middle line:84% It's, well, that's the only thing you have to comment, 00:22:49.060 --> 00:22:53.000 align:middle line:84% because otherwise, it's always a non-negative number. 00:22:53.000 --> 00:22:54.180 align:middle line:90% But it could be 0. 00:22:54.180 --> 00:22:55.980 align:middle line:90% It could be infinity. 00:22:55.980 --> 00:22:58.100 align:middle line:84% But if it's any other number, then it 00:22:58.100 --> 00:22:59.600 align:middle line:84% is just what you think it should be. 00:22:59.600 --> 00:23:02.460 align:middle line:90% It's 1 over that number. 00:23:02.460 --> 00:23:09.940 align:middle line:84% So r here is the radius of convergence, OK? 00:23:09.940 --> 00:23:12.980 align:middle line:84% And now the thing is that now we have 00:23:12.980 --> 00:23:15.010 align:middle line:90% the following symbol, picture. 00:23:15.010 --> 00:23:19.220 align:middle line:90% 00:23:19.220 --> 00:23:21.920 align:middle line:90% So now we have that. 00:23:21.920 --> 00:23:23.510 align:middle line:90% So you have this power series. 00:23:23.510 --> 00:23:28.320 align:middle line:90% 00:23:28.320 --> 00:23:30.660 align:middle line:84% Again, it doesn't matter what you start with. 00:23:30.660 --> 00:23:34.080 align:middle line:84% Some matters what you start with, but not the convergence. 00:23:34.080 --> 00:23:36.240 align:middle line:90% So you have this power series. 00:23:36.240 --> 00:23:41.860 align:middle line:84% You are forming r, this is the radius of convergence. 00:23:41.860 --> 00:23:47.520 align:middle line:90% 00:23:47.520 --> 00:23:50.400 align:middle line:84% And then the claim is-- and again, this could be 0. 00:23:50.400 --> 00:23:52.040 align:middle line:90% It could be infinity. 00:23:52.040 --> 00:23:55.240 align:middle line:84% But often, it's just a real number, 00:23:55.240 --> 00:23:58.480 align:middle line:84% a non-negative real number, positive real number. 00:23:58.480 --> 00:24:04.640 align:middle line:84% And so, it's saying that if you take the interval here around 0, 00:24:04.640 --> 00:24:12.120 align:middle line:84% in the interval from minus r to r, then inside this interval 00:24:12.120 --> 00:24:16.320 align:middle line:84% here, inside here when x is in here, 00:24:16.320 --> 00:24:22.840 align:middle line:84% then the power series is convergent. 00:24:22.840 --> 00:24:25.610 align:middle line:90% 00:24:25.610 --> 00:24:32.510 align:middle line:84% And when x is here or here, power series is divergent. 00:24:32.510 --> 00:24:40.210 align:middle line:90% 00:24:40.210 --> 00:24:45.810 align:middle line:84% And then when x is here or here at those two points, 00:24:45.810 --> 00:24:50.930 align:middle line:84% then it's unclear and further test is needed. 00:24:50.930 --> 00:24:52.060 align:middle line:90% Further test is needed. 00:24:52.060 --> 00:24:56.530 align:middle line:90% 00:24:56.530 --> 00:24:57.460 align:middle line:90% Tests are needed. 00:24:57.460 --> 00:25:01.730 align:middle line:90% 00:25:01.730 --> 00:25:03.370 align:middle line:90% OK. 00:25:03.370 --> 00:25:04.650 align:middle line:90% Yeah? 00:25:04.650 --> 00:25:07.770 align:middle line:84% AUDIENCE: Starting from the last step on the left 00:25:07.770 --> 00:25:10.210 align:middle line:84% to get to this equation for r, are you 00:25:10.210 --> 00:25:14.450 align:middle line:84% just essentially saying that less than or equal to 1 or less 00:25:14.450 --> 00:25:15.870 align:middle line:90% than or equal to 1? 00:25:15.870 --> 00:25:18.530 align:middle line:84% But then we only care about one x less than 1 00:25:18.530 --> 00:25:19.750 align:middle line:90% because we need that. 00:25:19.750 --> 00:25:21.333 align:middle line:84% TOBIAS COLDING: Yeah, so that's right. 00:25:21.333 --> 00:25:22.490 align:middle line:90% So let's come to the proof. 00:25:22.490 --> 00:25:22.990 align:middle line:90% Yeah. 00:25:22.990 --> 00:25:24.610 align:middle line:84% So this is what you're talking about. 00:25:24.610 --> 00:25:27.650 align:middle line:90% 00:25:27.650 --> 00:25:33.510 align:middle line:84% So this is the radius of convergence. 00:25:33.510 --> 00:25:39.570 align:middle line:84% And so as you say, all we need to do is-- 00:25:39.570 --> 00:25:44.650 align:middle line:84% we know from the more fancy version of the ratio test, 00:25:44.650 --> 00:25:49.850 align:middle line:84% that is, in terms of limsup, we know that-- 00:25:49.850 --> 00:25:53.470 align:middle line:90% So proof here. 00:25:53.470 --> 00:25:59.200 align:middle line:84% Power series is convergent by the ratio test. 00:25:59.200 --> 00:26:03.870 align:middle line:90% 00:26:03.870 --> 00:26:05.110 align:middle line:90% Not ratio test, sorry. 00:26:05.110 --> 00:26:09.190 align:middle line:90% The root test, by the root test. 00:26:09.190 --> 00:26:14.310 align:middle line:84% So again, the version, the limsup version-- 00:26:14.310 --> 00:26:17.350 align:middle line:84% not the one I talked about last time, but this one we just 00:26:17.350 --> 00:26:19.110 align:middle line:90% talked about-- 00:26:19.110 --> 00:26:23.840 align:middle line:84% by the limsup version of the root test. 00:26:23.840 --> 00:26:28.640 align:middle line:84% Then the power series is convergent if this thing here-- 00:26:28.640 --> 00:26:30.840 align:middle line:84% so this means that this thing here-- 00:26:30.840 --> 00:26:41.960 align:middle line:84% so if this thing here, limsup of an n-th root. 00:26:41.960 --> 00:26:47.160 align:middle line:84% If this thing here is less than 1, it's divergent. 00:26:47.160 --> 00:27:01.160 align:middle line:84% If this thing here is bigger than 1, 00:27:01.160 --> 00:27:15.420 align:middle line:84% and it's inconclusive if this thing here is equal to 1 here. 00:27:15.420 --> 00:27:16.460 align:middle line:90% It's inconclusive. 00:27:16.460 --> 00:27:20.460 align:middle line:90% 00:27:20.460 --> 00:27:27.500 align:middle line:84% But now you see that this here just means that the norm of x 00:27:27.500 --> 00:27:30.560 align:middle line:84% should be smaller than 1 over that. 00:27:30.560 --> 00:27:36.660 align:middle line:84% So this here is the same as saying that the norm of x 00:27:36.660 --> 00:27:42.580 align:middle line:84% is smaller than r, the radius of convergence. 00:27:42.580 --> 00:27:46.620 align:middle line:84% This is equivalent to saying that the norm of x 00:27:46.620 --> 00:27:51.660 align:middle line:90% is bigger than r. 00:27:51.660 --> 00:27:56.500 align:middle line:84% And this here is equivalent to saying that the norm of x 00:27:56.500 --> 00:27:57.620 align:middle line:90% is equal to r. 00:27:57.620 --> 00:28:07.500 align:middle line:90% 00:28:07.500 --> 00:28:10.440 align:middle line:84% So that's the radius of convergence. 00:28:10.440 --> 00:28:12.700 align:middle line:90% So this is, like, a priori. 00:28:12.700 --> 00:28:16.540 align:middle line:84% This is not at all clear that it couldn't 00:28:16.540 --> 00:28:18.630 align:middle line:84% be-- that a power series couldn't 00:28:18.630 --> 00:28:23.030 align:middle line:84% be convergent for some sort of discrete set of x. 00:28:23.030 --> 00:28:24.550 align:middle line:90% But that's not the case. 00:28:24.550 --> 00:28:26.930 align:middle line:84% It's convergent on some interval. 00:28:26.930 --> 00:28:31.310 align:middle line:84% It's unclear at the end of that interval, 00:28:31.310 --> 00:28:33.510 align:middle line:84% but it is convergence on the interval, 00:28:33.510 --> 00:28:36.030 align:middle line:84% and divergent outside that interval. 00:28:36.030 --> 00:28:43.430 align:middle line:84% Now that interval could be just 0 or it could be all over. 00:28:43.430 --> 00:28:45.860 align:middle line:84% But it is also symmetric around zero. 00:28:45.860 --> 00:28:48.810 align:middle line:90% 00:28:48.810 --> 00:28:52.310 align:middle line:84% AUDIENCE: But when we bring that over, isn't it 1 over? 00:28:52.310 --> 00:28:53.480 align:middle line:90% TOBIAS COLDING: Right. 00:28:53.480 --> 00:28:56.110 align:middle line:90% So it's 1 over this here. 00:28:56.110 --> 00:29:00.490 align:middle line:84% But remember how r was defined. r is up there. 00:29:00.490 --> 00:29:00.990 align:middle line:90% Yeah. 00:29:00.990 --> 00:29:04.990 align:middle line:90% 00:29:04.990 --> 00:29:07.710 align:middle line:90% OK. 00:29:07.710 --> 00:29:15.510 align:middle line:84% Now let me just say a word about why 00:29:15.510 --> 00:29:20.130 align:middle line:84% does the proof from last time, why 00:29:20.130 --> 00:29:22.930 align:middle line:90% does it really prove this more? 00:29:22.930 --> 00:29:27.370 align:middle line:84% I mean, the argument we gave last time, 00:29:27.370 --> 00:29:31.250 align:middle line:90% why does it give you this? 00:29:31.250 --> 00:29:33.070 align:middle line:84% I mean, it's much more satisfying, 00:29:33.070 --> 00:29:36.650 align:middle line:84% this version of the root test, because it 00:29:36.650 --> 00:29:39.650 align:middle line:90% applies to any series. 00:29:39.650 --> 00:29:41.370 align:middle line:84% Because otherwise, we talked about 00:29:41.370 --> 00:29:43.790 align:middle line:84% whether some ratio was convergent, 00:29:43.790 --> 00:29:45.310 align:middle line:90% but it may not converge. 00:29:45.310 --> 00:29:48.050 align:middle line:90% So this is much more satisfying. 00:29:48.050 --> 00:29:53.010 align:middle line:84% So suppose, again, you have a series like this. 00:29:53.010 --> 00:29:57.250 align:middle line:84% And so now you're looking at limsup. 00:29:57.250 --> 00:30:01.570 align:middle line:84% So this is, again, this is the root test. 00:30:01.570 --> 00:30:06.670 align:middle line:90% Root test version. 00:30:06.670 --> 00:30:10.970 align:middle line:90% Let's call it version 2.0. 00:30:10.970 --> 00:30:15.820 align:middle line:84% And so you take a series serious like this, 00:30:15.820 --> 00:30:21.820 align:middle line:84% and it's looking at limsup, and then 00:30:21.820 --> 00:30:27.440 align:middle line:84% the n-th root of the absolute value of the n. 00:30:27.440 --> 00:30:30.020 align:middle line:90% 00:30:30.020 --> 00:30:36.940 align:middle line:84% And again, it's saying that if this thing here is less than 1, 00:30:36.940 --> 00:30:37.900 align:middle line:90% it's convergent. 00:30:37.900 --> 00:30:40.860 align:middle line:90% 00:30:40.860 --> 00:30:46.700 align:middle line:84% If it's bigger than 1, it's divergent. 00:30:46.700 --> 00:30:49.760 align:middle line:84% And if it's equal to 1, it's inconclusive. 00:30:49.760 --> 00:30:56.160 align:middle line:90% 00:30:56.160 --> 00:31:02.160 align:middle line:84% Now, let's try to see why this here is the case. 00:31:02.160 --> 00:31:21.240 align:middle line:84% So proof of root test version 2.0. 00:31:21.240 --> 00:31:25.500 align:middle line:84% Maybe I won't do all of it, but let's just look at. 00:31:25.500 --> 00:31:28.040 align:middle line:84% Because in a minute, you see that it's really kind 00:31:28.040 --> 00:31:30.080 align:middle line:90% of super similar to last time. 00:31:30.080 --> 00:31:43.200 align:middle line:84% So suppose that this limsup of the n-th root here, of d-n, 00:31:43.200 --> 00:31:50.080 align:middle line:84% suppose that this thing here is equal to some d, which 00:31:50.080 --> 00:31:53.200 align:middle line:90% is strictly less than 1. 00:31:53.200 --> 00:31:54.280 align:middle line:90% OK? 00:31:54.280 --> 00:32:02.800 align:middle line:84% So this means that there exists-- 00:32:02.800 --> 00:32:18.730 align:middle line:84% so if you will, choose d bar with d less than d bar strictly 00:32:18.730 --> 00:32:21.310 align:middle line:90% less than 1. 00:32:21.310 --> 00:32:22.980 align:middle line:90% So I can choose it like this. 00:32:22.980 --> 00:32:26.530 align:middle line:90% 00:32:26.530 --> 00:32:30.810 align:middle line:84% But now you see that if I choose it like this, then 00:32:30.810 --> 00:32:31.500 align:middle line:90% there exists-- 00:32:31.500 --> 00:32:34.530 align:middle line:90% 00:32:34.530 --> 00:32:40.970 align:middle line:84% so remember, let's just remind ourselves that-- 00:32:40.970 --> 00:32:44.050 align:middle line:90% how was the limsup defined? 00:32:44.050 --> 00:32:49.650 align:middle line:84% The limsup was defined to be that you 00:32:49.650 --> 00:32:51.790 align:middle line:90% take this new sequence-- 00:32:51.790 --> 00:32:55.930 align:middle line:84% So you're taking this sequence here. 00:32:55.930 --> 00:33:00.330 align:middle line:84% So you're taking these guys here. 00:33:00.330 --> 00:33:04.010 align:middle line:84% So you're looking at now these guys here, 00:33:04.010 --> 00:33:10.870 align:middle line:84% and you define a sequence that is like the sup, 00:33:10.870 --> 00:33:17.750 align:middle line:84% starting at the n-th guy, and like this here. 00:33:17.750 --> 00:33:21.830 align:middle line:84% So this here is n plus 1, et cetera. 00:33:21.830 --> 00:33:24.170 align:middle line:90% Right? 00:33:24.170 --> 00:33:27.762 align:middle line:90% This is the sup of the tail. 00:33:27.762 --> 00:33:30.910 align:middle line:90% And so that was sup on-- 00:33:30.910 --> 00:33:32.550 align:middle line:84% as you're going further out, you're 00:33:32.550 --> 00:33:34.730 align:middle line:90% taking sup over smaller sets. 00:33:34.730 --> 00:33:38.550 align:middle line:90% So this sequence going down. 00:33:38.550 --> 00:33:42.230 align:middle line:84% So now if this limit here is sup d, 00:33:42.230 --> 00:33:45.030 align:middle line:84% which is strictly less than d bar, 00:33:45.030 --> 00:33:47.230 align:middle line:90% then it means that they exist. 00:33:47.230 --> 00:33:55.110 align:middle line:84% So there exists capital N. So that if little n is bigger 00:33:55.110 --> 00:33:58.590 align:middle line:84% than capital N, then all of these guys 00:33:58.590 --> 00:34:02.880 align:middle line:90% here is less than d bar, right? 00:34:02.880 --> 00:34:05.490 align:middle line:90% 00:34:05.490 --> 00:34:08.800 align:middle line:84% So if this here, then all of these guys 00:34:08.800 --> 00:34:12.659 align:middle line:84% here is less than or equal to d bar. 00:34:12.659 --> 00:34:17.150 align:middle line:84% This is if little n, if you're starting that far out, right? 00:34:17.150 --> 00:34:21.260 align:middle line:90% 00:34:21.260 --> 00:34:24.199 align:middle line:84% So you have that when n is bigger than this, 00:34:24.199 --> 00:34:25.540 align:middle line:90% you have that this. 00:34:25.540 --> 00:34:30.560 align:middle line:84% So you have that for all these guys here, d-n, 00:34:30.560 --> 00:34:35.100 align:middle line:84% the n-th root is less than d bar-- less than 00:34:35.100 --> 00:34:39.440 align:middle line:90% or equal to d bar. 00:34:39.440 --> 00:34:43.780 align:middle line:84% But now you take the n-th power on both sides, 00:34:43.780 --> 00:34:49.280 align:middle line:84% so you have that the n absolute value is less than d bar, 00:34:49.280 --> 00:34:51.159 align:middle line:90% the n power. 00:34:51.159 --> 00:34:55.900 align:middle line:84% This here is, d, this isn't even actually non-negative, 00:34:55.900 --> 00:34:58.520 align:middle line:84% so you don't even need this absolute value sign. 00:34:58.520 --> 00:35:00.040 align:middle line:90% So you're having this. 00:35:00.040 --> 00:35:03.680 align:middle line:84% But this is a number that is strictly less than 1, 00:35:03.680 --> 00:35:06.686 align:middle line:84% so this here is a geometric series. 00:35:06.686 --> 00:35:22.880 align:middle line:84% Geometric series where d bar is less than 1. 00:35:22.880 --> 00:35:26.180 align:middle line:84% So it is a convergent geometric series, right? 00:35:26.180 --> 00:35:29.140 align:middle line:84% So this is, you see, that it's basically the same proof we 00:35:29.140 --> 00:35:29.880 align:middle line:90% did last time. 00:35:29.880 --> 00:35:32.633 align:middle line:84% But last time, we were assuming that some ratio was converging 00:35:32.633 --> 00:35:37.720 align:middle line:84% or we were assuming that this here converges to something, 00:35:37.720 --> 00:35:39.620 align:middle line:84% but you don't need it to converge. 00:35:39.620 --> 00:35:46.580 align:middle line:84% You just need that the limsup is less than 1. 00:35:46.580 --> 00:35:51.300 align:middle line:84% And so likewise, if you examine what this case here say, then 00:35:51.300 --> 00:35:54.900 align:middle line:84% just like last time, it's going to say that if this here 00:35:54.900 --> 00:35:58.820 align:middle line:84% is bigger than 1, then the ordinary element doesn't even 00:35:58.820 --> 00:36:00.806 align:middle line:90% go to 0. 00:36:00.806 --> 00:36:05.650 align:middle line:84% So it fails the most basic test for being convergent 00:36:05.650 --> 00:36:08.390 align:middle line:84% And that was just like last time, also. 00:36:08.390 --> 00:36:10.670 align:middle line:90% And again, the advantage-- 00:36:10.670 --> 00:36:12.990 align:middle line:84% I think you were the one that was asking yesterday. 00:36:12.990 --> 00:36:16.570 align:middle line:84% I mean, last time this was the ratio test and the root test, 00:36:16.570 --> 00:36:18.910 align:middle line:84% and like in the ratio test, it's something 00:36:18.910 --> 00:36:25.710 align:middle line:84% about that the next element is going down with some fraction. 00:36:25.710 --> 00:36:29.570 align:middle line:84% And you very much see that this here works for all series. 00:36:29.570 --> 00:36:31.950 align:middle line:84% And it's really just because you don't 00:36:31.950 --> 00:36:34.350 align:middle line:84% compare the element to the next, but you're 00:36:34.350 --> 00:36:35.960 align:middle line:90% comparing it to this fixed. 00:36:35.960 --> 00:36:38.950 align:middle line:90% 00:36:38.950 --> 00:36:41.070 align:middle line:84% series, which is a geometric series. 00:36:41.070 --> 00:36:45.750 align:middle line:90% 00:36:45.750 --> 00:36:59.510 align:middle line:84% So that's the real version, so to speak, of the root test. 00:36:59.510 --> 00:37:05.400 align:middle line:84% So now let's talk a little bit more about power series. 00:37:05.400 --> 00:37:11.610 align:middle line:90% 00:37:11.610 --> 00:37:17.290 align:middle line:84% So like this one function that is particularly important, 00:37:17.290 --> 00:37:20.870 align:middle line:84% and I mean, all of these basic functions are, of course, 00:37:20.870 --> 00:37:23.850 align:middle line:84% important, but they are somewhat similar. 00:37:23.850 --> 00:37:27.880 align:middle line:84% And the one I want to talk about is the exponential function. 00:37:27.880 --> 00:37:36.170 align:middle line:90% 00:37:36.170 --> 00:37:39.230 align:middle line:84% And there's various way that you could define, 00:37:39.230 --> 00:37:41.730 align:middle line:84% so what is the exponential function? 00:37:41.730 --> 00:37:43.930 align:middle line:84% There's various ways that you could define 00:37:43.930 --> 00:37:45.310 align:middle line:90% the exponential function. 00:37:45.310 --> 00:37:48.630 align:middle line:84% You can define it, like, two that comes to mind. 00:37:48.630 --> 00:37:50.170 align:middle line:90% But there's others, too. 00:37:50.170 --> 00:37:53.530 align:middle line:84% One is that you could define it to be a function that's 00:37:53.530 --> 00:37:56.490 align:middle line:84% satisfying a certain differential equation, 00:37:56.490 --> 00:37:59.130 align:middle line:84% and that would be a good way of defining it. 00:37:59.130 --> 00:38:00.850 align:middle line:84% Another way of defining it, which 00:38:00.850 --> 00:38:02.780 align:middle line:84% is more relevant to what we're doing here, 00:38:02.780 --> 00:38:06.220 align:middle line:84% is defining it as a power series. 00:38:06.220 --> 00:38:18.220 align:middle line:84% So I'm looking at e to the x, and I'm looking at e to the x. 00:38:18.220 --> 00:38:23.822 align:middle line:84% So I define E, capital E of x, as this power series that we 00:38:23.822 --> 00:38:24.530 align:middle line:90% looked at before. 00:38:24.530 --> 00:38:29.792 align:middle line:90% 00:38:29.792 --> 00:38:34.660 align:middle line:84% Now I want to prove that this here is really 00:38:34.660 --> 00:38:36.380 align:middle line:90% what you should think of. 00:38:36.380 --> 00:38:38.810 align:middle line:84% This here is the exponential function. 00:38:38.810 --> 00:38:41.940 align:middle line:90% 00:38:41.940 --> 00:38:44.520 align:middle line:84% But what does it mean that is is exponential? 00:38:44.520 --> 00:38:54.440 align:middle line:84% I want to show that this is the exponential function. 00:38:54.440 --> 00:38:56.460 align:middle line:90% So what do I mean by? 00:38:56.460 --> 00:39:01.640 align:middle line:84% I mean, of course I could define any function 00:39:01.640 --> 00:39:04.160 align:middle line:84% to be the exponential function, but there 00:39:04.160 --> 00:39:07.720 align:middle line:84% are certain properties of the exponential function 00:39:07.720 --> 00:39:09.340 align:middle line:90% that you would want. 00:39:09.340 --> 00:39:13.520 align:middle line:90% 00:39:13.520 --> 00:39:23.400 align:middle line:84% Well, the first thing I observe is that E of 0 is equal to-- 00:39:23.400 --> 00:39:30.720 align:middle line:84% if x is 0, then all of these are 0, except when n is equal to 0. 00:39:30.720 --> 00:39:36.180 align:middle line:84% And then x to the 0, this here, is, by definition, one. 00:39:36.180 --> 00:39:41.060 align:middle line:84% And this is 0, factorial is by definition one. 00:39:41.060 --> 00:39:46.760 align:middle line:84% So this here is 1, by definition. 00:39:46.760 --> 00:39:50.800 align:middle line:84% And we know already, by the way, that this here we have already 00:39:50.800 --> 00:39:54.520 align:middle line:84% seen that the radius of convergence for this power 00:39:54.520 --> 00:40:03.690 align:middle line:84% series, of convergence is equal to infinity. 00:40:03.690 --> 00:40:07.330 align:middle line:84% We have already seen that, so it's a well-defined function 00:40:07.330 --> 00:40:08.390 align:middle line:90% for all x. 00:40:08.390 --> 00:40:14.490 align:middle line:90% 00:40:14.490 --> 00:40:27.530 align:middle line:84% So next time we will show that if you define it 00:40:27.530 --> 00:40:31.850 align:middle line:84% like that, that this function here, defined like that, 00:40:31.850 --> 00:40:40.470 align:middle line:84% has a property that x of E to x plus y is equal to this product. 00:40:40.470 --> 00:40:49.670 align:middle line:90% 00:40:49.670 --> 00:40:52.850 align:middle line:84% Well, the first thing I can observe is-- 00:40:52.850 --> 00:40:56.090 align:middle line:84% I haven't really gotten to the property yet, but at the moment 00:40:56.090 --> 00:40:59.740 align:middle line:84% I'm just examining that particular function, 00:40:59.740 --> 00:41:03.700 align:middle line:90% some properties that it has. 00:41:03.700 --> 00:41:05.540 align:middle line:84% And so the first thing I will observe 00:41:05.540 --> 00:41:10.480 align:middle line:84% is that if x is positive or non-negative, right, 00:41:10.480 --> 00:41:12.800 align:middle line:84% then all of these numbers are non-negative. 00:41:12.800 --> 00:41:14.800 align:middle line:84% So you're summing non-negative numbers. 00:41:14.800 --> 00:41:17.020 align:middle line:90% So the sum is non-negative. 00:41:17.020 --> 00:41:25.820 align:middle line:84% So if x is bigger or equal to 0, then it implies that E of x 00:41:25.820 --> 00:41:30.620 align:middle line:84% is bigger, is actually strictly positive. 00:41:30.620 --> 00:41:34.800 align:middle line:84% In fact, the smallest one is where x was equal to 0. 00:41:34.800 --> 00:41:38.140 align:middle line:84% But in that case, you got one, right? 00:41:38.140 --> 00:41:43.380 align:middle line:84% Now if x is negative, then it's not super clear. 00:41:43.380 --> 00:41:45.820 align:middle line:90% But if x is negative-- 00:41:45.820 --> 00:41:48.700 align:middle line:84% some of these numbers that you'll be summing are positive 00:41:48.700 --> 00:41:51.740 align:middle line:84% and some of them are negative, so it's not at all clear 00:41:51.740 --> 00:41:53.780 align:middle line:90% what happened to the sum. 00:41:53.780 --> 00:41:57.580 align:middle line:84% But if you grant me this for the moment, 00:41:57.580 --> 00:42:01.920 align:middle line:84% this relationship here, then we would have that E 00:42:01.920 --> 00:42:04.800 align:middle line:90% to the x minus E-- 00:42:04.800 --> 00:42:08.500 align:middle line:90% minus, sorry, E in x minus x-- 00:42:08.500 --> 00:42:10.190 align:middle line:90% So that's the same as-- 00:42:10.190 --> 00:42:14.160 align:middle line:90% 00:42:14.160 --> 00:42:19.120 align:middle line:84% so E of 0, which I already know is one, 00:42:19.120 --> 00:42:24.960 align:middle line:84% this here can think about that as x minus x. 00:42:24.960 --> 00:42:29.270 align:middle line:84% Or, if you will, to think about it as x plus minus x. 00:42:29.270 --> 00:42:31.840 align:middle line:90% 00:42:31.840 --> 00:42:36.120 align:middle line:84% And now, if you grant me this thing here 00:42:36.120 --> 00:42:39.880 align:middle line:84% that we prove next time, then I can write this 00:42:39.880 --> 00:42:46.522 align:middle line:90% as E of x times E of minus x. 00:42:46.522 --> 00:42:47.022 align:middle line:90% Right? 00:42:47.022 --> 00:42:53.492 align:middle line:90% 00:42:53.492 --> 00:42:59.970 align:middle line:84% And now if x here is negative, then of course minus x 00:42:59.970 --> 00:43:00.990 align:middle line:90% is positive. 00:43:00.990 --> 00:43:04.770 align:middle line:90% 00:43:04.770 --> 00:43:09.490 align:middle line:84% So now I know by this thing here that this here is positive. 00:43:09.490 --> 00:43:13.190 align:middle line:84% And so, since the product of these two things is 1, 00:43:13.190 --> 00:43:16.730 align:middle line:84% this here must be strictly positive as well, right? 00:43:16.730 --> 00:43:22.470 align:middle line:84% So this implies that E to the x is also strictly positive. 00:43:22.470 --> 00:43:26.098 align:middle line:90% 00:43:26.098 --> 00:43:28.510 align:middle line:84% So this is just a couple of observations. 00:43:28.510 --> 00:43:30.530 align:middle line:84% Now, of course, I'm using that relationship 00:43:30.530 --> 00:43:32.130 align:middle line:84% that I haven't established yet, but I 00:43:32.130 --> 00:43:35.170 align:middle line:90% will establish that next time. 00:43:35.170 --> 00:43:46.690 align:middle line:84% The next thing is that I want to define the number E, 00:43:46.690 --> 00:43:50.130 align:middle line:90% so E, that is 2.7, whatever. 00:43:50.130 --> 00:43:52.690 align:middle line:90% I don't remember. 00:43:52.690 --> 00:43:55.270 align:middle line:90% So E we define-- 00:43:55.270 --> 00:44:02.990 align:middle line:84% so little e, so I define it as this function here evaluated 00:44:02.990 --> 00:44:04.030 align:middle line:90% in one. 00:44:04.030 --> 00:44:04.530 align:middle line:90% OK. 00:44:04.530 --> 00:44:10.630 align:middle line:90% 00:44:10.630 --> 00:44:13.230 align:middle line:84% So I have my function, capital E, 00:44:13.230 --> 00:44:19.230 align:middle line:84% but I want to define a second function that is e. 00:44:19.230 --> 00:44:32.190 align:middle line:84% So I want to define a second function, and that 00:44:32.190 --> 00:44:36.630 align:middle line:84% is e to the x, like this, and that 00:44:36.630 --> 00:44:40.150 align:middle line:84% should have the kind of properties 00:44:40.150 --> 00:44:43.370 align:middle line:90% that you think it should have. 00:44:43.370 --> 00:44:48.330 align:middle line:84% But first, before I do that, let me just think. 00:44:48.330 --> 00:44:58.280 align:middle line:84% If you take e to the power m here, 00:44:58.280 --> 00:45:00.790 align:middle line:90% let's say that m is an integer. 00:45:00.790 --> 00:45:07.280 align:middle line:90% 00:45:07.280 --> 00:45:15.830 align:middle line:84% This here should just be e times e m times. 00:45:15.830 --> 00:45:20.720 align:middle line:90% 00:45:20.720 --> 00:45:27.280 align:middle line:84% Note that this here, I can, of course, write as e. 00:45:27.280 --> 00:45:35.270 align:middle line:84% This is what e is, so I can write it as this here, m times. 00:45:35.270 --> 00:45:38.000 align:middle line:90% 00:45:38.000 --> 00:45:43.400 align:middle line:84% But now I can use this here that I will prove next time again. 00:45:43.400 --> 00:45:50.340 align:middle line:84% So if I take E of 1 times E of 1, then that's E of 2. 00:45:50.340 --> 00:45:56.140 align:middle line:84% And if I take E three times, that is E of 3. 00:45:56.140 --> 00:46:03.580 align:middle line:84% So this thing here is actually the same as capital E of m, 00:46:03.580 --> 00:46:05.220 align:middle line:90% right? 00:46:05.220 --> 00:46:08.500 align:middle line:90% So I have that. 00:46:08.500 --> 00:46:16.180 align:middle line:84% By definition, e to the 0, this here I'm setting equal to 1. 00:46:16.180 --> 00:46:24.180 align:middle line:84% But I already have that one was equal to this capital E in 0. 00:46:24.180 --> 00:46:26.880 align:middle line:90% 00:46:26.880 --> 00:46:30.390 align:middle line:84% Now what about if m here is a negative integer? 00:46:30.390 --> 00:46:41.020 align:middle line:90% 00:46:41.020 --> 00:46:45.020 align:middle line:84% So suppose that m here is a negative integer. 00:46:45.020 --> 00:46:48.000 align:middle line:90% Suppose that m here is negative. 00:46:48.000 --> 00:46:52.190 align:middle line:90% 00:46:52.190 --> 00:46:56.230 align:middle line:90% What should e to the m here be? 00:46:56.230 --> 00:47:04.510 align:middle line:84% This here is just 1 over e to the minus m. 00:47:04.510 --> 00:47:11.330 align:middle line:84% OK, minus m is a positive integer, so this makes sense. 00:47:11.330 --> 00:47:14.310 align:middle line:90% 00:47:14.310 --> 00:47:21.230 align:middle line:84% Now, if you're looking at e to the-- again, 00:47:21.230 --> 00:47:24.970 align:middle line:84% using this relationship here, e-- 00:47:24.970 --> 00:47:27.750 align:middle line:84% and in fact, the one we just did. 00:47:27.750 --> 00:47:32.590 align:middle line:84% So this is on integers, and this here 00:47:32.590 --> 00:47:37.700 align:middle line:84% is now equal to E of 0, which is equal to 1. 00:47:37.700 --> 00:47:40.870 align:middle line:90% 00:47:40.870 --> 00:47:44.510 align:middle line:84% And this is, by that relationship over there, 00:47:44.510 --> 00:47:49.490 align:middle line:90% it's E of m times E of minus m. 00:47:49.490 --> 00:47:55.490 align:middle line:84% And so you see that this implies that E of minus M 00:47:55.490 --> 00:47:59.370 align:middle line:84% is equal to 1 over E of m, by just 00:47:59.370 --> 00:48:02.170 align:middle line:90% dividing this on both sides. 00:48:02.170 --> 00:48:07.810 align:middle line:84% So you have this, and this thing here, but this thing here-- 00:48:07.810 --> 00:48:10.210 align:middle line:84% sorry, this is not what I wanted. 00:48:10.210 --> 00:48:13.590 align:middle line:84% If I take m negative, I just use the other one. 00:48:13.590 --> 00:48:14.610 align:middle line:90% Sorry. 00:48:14.610 --> 00:48:23.990 align:middle line:84% E of M is equal to 1 over E of minus m, like that. 00:48:23.990 --> 00:48:26.930 align:middle line:90% 00:48:26.930 --> 00:48:29.530 align:middle line:84% M was supposed to be negative, so this 00:48:29.530 --> 00:48:32.930 align:middle line:84% means that minus M is a positive integer. 00:48:32.930 --> 00:48:36.310 align:middle line:84% And this one here is then this one. 00:48:36.310 --> 00:48:39.050 align:middle line:84% I know that this here already agree 00:48:39.050 --> 00:48:47.420 align:middle line:90% with this little function of e. 00:48:47.420 --> 00:48:52.740 align:middle line:84% So you see that this thing here agrees with e to the power m. 00:48:52.740 --> 00:48:55.995 align:middle line:90% 00:48:55.995 --> 00:49:02.220 align:middle line:84% Because we define it like this, and now just checking that also 00:49:02.220 --> 00:49:05.140 align:middle line:84% on the negative integer, this function here 00:49:05.140 --> 00:49:07.500 align:middle line:90% and this function here agree. 00:49:07.500 --> 00:49:08.740 align:middle line:90% OK. 00:49:08.740 --> 00:49:12.140 align:middle line:84% So I now have these two different functions, 00:49:12.140 --> 00:49:14.280 align:middle line:84% but one I haven't really defined yet. 00:49:14.280 --> 00:49:17.460 align:middle line:84% I've only defined it now on all integers. 00:49:17.460 --> 00:49:21.940 align:middle line:84% But so far, these two functions agree. 00:49:21.940 --> 00:49:28.060 align:middle line:84% Now what I also should have is that I should have-- 00:49:28.060 --> 00:49:30.330 align:middle line:84% so suppose you're looking at a rational number. 00:49:30.330 --> 00:49:34.700 align:middle line:90% 00:49:34.700 --> 00:49:36.680 align:middle line:84% So suppose you take a rational number. 00:49:36.680 --> 00:49:39.000 align:middle line:84% Remember, we usually write a rational number. 00:49:39.000 --> 00:49:42.540 align:middle line:84% So suppose that q here is a rational number. 00:49:42.540 --> 00:49:52.460 align:middle line:84% We usually write a rational number as an element in set. 00:49:52.460 --> 00:49:53.540 align:middle line:90% So it could be negative. 00:49:53.540 --> 00:49:54.960 align:middle line:90% It could be 0. 00:49:54.960 --> 00:50:00.280 align:middle line:84% And n here is a positive integer, so 00:50:00.280 --> 00:50:01.460 align:middle line:90% a fraction like this. 00:50:01.460 --> 00:50:05.200 align:middle line:90% 00:50:05.200 --> 00:50:10.560 align:middle line:90% Now what should e of q be? 00:50:10.560 --> 00:50:13.360 align:middle line:90% 00:50:13.360 --> 00:50:17.880 align:middle line:84% Well, e of q, this here should be-- 00:50:17.880 --> 00:50:20.080 align:middle line:84% how you would think about it is that it really 00:50:20.080 --> 00:50:21.540 align:middle line:90% should be something. 00:50:21.540 --> 00:50:24.120 align:middle line:90% 00:50:24.120 --> 00:50:25.620 align:middle line:90% This here should be a number. 00:50:25.620 --> 00:50:29.360 align:middle line:84% Should be a number, alpha, and this should have 00:50:29.360 --> 00:50:33.200 align:middle line:90% the property that is positive. 00:50:33.200 --> 00:50:38.280 align:middle line:84% And it should have the property, so let's just write it. 00:50:38.280 --> 00:50:43.250 align:middle line:90% So e of q, this is m over n. 00:50:43.250 --> 00:50:47.410 align:middle line:84% It should have the property that if you raise it-- 00:50:47.410 --> 00:50:52.490 align:middle line:84% alpha here, suppose you were raising it to the power n, 00:50:52.490 --> 00:50:55.130 align:middle line:84% then you should get this number that we have already 00:50:55.130 --> 00:50:57.330 align:middle line:90% defined, right? 00:50:57.330 --> 00:51:04.422 align:middle line:84% So it should be positive n when you raise it like this, 00:51:04.422 --> 00:51:07.010 align:middle line:84% and then it should give you that. 00:51:07.010 --> 00:51:09.410 align:middle line:84% And just like with the square root of 2, 00:51:09.410 --> 00:51:12.930 align:middle line:84% there's just one element with this property. 00:51:12.930 --> 00:51:15.710 align:middle line:90% OK, So that's defined. 00:51:15.710 --> 00:51:21.810 align:middle line:84% So you see this way we have now defined this function, e, 00:51:21.810 --> 00:51:24.490 align:middle line:90% on all rational numbers. 00:51:24.490 --> 00:51:26.710 align:middle line:90% But now we need to check. 00:51:26.710 --> 00:51:29.010 align:middle line:84% We need to check that-- this thing here-- 00:51:29.010 --> 00:51:43.510 align:middle line:84% we need to check that e of q is equal like that. 00:51:43.510 --> 00:51:44.770 align:middle line:90% So I need to check that. 00:51:44.770 --> 00:51:47.450 align:middle line:90% 00:51:47.450 --> 00:51:57.190 align:middle line:84% So I want to say that e to the q is that alpha, right? 00:51:57.190 --> 00:52:02.110 align:middle line:84% So this means that I want to show. 00:52:02.110 --> 00:52:08.710 align:middle line:84% So I need to show two things, e to q is positive-- well, 00:52:08.710 --> 00:52:10.510 align:middle line:90% I've already checked that. 00:52:10.510 --> 00:52:14.130 align:middle line:84% We checked that E, capital E at anything is positive, 00:52:14.130 --> 00:52:15.750 align:middle line:90% so this is OK. 00:52:15.750 --> 00:52:18.070 align:middle line:84% The second thing I need to check is 00:52:18.070 --> 00:52:25.710 align:middle line:84% that it has the property that if you raise it to the n power, 00:52:25.710 --> 00:52:33.110 align:middle line:84% then you're getting e to the m, right? 00:52:33.110 --> 00:52:36.730 align:middle line:84% Now e to the m, that's an integer. 00:52:36.730 --> 00:52:38.810 align:middle line:90% It could be negative. 00:52:38.810 --> 00:52:42.360 align:middle line:84% But e to any negative integer, whether it's 00:52:42.360 --> 00:52:46.920 align:middle line:84% positive or negative, we already checked that this 00:52:46.920 --> 00:52:49.572 align:middle line:90% is equal to capital e of m. 00:52:49.572 --> 00:52:53.120 align:middle line:90% 00:52:53.120 --> 00:52:56.580 align:middle line:90% So now I need to show this here. 00:52:56.580 --> 00:52:57.740 align:middle line:90% So now what is this? 00:52:57.740 --> 00:52:59.100 align:middle line:90% So what do I need to check? 00:52:59.100 --> 00:53:09.920 align:middle line:84% So I need to check that e to the m over n, 00:53:09.920 --> 00:53:15.960 align:middle line:84% this thing here, raised to the power n is equal to this. 00:53:15.960 --> 00:53:18.040 align:middle line:90% But what is this here? 00:53:18.040 --> 00:53:25.930 align:middle line:84% This here is just e to the m over n, multiply itself n times. 00:53:25.930 --> 00:53:29.520 align:middle line:90% 00:53:29.520 --> 00:53:31.070 align:middle line:90% So this is now n times. 00:53:31.070 --> 00:53:34.920 align:middle line:90% 00:53:34.920 --> 00:53:39.700 align:middle line:84% But now I can use, again, this thing that I proved, next time, 00:53:39.700 --> 00:53:44.820 align:middle line:84% that when I do it like here, I'm just doing it 2 times, 00:53:44.820 --> 00:53:48.420 align:middle line:84% and here I'm doing it for different x and y. 00:53:48.420 --> 00:53:51.620 align:middle line:84% But if I do it n times with the same x, 00:53:51.620 --> 00:53:56.244 align:middle line:84% then I'm just adding that x together n times. 00:53:56.244 --> 00:53:59.620 align:middle line:90% 00:53:59.620 --> 00:54:03.900 align:middle line:84% And so this thing here is the same as e, 00:54:03.900 --> 00:54:09.340 align:middle line:84% by that rule over there, n times m over n. 00:54:09.340 --> 00:54:15.380 align:middle line:84% But this is just E of m, as it should be. 00:54:15.380 --> 00:54:18.460 align:middle line:84% So you see that these two functions here. 00:54:18.460 --> 00:54:32.150 align:middle line:84% So the conclusion now is that this function here 00:54:32.150 --> 00:54:42.730 align:middle line:84% is equal to this function here for all q for rational numbers. 00:54:42.730 --> 00:54:50.270 align:middle line:84% Now, what other property should this exponential-- so this here, 00:54:50.270 --> 00:54:52.310 align:middle line:84% so far, this is kind of the property 00:54:52.310 --> 00:54:54.490 align:middle line:84% that you want of the exponential function. 00:54:54.490 --> 00:54:58.910 align:middle line:84% But what other properties should the exponential function have? 00:54:58.910 --> 00:55:01.270 align:middle line:84% You should have one other property, 00:55:01.270 --> 00:55:03.590 align:middle line:84% and that's enough to determine it, 00:55:03.590 --> 00:55:05.610 align:middle line:84% and that is it should be continuous. 00:55:05.610 --> 00:55:11.430 align:middle line:84% So one more property is that this function e to the x, 00:55:11.430 --> 00:55:20.750 align:middle line:84% when x is defined like this, this function should 00:55:20.750 --> 00:55:27.110 align:middle line:84% be continuous, and we'll talk a little bit more 00:55:27.110 --> 00:55:31.460 align:middle line:84% about continuous function in just a second. 00:55:31.460 --> 00:55:39.050 align:middle line:84% Now that I have a theorem that we'll prove next time, one 00:55:39.050 --> 00:55:41.010 align:middle line:90% of the next times. 00:55:41.010 --> 00:55:47.410 align:middle line:84% And that is that, suppose you take two functions, 00:55:47.410 --> 00:55:52.290 align:middle line:84% so we then can appeal to the following theorem. 00:55:52.290 --> 00:56:07.970 align:middle line:84% If f and g are two continuous functions 00:56:07.970 --> 00:56:21.930 align:middle line:90% on r that agree on all of q. 00:56:21.930 --> 00:56:24.450 align:middle line:84% So if you have two continuous functions, 00:56:24.450 --> 00:56:27.910 align:middle line:84% and if you restrict it to the rational number, 00:56:27.910 --> 00:56:30.850 align:middle line:84% then the two functions are the same. 00:56:30.850 --> 00:56:32.810 align:middle line:84% If they are both continuous, then the function 00:56:32.810 --> 00:56:34.300 align:middle line:90% has to be the same. 00:56:34.300 --> 00:56:40.580 align:middle line:84% So this implies that f is equal to g everywhere, 00:56:40.580 --> 00:56:42.690 align:middle line:84% not just on the rational function. 00:56:42.690 --> 00:56:43.710 align:middle line:90% No, rational numbers. 00:56:43.710 --> 00:56:50.440 align:middle line:90% 00:56:50.440 --> 00:56:54.820 align:middle line:84% So we prove that either next time or the following time. 00:56:54.820 --> 00:56:59.260 align:middle line:84% Let me just go back and remind you 00:56:59.260 --> 00:57:02.180 align:middle line:84% about what it means for a function to be continuous. 00:57:02.180 --> 00:57:05.470 align:middle line:90% 00:57:05.470 --> 00:57:07.280 align:middle line:90% So this is the following. 00:57:07.280 --> 00:57:18.340 align:middle line:90% 00:57:18.340 --> 00:57:19.600 align:middle line:90% So continuous. 00:57:19.600 --> 00:57:25.460 align:middle line:90% 00:57:25.460 --> 00:57:31.460 align:middle line:84% So suppose you take a function, and the function, 00:57:31.460 --> 00:57:34.150 align:middle line:84% it doesn't even need to be defined on an interval. 00:57:34.150 --> 00:57:36.880 align:middle line:90% 00:57:36.880 --> 00:57:39.380 align:middle line:84% But you may as well assume that it's an interval, 00:57:39.380 --> 00:57:43.800 align:middle line:84% but it doesn't need to be defined on the interval. 00:57:43.800 --> 00:57:53.400 align:middle line:84% So a function here, f, is said to be 00:57:53.400 --> 00:58:04.240 align:middle line:84% continuous at x here in the domain where it's defined. 00:58:04.240 --> 00:58:13.320 align:middle line:84% If for all epsilon greater than 0, 00:58:13.320 --> 00:58:25.960 align:middle line:84% there exists delta greater than zero, such that if x minus x-0 00:58:25.960 --> 00:58:29.440 align:middle line:84% is smaller than delta, then the images 00:58:29.440 --> 00:58:31.590 align:middle line:90% are epsilon close to each other. 00:58:31.590 --> 00:58:37.690 align:middle line:90% 00:58:37.690 --> 00:58:41.610 align:middle line:84% And so again this delta typically 00:58:41.610 --> 00:58:45.910 align:middle line:84% depends not just on epsilon, but also on x-0. 00:58:45.910 --> 00:58:50.010 align:middle line:90% 00:58:50.010 --> 00:58:55.210 align:middle line:84% So it's continuous at a point x-0 if this is the case, 00:58:55.210 --> 00:59:09.490 align:middle line:84% and the function, f, here is said to be continuous if it 00:59:09.490 --> 00:59:16.320 align:middle line:90% is continuous at all points. 00:59:16.320 --> 00:59:25.535 align:middle line:90% 00:59:25.535 --> 00:59:32.950 align:middle line:84% Now let's just look at two simple examples, 00:59:32.950 --> 00:59:37.150 align:middle line:84% and then once we have those, then 00:59:37.150 --> 00:59:40.910 align:middle line:84% together with some algebraic rules that 00:59:40.910 --> 00:59:45.870 align:middle line:84% are similar to the algebraic rules that we had for sequences, 00:59:45.870 --> 00:59:51.670 align:middle line:84% then we get a huge number of examples. 00:59:51.670 --> 00:59:55.030 align:middle line:84% We get a large collection of examples 00:59:55.030 --> 00:59:57.670 align:middle line:90% of continuous functions. 00:59:57.670 --> 01:00:00.390 align:middle line:84% So the first example is the simplest 01:00:00.390 --> 01:00:05.840 align:middle line:84% of all examples, where f here is the constant function. 01:00:05.840 --> 01:00:12.190 align:middle line:90% 01:00:12.190 --> 01:00:17.630 align:middle line:84% So I'm thinking about f here is defined from R into R. 01:00:17.630 --> 01:00:23.430 align:middle line:84% And for each x in R, it's equal to the same constant. 01:00:23.430 --> 01:00:25.590 align:middle line:90% So that's f. 01:00:25.590 --> 01:00:29.240 align:middle line:84% I want to show that this f is a constant function, 01:00:29.240 --> 01:00:36.052 align:middle line:84% is continuous, so proof of continuity. 01:00:36.052 --> 01:00:46.560 align:middle line:90% 01:00:46.560 --> 01:00:52.960 align:middle line:84% So given epsilon greater than 0, at this point 01:00:52.960 --> 01:00:55.220 align:middle line:90% you can pick any delta you want. 01:00:55.220 --> 01:00:56.640 align:middle line:90% I'm just picking. 01:00:56.640 --> 01:01:00.920 align:middle line:84% So let delta be equal to 1, but you 01:01:00.920 --> 01:01:02.700 align:middle line:90% could pick anything you want. 01:01:02.700 --> 01:01:06.960 align:middle line:84% It has to be positive, but you can pick anything you want. 01:01:06.960 --> 01:01:14.320 align:middle line:84% Then you have that if x minus x-0 is less than 1, 01:01:14.320 --> 01:01:20.600 align:middle line:84% well then of course, f of x minus f of x-0, 01:01:20.600 --> 01:01:23.280 align:middle line:90% both of these numbers is c. 01:01:23.280 --> 01:01:28.540 align:middle line:84% So the difference is c minus c, so this is 0. 01:01:28.540 --> 01:01:32.180 align:middle line:84% So this is surely less than epsilon. 01:01:32.180 --> 01:01:36.400 align:middle line:84% So this shows that the function is continuous at all points. 01:01:36.400 --> 01:01:39.980 align:middle line:84% And you see that you could pick delta anything you want, 01:01:39.980 --> 01:01:44.360 align:middle line:84% and, of course, in particular, it doesn't even depend on x-0. 01:01:44.360 --> 01:01:47.470 align:middle line:84% It doesn't depend on epsilon, and it doesn't depend on x-0. 01:01:47.470 --> 01:01:50.120 align:middle line:90% 01:01:50.120 --> 01:01:55.100 align:middle line:84% So the constant functions are continuous. 01:01:55.100 --> 01:01:58.900 align:middle line:90% What about the next symbol? 01:01:58.900 --> 01:02:04.340 align:middle line:84% The next simplest function would be the linear function, or just 01:02:04.340 --> 01:02:07.180 align:middle line:90% x function, x. 01:02:07.180 --> 01:02:21.740 align:middle line:84% So suppose you're looking at the function f on R to R, 01:02:21.740 --> 01:02:25.910 align:middle line:90% and where f of x is equal to x. 01:02:25.910 --> 01:02:27.890 align:middle line:90% So this is also continuous. 01:02:27.890 --> 01:02:32.270 align:middle line:90% 01:02:32.270 --> 01:02:33.570 align:middle line:90% Also continuous. 01:02:33.570 --> 01:02:37.310 align:middle line:90% 01:02:37.310 --> 01:02:42.200 align:middle line:84% And the proof of that, given epsilon. 01:02:42.200 --> 01:02:44.990 align:middle line:90% 01:02:44.990 --> 01:02:50.030 align:middle line:90% set delta equal to epsilon. 01:02:50.030 --> 01:02:56.870 align:middle line:84% Then you see that if x minus x-0 is less than epsilon, 01:02:56.870 --> 01:03:05.590 align:middle line:84% then f of x minus f of x-0, but f of x is equal to x, 01:03:05.590 --> 01:03:08.550 align:middle line:90% and f of x-0 is equal to x zero. 01:03:08.550 --> 01:03:13.000 align:middle line:84% So this thing here is, of course, less than epsilon, 01:03:13.000 --> 01:03:13.500 align:middle line:90% right? 01:03:13.500 --> 01:03:17.630 align:middle line:90% 01:03:17.630 --> 01:03:20.710 align:middle line:90% So that's kind of trivial. 01:03:20.710 --> 01:03:27.690 align:middle line:84% Now the next thing is that we have the algebraic property 01:03:27.690 --> 01:03:29.890 align:middle line:90% of continuous functions. 01:03:29.890 --> 01:03:32.200 align:middle line:84% So we have some algebraic property. 01:03:32.200 --> 01:03:48.290 align:middle line:90% 01:03:48.290 --> 01:03:54.380 align:middle line:84% Properties of continuous function. 01:03:54.380 --> 01:04:00.130 align:middle line:90% 01:04:00.130 --> 01:04:03.490 align:middle line:84% And I will prove a couple of these, and the proofs you'll 01:04:03.490 --> 01:04:05.050 align:middle line:90% see in just a second. 01:04:05.050 --> 01:04:09.650 align:middle line:90% 01:04:09.650 --> 01:04:11.190 align:middle line:84% And there's good reasons for that. 01:04:11.190 --> 01:04:13.290 align:middle line:84% They're very similar to the property 01:04:13.290 --> 01:04:17.490 align:middle line:84% we had for limits of sequences, and the proofs 01:04:17.490 --> 01:04:20.770 align:middle line:84% are very similar to those proofs. 01:04:20.770 --> 01:04:32.740 align:middle line:84% So the first one is that if f and g are continuous, 01:04:32.740 --> 01:04:36.680 align:middle line:84% then the sum of these two functions is also continuous. 01:04:36.680 --> 01:04:43.660 align:middle line:90% 01:04:43.660 --> 01:05:02.340 align:middle line:84% The next one is that if f is continuous and c is a constant, 01:05:02.340 --> 01:05:04.720 align:middle line:90% then C times f is continuous. 01:05:04.720 --> 01:05:09.780 align:middle line:90% 01:05:09.780 --> 01:05:14.800 align:middle line:84% The third property is that, again, 01:05:14.800 --> 01:05:27.680 align:middle line:84% if f comma g are continuous, then the product 01:05:27.680 --> 01:05:29.580 align:middle line:90% f times g is continuous. 01:05:29.580 --> 01:05:34.600 align:middle line:90% 01:05:34.600 --> 01:05:46.960 align:middle line:84% The fourth property is that if f is continuous, 01:05:46.960 --> 01:05:57.580 align:middle line:84% and f is never equal to 0, then one over f is continuous. 01:05:57.580 --> 01:06:02.480 align:middle line:90% 01:06:02.480 --> 01:06:05.660 align:middle line:84% And the fifth property, that's a little bit different. 01:06:05.660 --> 01:06:08.760 align:middle line:84% We didn't have that for sequences. 01:06:08.760 --> 01:06:13.520 align:middle line:84% The fifth property is that if you take a map, f of g, 01:06:13.520 --> 01:06:17.400 align:middle line:84% take a function, f, you think about it 01:06:17.400 --> 01:06:20.370 align:middle line:84% as a function from R to R, and then you 01:06:20.370 --> 01:06:25.970 align:middle line:84% have another function, g, from R to R. 01:06:25.970 --> 01:06:30.650 align:middle line:84% And now you could look at where you check first. 01:06:30.650 --> 01:06:33.250 align:middle line:84% You could look at the composition. 01:06:33.250 --> 01:06:35.570 align:middle line:84% So you can look at the function h 01:06:35.570 --> 01:06:41.450 align:middle line:84% of x, which is where you first apply f to x, 01:06:41.450 --> 01:06:45.150 align:middle line:90% and then you apply g to it. 01:06:45.150 --> 01:06:53.370 align:middle line:84% So if both f, so if this here and the function g 01:06:53.370 --> 01:07:00.490 align:middle line:84% are continuous, then the composition 01:07:00.490 --> 01:07:02.150 align:middle line:90% here is also continuous. 01:07:02.150 --> 01:07:07.930 align:middle line:90% 01:07:07.930 --> 01:07:09.930 align:middle line:84% When you have a composition of function, 01:07:09.930 --> 01:07:13.650 align:middle line:84% you always have to be careful that it makes sense. 01:07:13.650 --> 01:07:18.710 align:middle line:84% So to make sense means that the function, g, the one that-- 01:07:18.710 --> 01:07:19.630 align:middle line:90% this thing-- 01:07:19.630 --> 01:07:23.950 align:middle line:84% function that you apply, is that the image of-- 01:07:23.950 --> 01:07:25.830 align:middle line:84% when you use the first function, you 01:07:25.830 --> 01:07:29.950 align:middle line:84% have to make sure that what you get here 01:07:29.950 --> 01:07:34.590 align:middle line:84% is at some point where g is defined. 01:07:34.590 --> 01:07:40.710 align:middle line:84% So just for simplicity, I just looked at maps, into functions 01:07:40.710 --> 01:07:43.855 align:middle line:84% into R where g also was defined on all of that. 01:07:43.855 --> 01:07:45.730 align:middle line:84% But of course, it doesn't have to be defined, 01:07:45.730 --> 01:07:48.750 align:middle line:84% but you just have to make sure that g is 01:07:48.750 --> 01:07:51.650 align:middle line:90% defined where f and x end up. 01:07:51.650 --> 01:07:54.610 align:middle line:90% 01:07:54.610 --> 01:07:59.990 align:middle line:84% I'll talk a little bit about how to prove these, 01:07:59.990 --> 01:08:03.310 align:middle line:84% but it's very similar to what we did for sequences. 01:08:03.310 --> 01:08:08.550 align:middle line:84% Let me just apply it, so let me look at an example 01:08:08.550 --> 01:08:09.970 align:middle line:90% where I apply it. 01:08:09.970 --> 01:08:18.319 align:middle line:90% 01:08:18.319 --> 01:08:22.220 align:middle line:84% So suppose you take x squared plus 1. 01:08:22.220 --> 01:08:23.899 align:middle line:84% Suppose you take this function here. 01:08:23.899 --> 01:08:25.549 align:middle line:84% I'll take a look at two examples. 01:08:25.549 --> 01:08:29.399 align:middle line:90% 01:08:29.399 --> 01:08:32.520 align:middle line:84% Suppose you take this function here. 01:08:32.520 --> 01:08:35.680 align:middle line:84% I claim that this here is continuous. 01:08:35.680 --> 01:08:42.520 align:middle line:84% I'm not going to go through the proof because we already-- 01:08:42.520 --> 01:08:47.359 align:middle line:84% actually, so this here proves at least some of these properties. 01:08:47.359 --> 01:08:54.720 align:middle line:84% But once you have that, then you have that g of x equal to x. 01:08:54.720 --> 01:08:57.120 align:middle line:90% This here is continuous. 01:08:57.120 --> 01:09:02.279 align:middle line:84% This is what we proved just a little while ago. 01:09:02.279 --> 01:09:04.700 align:middle line:84% And if you take the function that is constant, 01:09:04.700 --> 01:09:07.920 align:middle line:84% and in this case constant 1, we already 01:09:07.920 --> 01:09:10.960 align:middle line:90% proved that that's continuous. 01:09:10.960 --> 01:09:16.140 align:middle line:84% Now this means that if you take a continuous function 01:09:16.140 --> 01:09:20.660 align:middle line:84% by whatever, rule three or something like that over there, 01:09:20.660 --> 01:09:25.260 align:middle line:84% and you take the function where it's multiplied 01:09:25.260 --> 01:09:29.420 align:middle line:90% by itself, this function here. 01:09:29.420 --> 01:09:31.740 align:middle line:90% Because g is continuous. 01:09:31.740 --> 01:09:33.880 align:middle line:84% This function here is also continuous. 01:09:33.880 --> 01:09:39.819 align:middle line:90% 01:09:39.819 --> 01:09:48.779 align:middle line:84% And then once you have that, you can add it, g squared plus h. 01:09:48.779 --> 01:09:51.080 align:middle line:84% You now have established that this is continuous. 01:09:51.080 --> 01:09:53.140 align:middle line:84% This is a constant function, so we already 01:09:53.140 --> 01:09:54.720 align:middle line:90% proved that this is continuous. 01:09:54.720 --> 01:09:58.440 align:middle line:84% So the sum, by rule one over there, this here is continuous. 01:09:58.440 --> 01:10:10.160 align:middle line:90% 01:10:10.160 --> 01:10:14.710 align:middle line:84% So this show that this function here, this quadratic function, 01:10:14.710 --> 01:10:15.870 align:middle line:90% is continuous. 01:10:15.870 --> 01:10:17.870 align:middle line:84% But in fact, of course, this argument 01:10:17.870 --> 01:10:24.270 align:middle line:84% gives you that any polynomial, any polynomial is continuous. 01:10:24.270 --> 01:10:27.910 align:middle line:84% So this here proves that any polynomial-- 01:10:27.910 --> 01:10:34.210 align:middle line:90% 01:10:34.210 --> 01:10:39.390 align:middle line:84% So if you're using 1, 2, and 3 then 01:10:39.390 --> 01:10:42.110 align:middle line:84% it's just starting at the constant function. 01:10:42.110 --> 01:10:48.330 align:middle line:84% And the function x, you can get to any polynomial, 01:10:48.330 --> 01:10:50.410 align:middle line:90% and using these operations. 01:10:50.410 --> 01:10:53.950 align:middle line:84% And so this means that any polynomial is continuous, right? 01:10:53.950 --> 01:11:02.090 align:middle line:84% So you have the following theorem, all polynomials. 01:11:02.090 --> 01:11:08.550 align:middle line:90% 01:11:08.550 --> 01:11:09.650 align:middle line:90% are continuous. 01:11:09.650 --> 01:11:15.236 align:middle line:90% 01:11:15.236 --> 01:11:20.850 align:middle line:84% And in fact, so this is just using 1, 2, and 3 over here. 01:11:20.850 --> 01:11:26.610 align:middle line:84% If you now also bring in four, then you have-- 01:11:26.610 --> 01:11:28.840 align:middle line:90% so here is proof. 01:11:28.840 --> 01:11:32.330 align:middle line:90% 01:11:32.330 --> 01:11:36.530 align:middle line:84% So it uses the two examples that we gave, 01:11:36.530 --> 01:11:53.450 align:middle line:84% and then in addition to that, it follows from the algebraic rules 01:11:53.450 --> 01:11:56.890 align:middle line:90% 1 through 3. 01:11:56.890 --> 01:12:01.930 align:middle line:84% If you bring in also 4, 5 is something slightly different. 01:12:01.930 --> 01:12:06.210 align:middle line:84% But if you bring in 4 here, then you 01:12:06.210 --> 01:12:08.800 align:middle line:84% get what's called any rational function. 01:12:08.800 --> 01:12:15.140 align:middle line:90% 01:12:15.140 --> 01:12:16.680 align:middle line:90% All rational functions. 01:12:16.680 --> 01:12:25.300 align:middle line:90% 01:12:25.300 --> 01:12:26.240 align:middle line:90% are continuous. 01:12:26.240 --> 01:12:27.870 align:middle line:90% So what is a rational function? 01:12:27.870 --> 01:12:32.500 align:middle line:90% 01:12:32.500 --> 01:12:34.300 align:middle line:84% A rational function is just where 01:12:34.300 --> 01:12:38.860 align:middle line:84% you're taking a polynomial, and then you're 01:12:38.860 --> 01:12:42.060 align:middle line:84% dividing it by another polynomial. 01:12:42.060 --> 01:12:45.500 align:middle line:84% So we know that any polynomial is continuous. 01:12:45.500 --> 01:12:47.500 align:middle line:90% If you take another polynomial-- 01:12:47.500 --> 01:12:53.080 align:middle line:84% and let's look at that other polynomial away from the roots, 01:12:53.080 --> 01:12:58.340 align:middle line:84% away from where it is 0, away from where it's 0. 01:12:58.340 --> 01:13:01.000 align:middle line:84% I mean, it's of course even continuous where it's 0, 01:13:01.000 --> 01:13:02.900 align:middle line:84% but you can't divide by it, right? 01:13:02.900 --> 01:13:10.600 align:middle line:84% So away from it, this function here is not 0 and is continuous. 01:13:10.600 --> 01:13:13.400 align:middle line:84% This here is continuous, rule four, 01:13:13.400 --> 01:13:28.260 align:middle line:84% so this is proof that the theorem above plus rule four. 01:13:28.260 --> 01:13:32.920 align:middle line:90% 01:13:32.920 --> 01:13:37.480 align:middle line:84% AUDIENCE: Earlier you were showing that the function of x 01:13:37.480 --> 01:13:41.240 align:middle line:84% shows your delta as a function of epsilon. 01:13:41.240 --> 01:13:42.240 align:middle line:90% TOBIAS COLDING: Exactly. 01:13:42.240 --> 01:13:45.180 align:middle line:84% In that case, I chose delta to be simply epsilon. 01:13:45.180 --> 01:13:45.680 align:middle line:90% Yeah. 01:13:45.680 --> 01:13:46.790 align:middle line:90% Yeah. 01:13:46.790 --> 01:13:48.040 align:middle line:90% AUDIENCE: And you can do that? 01:13:48.040 --> 01:13:50.000 align:middle line:84% TOBIAS COLDING: That's not a problem. 01:13:50.000 --> 01:13:54.160 align:middle line:84% If you give me an epsilon, then I have to pick a delta. 01:13:54.160 --> 01:13:58.160 align:middle line:84% And typically, the delta will depend on epsilon. 01:13:58.160 --> 01:14:01.560 align:middle line:84% Typically, it also depends on X. Let me give you 01:14:01.560 --> 01:14:04.080 align:middle line:90% an example of the last. 01:14:04.080 --> 01:14:07.892 align:middle line:84% I might not get to proving this, but that's fine. 01:14:07.892 --> 01:14:09.350 align:middle line:84% I mean, they're actually the proof. 01:14:09.350 --> 01:14:12.990 align:middle line:84% I'm not even sure I would go through it. 01:14:12.990 --> 01:14:18.050 align:middle line:84% Maybe if I have 10 minutes, I will talk about one of them. 01:14:18.050 --> 01:14:19.650 align:middle line:90% So let me give you an example. 01:14:19.650 --> 01:14:22.330 align:middle line:90% 01:14:22.330 --> 01:14:28.530 align:middle line:84% So typically, it's highly unusual in the first example 01:14:28.530 --> 01:14:31.790 align:middle line:84% with a constant function that it doesn't really matter. 01:14:31.790 --> 01:14:34.223 align:middle line:90% You could pick delta anything. 01:14:34.223 --> 01:14:37.650 align:middle line:84% Now suppose you're looking at the function 01:14:37.650 --> 01:14:40.730 align:middle line:90% f of x equal to 1 over x. 01:14:40.730 --> 01:14:44.510 align:middle line:84% This is a rational function, because it's a polynomial. 01:14:44.510 --> 01:14:46.930 align:middle line:84% In that case, the polynomial in the denominator 01:14:46.930 --> 01:14:54.290 align:middle line:84% is the constant function, 1, and then the denominator is x. 01:14:54.290 --> 01:14:59.530 align:middle line:84% Let's just look at this function where it's strictly positive. 01:14:59.530 --> 01:15:05.070 align:middle line:84% And then you see now the graph looks basically like this. 01:15:05.070 --> 01:15:08.610 align:middle line:84% So in particular, if you go close to 0-- 01:15:08.610 --> 01:15:09.650 align:middle line:90% it's not defined as 0. 01:15:09.650 --> 01:15:15.190 align:middle line:84% But if you go close to 0, then the image varies a lot. 01:15:15.190 --> 01:15:21.850 align:middle line:84% So if you take some fixed thing here, 01:15:21.850 --> 01:15:28.430 align:middle line:84% suppose you want to prove that it's continuous at some x zero. 01:15:28.430 --> 01:15:31.870 align:middle line:84% So this means that you have to pick some delta. 01:15:31.870 --> 01:15:35.230 align:middle line:84% So given an epsilon, you have to pick a delta 01:15:35.230 --> 01:15:39.030 align:middle line:84% so that if you are in this little interval, 01:15:39.030 --> 01:15:41.570 align:middle line:84% then the image lies in that little interval. 01:15:41.570 --> 01:15:45.790 align:middle line:84% But you see that as x-0 becomes very close to 0, 01:15:45.790 --> 01:15:50.870 align:middle line:84% then in any small interval, the function varies a lot. 01:15:50.870 --> 01:15:57.650 align:middle line:84% So this means that if you take some fixed epsilon, 01:15:57.650 --> 01:16:01.950 align:middle line:84% then you expect that if the x-0 was close to 0, 01:16:01.950 --> 01:16:04.100 align:middle line:84% then the delta had to be really, really, 01:16:04.100 --> 01:16:08.000 align:middle line:90% really small for this to work. 01:16:08.000 --> 01:16:12.940 align:middle line:84% So typically, delta depends not just on epsilon-- 01:16:12.940 --> 01:16:16.320 align:middle line:84% which, I mean, it's highly unusual it doesn't-- 01:16:16.320 --> 01:16:17.860 align:middle line:90% it also depends on x. 01:16:17.860 --> 01:16:21.440 align:middle line:84% And in fact, if it doesn't depend on x or x-0, 01:16:21.440 --> 01:16:23.260 align:middle line:90% then it has a particular name. 01:16:23.260 --> 01:16:24.340 align:middle line:90% We'll get to that later. 01:16:24.340 --> 01:16:26.360 align:middle line:90% It's called uniform continuity. 01:16:26.360 --> 01:16:30.420 align:middle line:84% Uniform continuity has some nice extra properties. 01:16:30.420 --> 01:16:33.560 align:middle line:90% 01:16:33.560 --> 01:16:36.120 align:middle line:84% And you will see later on that there's 01:16:36.120 --> 01:16:42.160 align:middle line:84% a good reason why the domain here did not include 0. 01:16:42.160 --> 01:16:46.360 align:middle line:84% If somehow this would have been a continuous function defined 01:16:46.360 --> 01:16:50.200 align:middle line:84% on an interval that included the end point, then actually 01:16:50.200 --> 01:16:52.460 align:middle line:84% it would automatically be uniform continuous. 01:16:52.460 --> 01:16:55.140 align:middle line:90% So it was a good reason. 01:16:55.140 --> 01:17:03.500 align:middle line:84% This example here, it didn't include the point 0. 01:17:03.500 --> 01:17:05.720 align:middle line:90% So I have three minutes left. 01:17:05.720 --> 01:17:10.120 align:middle line:84% Let me just talk about one particular of these. 01:17:10.120 --> 01:17:13.820 align:middle line:84% Like, the first one and the second one is super easy, 01:17:13.820 --> 01:17:16.240 align:middle line:90% and the third one is also easy. 01:17:16.240 --> 01:17:18.460 align:middle line:84% And it uses exactly the same trick 01:17:18.460 --> 01:17:20.380 align:middle line:84% that we looked at before, so let me just 01:17:20.380 --> 01:17:22.720 align:middle line:90% talk about that for a minute. 01:17:22.720 --> 01:17:27.460 align:middle line:84% Just because it's a little bit more interesting than 1 and 2, 01:17:27.460 --> 01:17:28.980 align:middle line:90% is barely interesting. 01:17:28.980 --> 01:17:34.620 align:middle line:84% So suppose I'm looking at the product of these two functions. 01:17:34.620 --> 01:17:37.660 align:middle line:84% And so remember that you give me an epsilon, 01:17:37.660 --> 01:17:39.480 align:middle line:90% I have to find a delta. 01:17:39.480 --> 01:17:44.500 align:middle line:84% So I have to find a delta so that this difference here 01:17:44.500 --> 01:17:46.820 align:middle line:90% is smaller. 01:17:46.820 --> 01:17:53.540 align:middle line:84% I want to find the delta, so that if x minus x-0 is smaller, 01:17:53.540 --> 01:17:56.300 align:middle line:84% if it's smaller than delta, then this difference here 01:17:56.300 --> 01:17:57.960 align:middle line:90% is smaller than epsilon, right? 01:17:57.960 --> 01:18:02.150 align:middle line:84% That's what I'm supposed to do to just knowing 01:18:02.150 --> 01:18:05.310 align:middle line:90% that f and g are continuous. 01:18:05.310 --> 01:18:10.750 align:middle line:84% And so just like when we had this for sequences, 01:18:10.750 --> 01:18:14.790 align:middle line:84% the similar thing for sequences, it's 01:18:14.790 --> 01:18:18.030 align:middle line:84% too complicated if you just look at it like this. 01:18:18.030 --> 01:18:22.410 align:middle line:84% So what you do is that you take f of x times g of x. 01:18:22.410 --> 01:18:26.310 align:middle line:84% You insert something, and it doesn't really 01:18:26.310 --> 01:18:32.110 align:middle line:84% matter which way you do it, but this is halfway between this 01:18:32.110 --> 01:18:35.690 align:middle line:84% and that where you take one point to be x-0, 01:18:35.690 --> 01:18:36.790 align:middle line:90% the others are x. 01:18:36.790 --> 01:18:41.790 align:middle line:84% Of course, you didn't have that, so you have to add it again. 01:18:41.790 --> 01:18:44.510 align:middle line:84% But now the thing is that what you've done here 01:18:44.510 --> 01:18:47.130 align:middle line:84% is you have split the difference, if you will. 01:18:47.130 --> 01:18:50.150 align:middle line:90% 01:18:50.150 --> 01:18:54.670 align:middle line:84% And so now when you want to estimate this, 01:18:54.670 --> 01:18:56.810 align:middle line:84% you estimate this thing here first. 01:18:56.810 --> 01:19:05.250 align:middle line:84% So this is g of x times f of x minus f of x-0. 01:19:05.250 --> 01:19:09.930 align:middle line:84% And you take the absolute value of this, 01:19:09.930 --> 01:19:18.690 align:middle line:84% and then here you can factor out f of x-0 and g 01:19:18.690 --> 01:19:23.190 align:middle line:90% of x minus g of x here. 01:19:23.190 --> 01:19:26.090 align:middle line:84% Now, you can do like we did before. 01:19:26.090 --> 01:19:30.890 align:middle line:84% This here is now fixed, so you can also take the absolute value 01:19:30.890 --> 01:19:34.170 align:middle line:90% sign like that. 01:19:34.170 --> 01:19:38.790 align:middle line:84% And now x-0 is fixed, so this is just some fixed number. 01:19:38.790 --> 01:19:40.650 align:middle line:84% So this means that this thing here 01:19:40.650 --> 01:19:43.410 align:middle line:84% has to be maybe even smaller, depending 01:19:43.410 --> 01:19:44.830 align:middle line:90% on what this constant is. 01:19:44.830 --> 01:19:48.370 align:middle line:84% But that's easy to achieve with some delta, right? 01:19:48.370 --> 01:19:50.690 align:middle line:84% Here, it's a little bit more complicated 01:19:50.690 --> 01:19:53.440 align:middle line:84% because this one here is not fixed. 01:19:53.440 --> 01:19:59.260 align:middle line:84% But you have to then say that once you are close to x-0, 01:19:59.260 --> 01:20:02.500 align:middle line:84% then this here is bound for this. 01:20:02.500 --> 01:20:05.180 align:middle line:84% You see, it's like what we did before. 01:20:05.180 --> 01:20:09.340 align:middle line:84% So we're out of time, so I'll just leave it there. 01:20:09.340 --> 01:20:11.460 align:middle line:84% So next time we'll talk more about this 01:20:11.460 --> 01:20:12.640 align:middle line:90% with exponential function. 01:20:12.640 --> 01:20:13.260 align:middle line:90% Yeah? 01:20:13.260 --> 01:20:14.677 align:middle line:84% AUDIENCE: So our rational function 01:20:14.677 --> 01:20:17.940 align:middle line:84% is said to be continuous or do we not call them continuous? 01:20:17.940 --> 01:20:19.120 align:middle line:90% TOBIAS COLDING: No. 01:20:19.120 --> 01:20:21.540 align:middle line:84% Any rational function is continuous. 01:20:21.540 --> 01:20:29.780 align:middle line:84% It's defined at points where the polynomial in the denominator 01:20:29.780 --> 01:20:35.806 align:middle line:84% away from the roots of the polynomial in the denominator. 01:20:35.806 --> 01:20:37.000 align:middle line:90% It's continuous. 01:20:37.000 --> 01:20:37.700 align:middle line:90% What? 01:20:37.700 --> 01:20:39.140 align:middle line:84% AUDIENCE: It's said to be non-continuous if the roots 01:20:39.140 --> 01:20:40.585 align:middle line:90% of the polynomial [INAUDIBLE]? 01:20:40.585 --> 01:20:41.460 align:middle line:90% TOBIAS COLDING: What? 01:20:41.460 --> 01:20:44.020 align:middle line:84% Yeah, I mean, any rational function is continuous. 01:20:44.020 --> 01:20:47.820 align:middle line:84% It's just not defined at the roots. 01:20:47.820 --> 01:20:53.000 align:middle line:84% OK, now for some polynomial, some rational functions, 01:20:53.000 --> 01:21:00.820 align:middle line:84% if you were silly and you wrote this thing here, 01:21:00.820 --> 01:21:02.460 align:middle line:90% this is a polynomial. 01:21:02.460 --> 01:21:03.540 align:middle line:90% That's a polynomial. 01:21:03.540 --> 01:21:07.560 align:middle line:84% This one has a root at 0, but really, this function here, you 01:21:07.560 --> 01:21:13.340 align:middle line:84% would probably think about as being the constant function 1. 01:21:13.340 --> 01:21:14.760 align:middle line:90% But it's not a pure-- 01:21:14.760 --> 01:21:19.540 align:middle line:84% it's continuous away from the root. 01:21:19.540 --> 01:21:22.340 align:middle line:84% It doesn't say whether it's continuous at the root. 01:21:22.340 --> 01:21:27.200 align:middle line:84% It's just not defined at the root of the denominator. 01:21:27.200 --> 01:21:28.400 align:middle line:90% Yep. 01:21:28.400 --> 01:21:30.440 align:middle line:84% AUDIENCE: [? One ?] point is not like that. 01:21:30.440 --> 01:21:32.260 align:middle line:84% You didn't say continuous or [INAUDIBLE]. 01:21:32.260 --> 01:21:32.840 align:middle line:90% TOBIAS COLDING: Yep. 01:21:32.840 --> 01:21:33.420 align:middle line:90% That's right. 01:21:33.420 --> 01:21:34.238 align:middle line:90% That's right. 01:21:34.238 --> 01:21:34.780 align:middle line:90% That's right. 01:21:34.780 --> 01:21:38.080 align:middle line:84% So if you have a polynomial in the denominator, 01:21:38.080 --> 01:21:42.440 align:middle line:84% there's finitely many points where you have roots. 01:21:42.440 --> 01:21:45.300 align:middle line:84% And at those points, it's absolutely not defined. 01:21:45.300 --> 01:21:48.240 align:middle line:84% You might be able to extend it in a continuous way 01:21:48.240 --> 01:21:49.500 align:middle line:90% to those points. 01:21:49.500 --> 01:21:52.950 align:middle line:84% But typically, it's not defined in those points. 01:21:52.950 --> 01:21:56.000 align:middle line:90%