WEBVTT 00:00:00.000 --> 00:00:01.964 align:middle line:90% [SQUEAKING] 00:00:01.964 --> 00:00:03.928 align:middle line:90% [RUSTLING] 00:00:03.928 --> 00:00:06.383 align:middle line:90% [CLICKING] 00:00:06.383 --> 00:00:10.320 align:middle line:90% 00:00:10.320 --> 00:00:12.160 align:middle line:84% TOBIAS COLDING: Last time, we talked about-- 00:00:12.160 --> 00:00:16.120 align:middle line:84% one of the things we talked about last time was series. 00:00:16.120 --> 00:00:19.080 align:middle line:90% So we'll continue on that today. 00:00:19.080 --> 00:00:22.127 align:middle line:84% So series is that you have a sequence. 00:00:22.127 --> 00:00:23.460 align:middle line:90% You're starting with a sequence. 00:00:23.460 --> 00:00:24.810 align:middle line:90% So a n is a sequence. 00:00:24.810 --> 00:00:28.600 align:middle line:90% 00:00:28.600 --> 00:00:31.020 align:middle line:84% And then you form a new sequence. 00:00:31.020 --> 00:00:32.350 align:middle line:90% So you form a new sequence. 00:00:32.350 --> 00:00:35.320 align:middle line:90% 00:00:35.320 --> 00:00:45.030 align:middle line:84% A new sequence from a n as follows. 00:00:45.030 --> 00:00:51.400 align:middle line:90% 00:00:51.400 --> 00:01:00.060 align:middle line:84% So the new sequence, let's call that Sn, and S1 is just a1, 00:01:00.060 --> 00:01:14.940 align:middle line:84% S2 is a1 plus a2, S3 is a1 plus a2 plus a3, and so on. 00:01:14.940 --> 00:01:21.720 align:middle line:84% So this is the sum of the first n elements. 00:01:21.720 --> 00:01:24.740 align:middle line:90% 00:01:24.740 --> 00:01:29.890 align:middle line:90% And you typically write a Sn. 00:01:29.890 --> 00:01:32.820 align:middle line:90% 00:01:32.820 --> 00:01:37.200 align:middle line:84% You write Sn as the sum here from-- 00:01:37.200 --> 00:01:39.760 align:middle line:84% and then, in this case, I'm starting at 1. 00:01:39.760 --> 00:01:47.740 align:middle line:84% But you could start at something else, i equal to 1, to n of ai. 00:01:47.740 --> 00:01:50.700 align:middle line:90% And so to say that's the-- 00:01:50.700 --> 00:01:52.420 align:middle line:90% so and you write the series. 00:01:52.420 --> 00:01:56.020 align:middle line:84% You also write-- so this is slight abuse of notation, 00:01:56.020 --> 00:02:06.430 align:middle line:84% that you write this infinite sum, sorry, as the series. 00:02:06.430 --> 00:02:08.229 align:middle line:90% So you think about this as-- 00:02:08.229 --> 00:02:09.729 align:middle line:90% you think about this thing here. 00:02:09.729 --> 00:02:12.070 align:middle line:84% When you write this thing here, you really 00:02:12.070 --> 00:02:16.150 align:middle line:90% think about the sequence Sn. 00:02:16.150 --> 00:02:27.750 align:middle line:84% So write this for the sequence Sn. 00:02:27.750 --> 00:02:32.720 align:middle line:90% And so the series converge. 00:02:32.720 --> 00:02:40.030 align:middle line:90% 00:02:40.030 --> 00:02:52.730 align:middle line:84% This means that the sequence Sn converges. 00:02:52.730 --> 00:02:57.450 align:middle line:90% 00:02:57.450 --> 00:03:00.910 align:middle line:90% And so the most-- 00:03:00.910 --> 00:03:04.890 align:middle line:84% so remember that a sequence converges-- so recall, 00:03:04.890 --> 00:03:09.010 align:middle line:84% by the Cauchy convergence theorem-- 00:03:09.010 --> 00:03:25.690 align:middle line:84% so recall by the Cauchy convergence theorem 00:03:25.690 --> 00:03:42.980 align:middle line:84% that Sn converge if and only if Sn is a Cauchy sequence. 00:03:42.980 --> 00:03:51.010 align:middle line:90% 00:03:51.010 --> 00:03:57.190 align:middle line:84% And a Cauchy sequence just means that if you go sufficiently far 00:03:57.190 --> 00:04:00.430 align:middle line:84% out and you're looking at this difference here, 00:04:00.430 --> 00:04:07.110 align:middle line:84% then this here is small, provided 00:04:07.110 --> 00:04:11.230 align:middle line:84% that m and n are larger than some capital 00:04:11.230 --> 00:04:17.003 align:middle line:84% N, where this here is chosen depending on epsilon. 00:04:17.003 --> 00:04:20.269 align:middle line:84% That's what it means to be a Cauchy sequence. 00:04:20.269 --> 00:04:30.790 align:middle line:84% So in particular, of course, if you take-- so in particular, 00:04:30.790 --> 00:04:37.020 align:middle line:90% if you take m to be n plus 1-- 00:04:37.020 --> 00:04:39.650 align:middle line:90% 00:04:39.650 --> 00:04:40.150 align:middle line:90% sorry. 00:04:40.150 --> 00:04:46.210 align:middle line:90% Let me say m to be n minus 1. 00:04:46.210 --> 00:04:47.950 align:middle line:90% Doesn't matter. 00:04:47.950 --> 00:04:54.520 align:middle line:84% So if I take it like this, then Sn minus Sm. 00:04:54.520 --> 00:05:00.560 align:middle line:84% Well, this here is some of the first n elements, 00:05:00.560 --> 00:05:03.360 align:middle line:84% and from that we have to subtract 00:05:03.360 --> 00:05:09.160 align:middle line:84% the sum of the first n minus 1 elements. 00:05:09.160 --> 00:05:11.100 align:middle line:84% So this here-- just leave this here. 00:05:11.100 --> 00:05:13.520 align:middle line:90% Cancel out all of these. 00:05:13.520 --> 00:05:14.990 align:middle line:84% And so you're just left with a n. 00:05:14.990 --> 00:05:17.600 align:middle line:90% 00:05:17.600 --> 00:05:24.920 align:middle line:84% So this means that if it is converging, then in particular, 00:05:24.920 --> 00:05:32.960 align:middle line:84% a n here has to go to 0 as n goes to infinity. 00:05:32.960 --> 00:05:35.720 align:middle line:84% Because in fact, this whole difference 00:05:35.720 --> 00:05:39.520 align:middle line:84% here, as long as m and n are sufficiently large, 00:05:39.520 --> 00:05:43.320 align:middle line:90% has to be as small as you want. 00:05:43.320 --> 00:05:46.600 align:middle line:84% And so in a particularly special case 00:05:46.600 --> 00:05:49.960 align:middle line:84% where this here is the one prior to that, 00:05:49.960 --> 00:05:53.380 align:middle line:84% then you see that the a n will have to go to 0. 00:05:53.380 --> 00:05:56.820 align:middle line:84% So again, if you see a series, the first thing-- 00:05:56.820 --> 00:05:58.440 align:middle line:90% you do don't apply any tests. 00:05:58.440 --> 00:06:02.540 align:middle line:84% The first thing that you check is that this element here 00:06:02.540 --> 00:06:05.360 align:middle line:90% goes to 0. 00:06:05.360 --> 00:06:09.390 align:middle line:84% Now what are the most important series? 00:06:09.390 --> 00:06:12.060 align:middle line:90% 00:06:12.060 --> 00:06:15.690 align:middle line:84% So the most important is the geometric series. 00:06:15.690 --> 00:06:20.980 align:middle line:90% 00:06:20.980 --> 00:06:23.020 align:middle line:84% And the reason it's the most important 00:06:23.020 --> 00:06:25.580 align:middle line:84% is that it can be used to determine-- 00:06:25.580 --> 00:06:29.100 align:middle line:84% it's easy to determine whether it's converging or not. 00:06:29.100 --> 00:06:34.020 align:middle line:84% And it can be used to determine using the test 00:06:34.020 --> 00:06:36.300 align:middle line:90% whether other series converge. 00:06:36.300 --> 00:06:40.380 align:middle line:84% And so the geometric series is that you take a sum. 00:06:40.380 --> 00:06:43.400 align:middle line:90% Typically, here, you start at 0. 00:06:43.400 --> 00:06:45.100 align:middle line:90% You don't have to start at 0. 00:06:45.100 --> 00:06:47.680 align:middle line:90% But of course, the limit. 00:06:47.680 --> 00:06:48.180 align:middle line:90% The limit. 00:06:48.180 --> 00:06:52.320 align:middle line:84% If you take a series, then, to say that it's converging, 00:06:52.320 --> 00:06:55.520 align:middle line:84% meaning that the sequence Sn converges, 00:06:55.520 --> 00:07:00.580 align:middle line:84% and the limit of these here is also. 00:07:00.580 --> 00:07:03.320 align:middle line:84% And that's where there's this abuse of notation. 00:07:03.320 --> 00:07:05.380 align:middle line:90% It's denoted by the same thing. 00:07:05.380 --> 00:07:11.640 align:middle line:84% So this thing here is playing the role of the sequence, 00:07:11.640 --> 00:07:15.160 align:middle line:84% but it's also denoting the limit. 00:07:15.160 --> 00:07:18.000 align:middle line:90% That's just a standard thing. 00:07:18.000 --> 00:07:20.030 align:middle line:84% So in this case, I was starting from 1. 00:07:20.030 --> 00:07:26.080 align:middle line:90% 00:07:26.080 --> 00:07:29.360 align:middle line:84% So now your matrix series is where 00:07:29.360 --> 00:07:34.440 align:middle line:84% you take some number, c, and then you multiply it by itself. 00:07:34.440 --> 00:07:40.100 align:middle line:84% This is some real number, and you multiply it with itself 00:07:40.100 --> 00:07:44.840 align:middle line:84% i times, where i runs from 0 to infinity. 00:07:44.840 --> 00:07:48.230 align:middle line:84% And when i is zero, this is meant to be 1. 00:07:48.230 --> 00:07:51.150 align:middle line:90% 00:07:51.150 --> 00:07:54.450 align:middle line:84% And the reason why I want to do it here at 0 00:07:54.450 --> 00:08:02.050 align:middle line:84% is that it's probably worth remembering that this here, it's 00:08:02.050 --> 00:08:03.690 align:middle line:90% easy to deduce. 00:08:03.690 --> 00:08:10.550 align:middle line:84% But it's such a standard fact that this here is converging. 00:08:10.550 --> 00:08:13.290 align:middle line:90% 00:08:13.290 --> 00:08:15.610 align:middle line:84% So the geometric series converges, 00:08:15.610 --> 00:08:20.490 align:middle line:84% provided that the absolute value of c is less than 1. 00:08:20.490 --> 00:08:23.300 align:middle line:84% And when it is, this here is the limit. 00:08:23.300 --> 00:08:26.130 align:middle line:90% 00:08:26.130 --> 00:08:33.750 align:middle line:84% So then you can ask if c here is 1, 00:08:33.750 --> 00:08:36.130 align:middle line:90% so if c is actually equal to 1. 00:08:36.130 --> 00:08:39.929 align:middle line:84% Not just the absolute value, but if c is actually equal to 1, 00:08:39.929 --> 00:08:42.710 align:middle line:84% then it's just 1 times itself a number of times. 00:08:42.710 --> 00:08:45.250 align:middle line:90% But that does not go to 0. 00:08:45.250 --> 00:08:47.650 align:middle line:90% So that's clearly diverging. 00:08:47.650 --> 00:08:51.570 align:middle line:84% And if it's minus 1 multiply itself by a number of times, 00:08:51.570 --> 00:08:54.350 align:middle line:84% then it just alternates between 1 and negative 1. 00:08:54.350 --> 00:08:55.870 align:middle line:90% So it doesn't go to 0. 00:08:55.870 --> 00:08:59.430 align:middle line:90% So it's divergent here. 00:08:59.430 --> 00:09:04.510 align:middle line:84% And in fact, as long as c here is bigger or equal to-- 00:09:04.510 --> 00:09:07.590 align:middle line:84% the absolute value is bigger or equal to 1, 00:09:07.590 --> 00:09:10.190 align:middle line:84% because you take this number and multiply it 00:09:10.190 --> 00:09:14.550 align:middle line:84% by itself a number of times, if the absolute value of c 00:09:14.550 --> 00:09:17.930 align:middle line:84% is bigger than 1, then it actually is even blowing up. 00:09:17.930 --> 00:09:22.510 align:middle line:84% It becomes larger and larger in absolute value. 00:09:22.510 --> 00:09:26.210 align:middle line:84% So it's divergent here and it's convergent. 00:09:26.210 --> 00:09:28.810 align:middle line:90% So that was kind of easy. 00:09:28.810 --> 00:09:32.270 align:middle line:84% And it was easy to determine why we did-- 00:09:32.270 --> 00:09:38.230 align:middle line:84% this little trick will determine what this limit was. 00:09:38.230 --> 00:09:41.350 align:middle line:84% And this is a trick that goes back, I think, certainly 00:09:41.350 --> 00:09:44.950 align:middle line:84% to the ancient Greeks, I mean, this idea. 00:09:44.950 --> 00:09:47.850 align:middle line:84% And you can do this with various other things, these things 00:09:47.850 --> 00:09:51.630 align:middle line:84% about adding things in different order kind of stuff, 00:09:51.630 --> 00:09:55.510 align:middle line:90% or multiplying by something. 00:09:55.510 --> 00:09:57.150 align:middle line:90% So that was the most important. 00:09:57.150 --> 00:10:00.030 align:middle line:84% And again, we'll come back to why it is the most important, 00:10:00.030 --> 00:10:03.410 align:middle line:84% but it's because it can be used to determine 00:10:03.410 --> 00:10:06.610 align:middle line:84% whether other series converge or not. 00:10:06.610 --> 00:10:17.090 align:middle line:84% Another classical, but that sits inside a family of series 00:10:17.090 --> 00:10:19.730 align:middle line:90% is the harmonic series. 00:10:19.730 --> 00:10:23.190 align:middle line:84% So this is an example of a series. 00:10:23.190 --> 00:10:30.770 align:middle line:84% It's one series that sits inside a larger family of series. 00:10:30.770 --> 00:10:35.170 align:middle line:84% And so the harmonic series is when you take 1 over n 00:10:35.170 --> 00:10:38.890 align:middle line:84% and you're summing here like that. 00:10:38.890 --> 00:10:44.470 align:middle line:84% This series here, and we saw that last time, is divergent. 00:10:44.470 --> 00:10:47.620 align:middle line:90% 00:10:47.620 --> 00:10:51.900 align:middle line:84% And in fact, the larger series that is sitting inside is the-- 00:10:51.900 --> 00:10:58.270 align:middle line:90% sits inside a larger family. 00:10:58.270 --> 00:11:03.220 align:middle line:90% 00:11:03.220 --> 00:11:06.140 align:middle line:84% And that larger family is where you're 00:11:06.140 --> 00:11:11.940 align:middle line:84% summing from 1 to infinity of 1 over n to the alpha. 00:11:11.940 --> 00:11:16.860 align:middle line:84% And here, alpha is taken to be bigger or equal to 1. 00:11:16.860 --> 00:11:19.580 align:middle line:84% And so in the case where alpha is equal to 1, 00:11:19.580 --> 00:11:21.280 align:middle line:90% that's a harmonic series. 00:11:21.280 --> 00:11:23.020 align:middle line:90% These ones diverge. 00:11:23.020 --> 00:11:27.460 align:middle line:84% But if alpha-- and so this here is for alpha equal to 1. 00:11:27.460 --> 00:11:29.690 align:middle line:84% It's diverged because it's the harmonic series. 00:11:29.690 --> 00:11:32.580 align:middle line:90% 00:11:32.580 --> 00:11:36.020 align:middle line:84% And when alpha is strictly bigger than 1, what we 00:11:36.020 --> 00:11:38.220 align:middle line:84% just-- last time, we just saw one example 00:11:38.220 --> 00:11:41.260 align:middle line:84% of this when alpha was equal to 2, 00:11:41.260 --> 00:11:43.080 align:middle line:90% and we saw that that converged. 00:11:43.080 --> 00:11:47.520 align:middle line:84% But in fact, you can prove that for any alpha strictly bigger 00:11:47.520 --> 00:11:49.910 align:middle line:90% than 1, it does converge. 00:11:49.910 --> 00:11:53.600 align:middle line:90% 00:11:53.600 --> 00:11:57.640 align:middle line:84% And the typical thing to use there is-- 00:11:57.640 --> 00:12:00.400 align:middle line:84% and we'll talk about that at some later lecture 00:12:00.400 --> 00:12:02.210 align:middle line:84% is what's called the integral test. 00:12:02.210 --> 00:12:09.680 align:middle line:90% 00:12:09.680 --> 00:12:22.470 align:middle line:84% Now there is-- when you have a series, then most, but not all-- 00:12:22.470 --> 00:12:25.280 align:middle line:90% 00:12:25.280 --> 00:12:30.800 align:middle line:84% most tests, but again, not all, determine really 00:12:30.800 --> 00:12:35.520 align:middle line:84% whether the series is what's called absolutely convergent. 00:12:35.520 --> 00:12:36.968 align:middle line:84% So what is absolutely convergent? 00:12:36.968 --> 00:12:38.260 align:middle line:90% We talked about that last time. 00:12:38.260 --> 00:12:41.480 align:middle line:90% But let me remind you. 00:12:41.480 --> 00:12:44.180 align:middle line:90% Absolute convergence. 00:12:44.180 --> 00:12:48.900 align:middle line:90% 00:12:48.900 --> 00:12:56.100 align:middle line:84% So if you take a series here, then here 00:12:56.100 --> 00:12:59.100 align:middle line:90% is absolutely convergent. 00:12:59.100 --> 00:13:00.880 align:middle line:90% Is absolutely convergent. 00:13:00.880 --> 00:13:09.140 align:middle line:90% 00:13:09.140 --> 00:13:17.380 align:middle line:84% This just means-- it means that the series where you're slapping 00:13:17.380 --> 00:13:22.180 align:middle line:84% on absolute value, so this thing here, 00:13:22.180 --> 00:13:24.520 align:middle line:84% means that this series here is convergent. 00:13:24.520 --> 00:13:29.980 align:middle line:90% 00:13:29.980 --> 00:13:32.360 align:middle line:90% And so it's easy to see. 00:13:32.360 --> 00:13:37.700 align:middle line:90% So easy to see. 00:13:37.700 --> 00:13:41.020 align:middle line:90% And we saw this last time. 00:13:41.020 --> 00:13:43.610 align:middle line:84% And it was just from the Cauchy convergence theorem. 00:13:43.610 --> 00:13:52.350 align:middle line:90% 00:13:52.350 --> 00:14:05.750 align:middle line:84% It is easy to see that absolute convergence implies convergence, 00:14:05.750 --> 00:14:07.750 align:middle line:90% but not the other way around. 00:14:07.750 --> 00:14:12.430 align:middle line:84% So Cauchy convergence theorem gives you this implication. 00:14:12.430 --> 00:14:17.350 align:middle line:84% Why it's not the case that you have the other way around? 00:14:17.350 --> 00:14:25.670 align:middle line:84% And that is because what's called the alternating series. 00:14:25.670 --> 00:14:29.550 align:middle line:90% So the alternating series. 00:14:29.550 --> 00:14:30.570 align:middle line:90% Alternating. 00:14:30.570 --> 00:14:34.550 align:middle line:90% 00:14:34.550 --> 00:14:38.270 align:middle line:84% And there's something called the alternating series test. 00:14:38.270 --> 00:14:41.810 align:middle line:84% But let me just explain to you in words 00:14:41.810 --> 00:14:44.530 align:middle line:84% why the alternating series converge. 00:14:44.530 --> 00:14:46.580 align:middle line:84% So the alternating harmonic series. 00:14:46.580 --> 00:14:54.330 align:middle line:90% 00:14:54.330 --> 00:15:01.330 align:middle line:84% So if you take this series here minus 1 to the n times-- 00:15:01.330 --> 00:15:07.410 align:middle line:84% over n and you sum here from 0 to-- 00:15:07.410 --> 00:15:11.490 align:middle line:84% sorry, not 0, because you're dividing by 0, so from 1 00:15:11.490 --> 00:15:16.090 align:middle line:84% to infinity, this here is called the alternating series. 00:15:16.090 --> 00:15:18.930 align:middle line:90% Alternating harmonic series. 00:15:18.930 --> 00:15:22.290 align:middle line:84% If you were slapping an absolute value sign on it, 00:15:22.290 --> 00:15:28.850 align:middle line:84% this means that a n here is minus 1 to the power n over n. 00:15:28.850 --> 00:15:34.930 align:middle line:84% So if you take the absolute value sign of a n, 00:15:34.930 --> 00:15:37.410 align:middle line:90% then this here is just 1. 00:15:37.410 --> 00:15:39.370 align:middle line:90% So this is 1 over n. 00:15:39.370 --> 00:15:43.270 align:middle line:84% So this here is the corresponding series when 00:15:43.270 --> 00:15:45.070 align:middle line:90% you're slapping absolute value. 00:15:45.070 --> 00:15:46.640 align:middle line:90% And this is the harmonic series. 00:15:46.640 --> 00:15:51.910 align:middle line:90% 00:15:51.910 --> 00:15:55.170 align:middle line:90% And this is divergent. 00:15:55.170 --> 00:15:57.830 align:middle line:90% That was what we just discussed. 00:15:57.830 --> 00:16:00.722 align:middle line:84% Now why is this series here convergent? 00:16:00.722 --> 00:16:04.590 align:middle line:84% We'll probably come back to that in a later lecture in a more 00:16:04.590 --> 00:16:08.060 align:middle line:84% general setting, but let me just explain this. 00:16:08.060 --> 00:16:11.910 align:middle line:90% 00:16:11.910 --> 00:16:21.030 align:middle line:84% And so the idea is that before you start summing, 00:16:21.030 --> 00:16:25.030 align:middle line:84% imagine that you have here-- before you start your sum, 00:16:25.030 --> 00:16:28.642 align:middle line:84% then you are in a neutral position, which is 1. 00:16:28.642 --> 00:16:30.030 align:middle line:90% 0, sorry. 00:16:30.030 --> 00:16:35.350 align:middle line:90% And then you do the first thing. 00:16:35.350 --> 00:16:38.210 align:middle line:90% You're looking at S1. 00:16:38.210 --> 00:16:43.280 align:middle line:84% But S1 will bring you down, because S1 00:16:43.280 --> 00:16:47.740 align:middle line:84% is just equal to minus 1 to the power 1. 00:16:47.740 --> 00:16:49.060 align:middle line:90% But that's minus 1. 00:16:49.060 --> 00:16:51.280 align:middle line:90% So it's minus 1 over n. 00:16:51.280 --> 00:16:53.800 align:middle line:90% This is the first element. 00:16:53.800 --> 00:16:56.040 align:middle line:84% So that's going to bring you down. 00:16:56.040 --> 00:16:59.720 align:middle line:84% Let me just exaggerate the picture 00:16:59.720 --> 00:17:04.359 align:middle line:84% and bring you down here to minus 1 over n. 00:17:04.359 --> 00:17:08.180 align:middle line:90% Minus 1 over-- n is equal to 1. 00:17:08.180 --> 00:17:11.160 align:middle line:90% So this is minus 1. 00:17:11.160 --> 00:17:15.000 align:middle line:90% This is how the series starts. 00:17:15.000 --> 00:17:18.680 align:middle line:90% And so the next element is S2. 00:17:18.680 --> 00:17:21.020 align:middle line:90% But S2 is minus 1. 00:17:21.020 --> 00:17:22.760 align:middle line:90% That was the first element. 00:17:22.760 --> 00:17:30.480 align:middle line:84% And then plus and then minus 1 squared over 2. 00:17:30.480 --> 00:17:33.920 align:middle line:84% So this is-- and you would be tempted, but it's not-- 00:17:33.920 --> 00:17:36.300 align:middle line:90% don't add them at this stage. 00:17:36.300 --> 00:17:40.610 align:middle line:84% But just looking at it like this. 00:17:40.610 --> 00:17:48.460 align:middle line:84% So the next-- so this here is S1. 00:17:48.460 --> 00:17:50.700 align:middle line:90% Let me just write it down here. 00:17:50.700 --> 00:17:52.580 align:middle line:90% This is S1. 00:17:52.580 --> 00:17:57.780 align:middle line:84% When you're looking at S2, then you're coming back towards 0. 00:17:57.780 --> 00:18:02.440 align:middle line:84% But you don't go as far up as you went down, 00:18:02.440 --> 00:18:05.580 align:middle line:90% because it's 1 over 2. 00:18:05.580 --> 00:18:06.080 align:middle line:90% Right. 00:18:06.080 --> 00:18:08.740 align:middle line:90% So this here is S2. 00:18:08.740 --> 00:18:12.300 align:middle line:84% And then you see the next one is-- 00:18:12.300 --> 00:18:13.360 align:middle line:90% the next one. 00:18:13.360 --> 00:18:18.500 align:middle line:84% If you start here, then you are supposed to-- so for S3, 00:18:18.500 --> 00:18:26.220 align:middle line:84% it's really S2 plus minus 1 to the power 3 over 3. 00:18:26.220 --> 00:18:32.100 align:middle line:84% So this is S2 minus, because minus 1 to the power 3 00:18:32.100 --> 00:18:35.400 align:middle line:90% is a negative sign like that. 00:18:35.400 --> 00:18:38.800 align:middle line:90% So the next one here-- 00:18:38.800 --> 00:18:42.460 align:middle line:84% so you're starting here before you have done any summation. 00:18:42.460 --> 00:18:44.000 align:middle line:90% You're starting at 0. 00:18:44.000 --> 00:18:45.440 align:middle line:90% You're brought down here. 00:18:45.440 --> 00:18:47.960 align:middle line:90% You come up here. 00:18:47.960 --> 00:18:50.300 align:middle line:84% But then in S3, you're going down, 00:18:50.300 --> 00:18:53.600 align:middle line:84% but you don't go as far down as the previous one. 00:18:53.600 --> 00:18:55.320 align:middle line:90% So you're going down here. 00:18:55.320 --> 00:18:58.220 align:middle line:84% And so you're really continuing this process. 00:18:58.220 --> 00:19:01.000 align:middle line:84% And you see that this interval gets smaller and smaller, 00:19:01.000 --> 00:19:03.380 align:middle line:84% and it just goes down, and then it come up. 00:19:03.380 --> 00:19:07.200 align:middle line:84% But it doesn't go as far up as you were before, 00:19:07.200 --> 00:19:08.700 align:middle line:90% and then so on and so forth. 00:19:08.700 --> 00:19:11.500 align:middle line:84% And so you could make this more precise. 00:19:11.500 --> 00:19:13.000 align:middle line:90% It's quite easy. 00:19:13.000 --> 00:19:16.460 align:middle line:84% And this shows you that this series is converging. 00:19:16.460 --> 00:19:20.680 align:middle line:84% And in fact, this is the basis for this alternating series 00:19:20.680 --> 00:19:21.180 align:middle line:90% test. 00:19:21.180 --> 00:19:24.440 align:middle line:84% And it works as long as you have alternating-- 00:19:24.440 --> 00:19:27.800 align:middle line:84% where the sign of the a n's are alternating 00:19:27.800 --> 00:19:30.240 align:middle line:84% and the absolute value goes to 0. 00:19:30.240 --> 00:19:32.840 align:middle line:84% Then this here, that's your alternating series. 00:19:32.840 --> 00:19:35.970 align:middle line:84% So I'll probably come back to that at some later stage. 00:19:35.970 --> 00:19:44.050 align:middle line:84% But this just shows you that if you take a series, then-- 00:19:44.050 --> 00:19:46.770 align:middle line:84% So the absolute convergence means 00:19:46.770 --> 00:19:48.990 align:middle line:84% that if you slap absolute value sign, 00:19:48.990 --> 00:19:51.930 align:middle line:84% then that new series also converges. 00:19:51.930 --> 00:19:55.065 align:middle line:84% Absolute convergence implies convergence, but not 00:19:55.065 --> 00:19:55.940 align:middle line:90% the other way around. 00:19:55.940 --> 00:19:58.750 align:middle line:90% 00:19:58.750 --> 00:20:03.910 align:middle line:90% So now the next thing is that-- 00:20:03.910 --> 00:20:07.200 align:middle line:84% so the next thing is that if you take a series-- 00:20:07.200 --> 00:20:11.370 align:middle line:90% 00:20:11.370 --> 00:20:16.730 align:middle line:84% suppose you take a series and suppose that you knew that-- 00:20:16.730 --> 00:20:24.610 align:middle line:84% so suppose you have a series like this and you knew that 00:20:24.610 --> 00:20:26.470 align:middle line:90% a n here-- 00:20:26.470 --> 00:20:29.750 align:middle line:84% all of the a n's were non-negative for all n. 00:20:29.750 --> 00:20:33.590 align:middle line:90% 00:20:33.590 --> 00:20:35.950 align:middle line:90% Then convergence here. 00:20:35.950 --> 00:20:36.950 align:middle line:90% Convergence. 00:20:36.950 --> 00:20:41.110 align:middle line:90% 00:20:41.110 --> 00:20:42.370 align:middle line:90% Just implied, right? 00:20:42.370 --> 00:20:47.550 align:middle line:84% So convergence is-- because the Sn is now the sum here 00:20:47.550 --> 00:20:48.890 align:middle line:90% of these guys here. 00:20:48.890 --> 00:20:51.470 align:middle line:90% 00:20:51.470 --> 00:20:53.830 align:middle line:90% That's Sn. 00:20:53.830 --> 00:20:55.850 align:middle line:84% And you see that this is, of course, 00:20:55.850 --> 00:20:58.150 align:middle line:84% less than or equal to-- because all of these guys 00:20:58.150 --> 00:21:03.030 align:middle line:84% are non-negative, it's less than or equal to where you summing. 00:21:03.030 --> 00:21:08.710 align:middle line:84% Not just up to n, but one more, because this is non-negative. 00:21:08.710 --> 00:21:11.190 align:middle line:90% And so this is Sn plus 1. 00:21:11.190 --> 00:21:14.910 align:middle line:84% So you see that if the elements are non-negative, 00:21:14.910 --> 00:21:18.610 align:middle line:90% then the sequence Sn is-- 00:21:18.610 --> 00:21:24.210 align:middle line:84% so Sn is monotone non-decreasing sequence. 00:21:24.210 --> 00:21:27.870 align:middle line:84% And so to say-- by the monotone convergence theorem, 00:21:27.870 --> 00:21:31.510 align:middle line:84% to say that this sequence, Sn, is convergent, 00:21:31.510 --> 00:21:33.950 align:middle line:90% is equivalent to saying that. 00:21:33.950 --> 00:21:47.500 align:middle line:84% So Sn converges if and only if Sn is bounded from above. 00:21:47.500 --> 00:21:58.290 align:middle line:90% 00:21:58.290 --> 00:21:58.790 align:middle line:90% OK. 00:21:58.790 --> 00:22:03.890 align:middle line:84% So now when you have a series and you have already 00:22:03.890 --> 00:22:08.730 align:middle line:84% checked that a n goes to 0, and you are then 00:22:08.730 --> 00:22:13.590 align:middle line:84% asked to determine whether the series is convergent or not, 00:22:13.590 --> 00:22:15.830 align:middle line:84% then there are these number of tests. 00:22:15.830 --> 00:22:18.150 align:middle line:90% And so there's a bunch of tests. 00:22:18.150 --> 00:22:24.750 align:middle line:90% So tests for convergence. 00:22:24.750 --> 00:22:30.090 align:middle line:90% 00:22:30.090 --> 00:22:32.220 align:middle line:90% And so there's a bunch of them. 00:22:32.220 --> 00:22:35.340 align:middle line:90% And let me talk about-- 00:22:35.340 --> 00:22:37.870 align:middle line:84% in this lecture, I'll just talk about the comparison test. 00:22:37.870 --> 00:22:40.740 align:middle line:90% 00:22:40.740 --> 00:22:46.040 align:middle line:84% And I mentioned them very briefly last time, 00:22:46.040 --> 00:22:48.020 align:middle line:84% but I'll talk a little bit more about them. 00:22:48.020 --> 00:22:49.570 align:middle line:90% And then there's the ratio test. 00:22:49.570 --> 00:22:56.980 align:middle line:90% 00:22:56.980 --> 00:22:59.120 align:middle line:90% And then there's the root test. 00:22:59.120 --> 00:23:01.180 align:middle line:90% So I'll talk about those three. 00:23:01.180 --> 00:23:02.740 align:middle line:90% But there are other tests. 00:23:02.740 --> 00:23:03.590 align:middle line:90% So root test. 00:23:03.590 --> 00:23:14.500 align:middle line:90% 00:23:14.500 --> 00:23:20.140 align:middle line:84% So now let's start with the comparison test. 00:23:20.140 --> 00:23:22.850 align:middle line:84% And there are two versions of this. 00:23:22.850 --> 00:23:27.060 align:middle line:90% 00:23:27.060 --> 00:23:30.660 align:middle line:90% One version is the following. 00:23:30.660 --> 00:23:40.460 align:middle line:84% Let's suppose that you have-- so comparison test version one. 00:23:40.460 --> 00:23:46.240 align:middle line:90% 00:23:46.240 --> 00:23:49.640 align:middle line:84% And so the first version of the comparison test 00:23:49.640 --> 00:23:50.780 align:middle line:90% is the following. 00:23:50.780 --> 00:23:56.460 align:middle line:84% Imagine that you have two series here, a n, and again, 00:23:56.460 --> 00:23:58.220 align:middle line:90% we talked about this last time. 00:23:58.220 --> 00:24:02.880 align:middle line:84% But let me just say a few more words, that you have this series 00:24:02.880 --> 00:24:04.840 align:middle line:90% and you have that series. 00:24:04.840 --> 00:24:11.080 align:middle line:84% And I'm assuming here that a n and the bn 00:24:11.080 --> 00:24:12.540 align:middle line:90% are non-negative numbers. 00:24:12.540 --> 00:24:17.800 align:middle line:84% And in fact, I'm assuming that bn is larger or equal to a n. 00:24:17.800 --> 00:24:20.460 align:middle line:84% In the comparison test, this version, 00:24:20.460 --> 00:24:28.460 align:middle line:84% the simple version of the comparison test is this here. 00:24:28.460 --> 00:24:38.780 align:middle line:84% If the series based on the bn is convergent, 00:24:38.780 --> 00:24:47.160 align:middle line:84% then the series of the a n's is also convergent. 00:24:47.160 --> 00:24:52.860 align:middle line:90% 00:24:52.860 --> 00:24:57.660 align:middle line:84% And so this version of the comparison test 00:24:57.660 --> 00:25:00.820 align:middle line:84% is used both in this way, but it's also 00:25:00.820 --> 00:25:05.720 align:middle line:84% used sometimes where you know that this series is divergent. 00:25:05.720 --> 00:25:08.380 align:middle line:84% And from that, you conclude that this one is just 00:25:08.380 --> 00:25:09.960 align:middle line:90% a negation of this statement. 00:25:09.960 --> 00:25:14.420 align:middle line:90% 00:25:14.420 --> 00:25:18.100 align:middle line:90% So let's see. 00:25:18.100 --> 00:25:24.900 align:middle line:84% So the proof here of that is actually easy. 00:25:24.900 --> 00:25:28.390 align:middle line:90% So why is this the case? 00:25:28.390 --> 00:25:38.670 align:middle line:84% Well, the thing is that you know by the Cauchy convergence 00:25:38.670 --> 00:25:39.170 align:middle line:90% theorem. 00:25:39.170 --> 00:25:46.790 align:middle line:90% 00:25:46.790 --> 00:25:51.790 align:middle line:84% And we know that to determine whether-- 00:25:51.790 --> 00:25:56.110 align:middle line:84% all we need to check to determine whether this sequence 00:25:56.110 --> 00:25:57.870 align:middle line:90% here is convergent-- 00:25:57.870 --> 00:26:08.230 align:middle line:84% all we need to figure out is if it is a Cauchy sequence. 00:26:08.230 --> 00:26:12.310 align:middle line:84% So this means that you want to say that-- 00:26:12.310 --> 00:26:23.110 align:middle line:84% so suppose you're looking at the sequence defined from a. 00:26:23.110 --> 00:26:28.010 align:middle line:84% And so this is just where I'm now summing the first n 00:26:28.010 --> 00:26:32.250 align:middle line:90% elements in a. 00:26:32.250 --> 00:26:35.630 align:middle line:90% And I just want to-- 00:26:35.630 --> 00:26:40.490 align:middle line:84% so I'm using the superscript a just 00:26:40.490 --> 00:26:44.250 align:middle line:84% to denote that it's coming from the sequence a n. 00:26:44.250 --> 00:26:48.670 align:middle line:84% So if you're looking at this, and so likewise, 00:26:48.670 --> 00:26:50.450 align:middle line:90% I look at this thing here. 00:26:50.450 --> 00:26:55.130 align:middle line:90% 00:26:55.130 --> 00:26:58.210 align:middle line:84% Now, of course, you have that suppose you're 00:26:58.210 --> 00:27:02.650 align:middle line:90% looking at Sna minus Sma. 00:27:02.650 --> 00:27:05.570 align:middle line:90% 00:27:05.570 --> 00:27:08.730 align:middle line:84% You want to show that this here-- 00:27:08.730 --> 00:27:11.630 align:middle line:84% you want to show that this here is a Cauchy sequence. 00:27:11.630 --> 00:27:15.010 align:middle line:84% So you're looking at this difference here. 00:27:15.010 --> 00:27:19.570 align:middle line:84% But this thing here is just this here difference. 00:27:19.570 --> 00:27:23.810 align:middle line:84% I'm assuming that m is less than n. 00:27:23.810 --> 00:27:25.710 align:middle line:90% Strictly less than m. 00:27:25.710 --> 00:27:27.030 align:middle line:90% n, sorry. 00:27:27.030 --> 00:27:30.510 align:middle line:84% This just means that this difference here 00:27:30.510 --> 00:27:35.250 align:middle line:84% is just where you're coming from m plus 1. 00:27:35.250 --> 00:27:42.416 align:middle line:84% So this is just the a m plus 1 up to a n. 00:27:42.416 --> 00:27:45.630 align:middle line:90% This is just this one. 00:27:45.630 --> 00:27:46.710 align:middle line:90% Yeah? 00:27:46.710 --> 00:27:50.730 align:middle line:84% AUDIENCE: When we're proving convergence, there's now-- 00:27:50.730 --> 00:27:52.870 align:middle line:84% just to give some motivation, is the reason 00:27:52.870 --> 00:27:55.613 align:middle line:84% that we're using Cauchy sequences? 00:27:55.613 --> 00:27:57.030 align:middle line:84% TOBIAS COLDING: At this stage, you 00:27:57.030 --> 00:27:59.770 align:middle line:84% could either use Cauchy sequence, that's what I started. 00:27:59.770 --> 00:28:02.810 align:middle line:84% You could also have used the monotone convergence theorem. 00:28:02.810 --> 00:28:04.394 align:middle line:90% Maybe that would be-- 00:28:04.394 --> 00:28:04.936 align:middle line:90% AUDIENCE: OK. 00:28:04.936 --> 00:28:05.550 align:middle line:84% Because I was just going to say, back 00:28:05.550 --> 00:28:07.633 align:middle line:84% to the definition of convergence that we initially 00:28:07.633 --> 00:28:10.252 align:middle line:84% had where we had the absolute value of a of n minus some-- 00:28:10.252 --> 00:28:11.710 align:middle line:84% TOBIAS COLDING: Yeah, that's right. 00:28:11.710 --> 00:28:13.690 align:middle line:84% AUDIENCE: Is it just like-- is it-- 00:28:13.690 --> 00:28:16.150 align:middle line:84% TOBIAS COLDING: So if you went all the way back 00:28:16.150 --> 00:28:18.350 align:middle line:84% to the definition, then you would 00:28:18.350 --> 00:28:20.640 align:middle line:84% have to determine what the limit was. 00:28:20.640 --> 00:28:21.430 align:middle line:84% AUDIENCE: Right, which we don't want. 00:28:21.430 --> 00:28:23.763 align:middle line:84% TOBIAS COLDING: And we don't want because we don't know. 00:28:23.763 --> 00:28:26.160 align:middle line:84% But just because we may know what the limit of b-- 00:28:26.160 --> 00:28:28.980 align:middle line:90% of the sequence based on b is. 00:28:28.980 --> 00:28:31.160 align:middle line:84% But this doesn't mean that you can figure out what 00:28:31.160 --> 00:28:32.120 align:middle line:90% the limit of the a-- 00:28:32.120 --> 00:28:34.920 align:middle line:84% AUDIENCE: So does that motivate why the sequences are 00:28:34.920 --> 00:28:35.940 align:middle line:90% more useful in general? 00:28:35.940 --> 00:28:36.580 align:middle line:84% TOBIAS COLDING: Yeah, that's right. 00:28:36.580 --> 00:28:36.940 align:middle line:90% That's right. 00:28:36.940 --> 00:28:37.482 align:middle line:90% That's right. 00:28:37.482 --> 00:28:38.460 align:middle line:90% Exactly, that's right. 00:28:38.460 --> 00:28:39.002 align:middle line:90% That's right. 00:28:39.002 --> 00:28:41.080 align:middle line:84% So the advantage with Cauchy sequence 00:28:41.080 --> 00:28:44.340 align:middle line:84% is that you don't have to figure out what the limit is. 00:28:44.340 --> 00:28:48.060 align:middle line:84% And it's often-- I mean, like in this case, you have no idea. 00:28:48.060 --> 00:28:51.980 align:middle line:84% I mean, it's not in terms of anything from the b's. 00:28:51.980 --> 00:28:56.240 align:middle line:84% And I should-- but let me just continue doing it like this. 00:28:56.240 --> 00:28:59.320 align:middle line:84% But just now when you asked, I should probably 00:28:59.320 --> 00:29:02.880 align:middle line:84% have done it using the monotone convergence theorem. 00:29:02.880 --> 00:29:04.700 align:middle line:84% But you can do that as an exercise. 00:29:04.700 --> 00:29:07.240 align:middle line:90% But it doesn't matter. 00:29:07.240 --> 00:29:11.980 align:middle line:90% So the difference here is-- 00:29:11.980 --> 00:29:14.320 align:middle line:90% this is what the difference is. 00:29:14.320 --> 00:29:17.040 align:middle line:84% The absolute value signs are really irrelevant, 00:29:17.040 --> 00:29:19.140 align:middle line:84% because these are all non-negative numbers. 00:29:19.140 --> 00:29:23.685 align:middle line:84% So I don't have to have the absolute value sign. 00:29:23.685 --> 00:29:27.200 align:middle line:84% And now this thing here is, of course, less than. 00:29:27.200 --> 00:29:29.950 align:middle line:84% But this here is less than the corresponding b's. 00:29:29.950 --> 00:29:34.420 align:middle line:90% 00:29:34.420 --> 00:29:39.660 align:middle line:84% And this is less than the corresponding b here like that. 00:29:39.660 --> 00:29:48.500 align:middle line:84% But this here is exactly Sn, the one based on bSm. 00:29:48.500 --> 00:29:50.060 align:middle line:90% Based on b. 00:29:50.060 --> 00:29:53.760 align:middle line:84% And this here, as long as n and m are sufficiently large, 00:29:53.760 --> 00:29:55.820 align:middle line:84% we can make this as small as we want. 00:29:55.820 --> 00:29:58.150 align:middle line:84% So therefore, this here was the smallest one. 00:29:58.150 --> 00:30:01.700 align:middle line:90% 00:30:01.700 --> 00:30:03.380 align:middle line:84% You could also have done it-- maybe 00:30:03.380 --> 00:30:05.540 align:middle line:84% let's just say that you could also 00:30:05.540 --> 00:30:10.040 align:middle line:84% have done it using the monotone convergence, because now, 00:30:10.040 --> 00:30:15.060 align:middle line:90% of course, you have that. 00:30:15.060 --> 00:30:16.900 align:middle line:90% Again, the same of thing. 00:30:16.900 --> 00:30:21.780 align:middle line:84% Say that if you take the Sn based on the a's. 00:30:21.780 --> 00:30:27.320 align:middle line:84% So that's a1 up to a n sum of these guys. 00:30:27.320 --> 00:30:31.110 align:middle line:84% But this individually is less than the corresponding b's. 00:30:31.110 --> 00:30:34.560 align:middle line:90% 00:30:34.560 --> 00:30:41.040 align:middle line:84% And this here is just the sequence Sn based on the b's. 00:30:41.040 --> 00:30:45.360 align:middle line:90% But these here converge. 00:30:45.360 --> 00:30:48.480 align:middle line:84% This is equivalent to saying that they are bounded. 00:30:48.480 --> 00:30:49.920 align:middle line:90% They are less than some c-- 00:30:49.920 --> 00:30:51.920 align:middle line:90% some real number c. 00:30:51.920 --> 00:30:53.660 align:middle line:90% And this here is for all a. 00:30:53.660 --> 00:30:57.040 align:middle line:90% 00:30:57.040 --> 00:30:59.120 align:middle line:90% For all n, sorry. 00:30:59.120 --> 00:31:01.560 align:middle line:84% But this means, of course, that those things here 00:31:01.560 --> 00:31:05.260 align:middle line:90% are bounded for all n. 00:31:05.260 --> 00:31:10.760 align:middle line:84% And this here is also a monotone sequence-- monotone increasing 00:31:10.760 --> 00:31:13.880 align:middle line:84% sequence that is not bounded from above. 00:31:13.880 --> 00:31:16.360 align:middle line:90% So therefore, it converges. 00:31:16.360 --> 00:31:19.690 align:middle line:84% So you can do it either of those two ways. 00:31:19.690 --> 00:31:22.530 align:middle line:90% So that's really easy to check. 00:31:22.530 --> 00:31:26.340 align:middle line:84% Let's see an example where we are using this test here. 00:31:26.340 --> 00:31:34.490 align:middle line:90% 00:31:34.490 --> 00:31:43.170 align:middle line:84% And so this could be that you're looking at something like-- 00:31:43.170 --> 00:31:45.880 align:middle line:84% you're looking at-- so here's an example. 00:31:45.880 --> 00:31:50.530 align:middle line:90% 00:31:50.530 --> 00:31:54.890 align:middle line:84% You're looking at a series that is true to the minus n 00:31:54.890 --> 00:32:00.550 align:middle line:84% over n, n equal to 1 to infinity. 00:32:00.550 --> 00:32:04.050 align:middle line:90% 00:32:04.050 --> 00:32:08.090 align:middle line:84% And you want to show-- you're being asked, 00:32:08.090 --> 00:32:14.210 align:middle line:90% is this series converging? 00:32:14.210 --> 00:32:17.170 align:middle line:90% 00:32:17.170 --> 00:32:22.750 align:middle line:84% So this is typically how it works. 00:32:22.750 --> 00:32:24.950 align:middle line:84% And so, of course, in this case, you 00:32:24.950 --> 00:32:30.830 align:middle line:84% should think about that this here is a sequence a n. 00:32:30.830 --> 00:32:33.750 align:middle line:90% And so you have to find the bn. 00:32:33.750 --> 00:32:37.430 align:middle line:84% But in this case, we just say-- so we're setting a. 00:32:37.430 --> 00:32:49.230 align:middle line:84% So answer set a n equal to 2 to minus n over 2 00:32:49.230 --> 00:32:53.930 align:middle line:84% and bn equal to 2 to the minus n. 00:32:53.930 --> 00:32:58.683 align:middle line:90% 00:32:58.683 --> 00:33:05.760 align:middle line:84% Then you have that a n bigger or equal to 0 is smaller than bn. 00:33:05.760 --> 00:33:09.350 align:middle line:90% 00:33:09.350 --> 00:33:17.130 align:middle line:84% And the sequence here based on the bn 00:33:17.130 --> 00:33:23.520 align:middle line:84% is a convergent geometric series. 00:33:23.520 --> 00:33:28.970 align:middle line:90% 00:33:28.970 --> 00:33:34.830 align:middle line:84% It's convergent between-- because c here, in this case, 00:33:34.830 --> 00:33:36.570 align:middle line:90% is 1 over 2. 00:33:36.570 --> 00:33:38.690 align:middle line:90% It's 1 over 2. 00:33:38.690 --> 00:33:41.170 align:middle line:90% So it's just 1 over 2. 00:33:41.170 --> 00:33:44.230 align:middle line:84% 1.5 is the same as 2 to the minus 1. 00:33:44.230 --> 00:33:47.450 align:middle line:90% 00:33:47.450 --> 00:33:49.550 align:middle line:90% And so this is convergent. 00:33:49.550 --> 00:33:52.910 align:middle line:90% 00:33:52.910 --> 00:33:55.490 align:middle line:90% So this is just so-- 00:33:55.490 --> 00:33:58.570 align:middle line:84% so typically, this could be used to-- that you're 00:33:58.570 --> 00:34:00.690 align:middle line:84% given a sequence and you're being asked, 00:34:00.690 --> 00:34:02.690 align:middle line:90% is convergent or not? 00:34:02.690 --> 00:34:06.170 align:middle line:84% And so you have to find the sequence 00:34:06.170 --> 00:34:09.330 align:middle line:84% and you're being asked whether it's convergent or not. 00:34:09.330 --> 00:34:14.489 align:middle line:84% And you have to find the comparison series. 00:34:14.489 --> 00:34:24.460 align:middle line:84% And then, in this case, say that this thing here is convergent. 00:34:24.460 --> 00:34:27.500 align:middle line:84% And so therefore, the sequence a n 00:34:27.500 --> 00:34:34.460 align:middle line:84% that was squeezed between that and 0 was also convergent. 00:34:34.460 --> 00:34:40.580 align:middle line:84% So this was the first version of the comparison test. 00:34:40.580 --> 00:34:47.300 align:middle line:84% Here is another more fancy version of comparing, 00:34:47.300 --> 00:34:50.089 align:middle line:84% where you're comparing two series. 00:34:50.089 --> 00:34:53.260 align:middle line:90% 00:34:53.260 --> 00:34:57.500 align:middle line:84% In fact, it's pretty much like that, except for this stuff 00:34:57.500 --> 00:34:59.540 align:middle line:90% we talked about. 00:34:59.540 --> 00:35:05.620 align:middle line:84% I mean, typically the tests are where you're comparing it 00:35:05.620 --> 00:35:09.060 align:middle line:90% with some other series. 00:35:09.060 --> 00:35:11.580 align:middle line:90% So here is version two. 00:35:11.580 --> 00:35:19.500 align:middle line:90% So comparison version two. 00:35:19.500 --> 00:35:26.225 align:middle line:90% 00:35:26.225 --> 00:35:31.480 align:middle line:84% And so you have, again, two series, a n. 00:35:31.480 --> 00:35:33.840 align:middle line:90% And again, it doesn't matter-- 00:35:33.840 --> 00:35:36.580 align:middle line:84% what the sum is matters where you're starting, 00:35:36.580 --> 00:35:41.320 align:middle line:84% but whether or not it converges doesn't matter where you start. 00:35:41.320 --> 00:35:45.120 align:middle line:84% So suppose that you have this series and you have this series 00:35:45.120 --> 00:35:45.620 align:middle line:90% here. 00:35:45.620 --> 00:35:48.840 align:middle line:90% 00:35:48.840 --> 00:36:00.200 align:middle line:84% And suppose that you know that the none of the bn's are 0. 00:36:00.200 --> 00:36:06.160 align:middle line:84% And you know that the a n divided by bn, 00:36:06.160 --> 00:36:11.080 align:middle line:84% that this ratio here, as n goes to infinity, 00:36:11.080 --> 00:36:17.140 align:middle line:84% converges to some L. And L here is-- 00:36:17.140 --> 00:36:22.980 align:middle line:84% L here is not 0 and is not infinity. 00:36:22.980 --> 00:36:29.060 align:middle line:84% I mean that it really is a limit, like that. 00:36:29.060 --> 00:36:32.610 align:middle line:84% So if that's the case, then the series-- 00:36:32.610 --> 00:36:39.380 align:middle line:90% 00:36:39.380 --> 00:36:52.810 align:middle line:84% then let's-- then let's have these as non-negative numbers. 00:36:52.810 --> 00:37:01.660 align:middle line:90% 00:37:01.660 --> 00:37:19.430 align:middle line:84% Then the series a n converges if and only if the series bn 00:37:19.430 --> 00:37:20.250 align:middle line:90% converges. 00:37:20.250 --> 00:37:30.270 align:middle line:90% 00:37:30.270 --> 00:37:33.270 align:middle line:90% And so an example of this-- 00:37:33.270 --> 00:37:36.790 align:middle line:84% so a typical example of this is where 00:37:36.790 --> 00:37:45.110 align:middle line:84% you're looking at the series 1 over n squared minus 1. 00:37:45.110 --> 00:37:46.990 align:middle line:90% Suppose you're looking at this. 00:37:46.990 --> 00:37:48.910 align:middle line:84% And now, just to make it so that-- 00:37:48.910 --> 00:37:53.070 align:middle line:84% make sure that it makes sense that you're not dividing by 0, 00:37:53.070 --> 00:37:56.590 align:middle line:84% you're looking at this thing here. 00:37:56.590 --> 00:38:02.270 align:middle line:84% Suppose that this is the series that you're given. 00:38:02.270 --> 00:38:04.490 align:middle line:90% Well, this here is-- 00:38:04.490 --> 00:38:07.050 align:middle line:84% so you wouldn't use the first comparison, 00:38:07.050 --> 00:38:08.770 align:middle line:84% although you could have used the first. 00:38:08.770 --> 00:38:11.855 align:middle line:84% But then it would be a little bit more complicated. 00:38:11.855 --> 00:38:14.230 align:middle line:84% Not super complicated, but a little bit more complicated, 00:38:14.230 --> 00:38:17.870 align:middle line:84% because this thing here this is-- 00:38:17.870 --> 00:38:20.230 align:middle line:84% so the first thing you observe is that this, of course, 00:38:20.230 --> 00:38:21.570 align:middle line:90% goes to 0. 00:38:21.570 --> 00:38:22.810 align:middle line:90% This goes to 0. 00:38:22.810 --> 00:38:26.250 align:middle line:84% But the second thing is that it's very close to a series 00:38:26.250 --> 00:38:28.250 align:middle line:90% that you know something about. 00:38:28.250 --> 00:38:32.610 align:middle line:84% It's very close to this series here. 00:38:32.610 --> 00:38:34.050 align:middle line:84% But again, it doesn't matter where 00:38:34.050 --> 00:38:38.250 align:middle line:84% I'm starting when you're asking whether it's converging or not. 00:38:38.250 --> 00:38:40.570 align:middle line:84% So it's very close to this series. 00:38:40.570 --> 00:38:44.850 align:middle line:90% But of course, this thing here-- 00:38:44.850 --> 00:38:47.650 align:middle line:84% this thing here is a smaller number. 00:38:47.650 --> 00:38:53.730 align:middle line:84% So this is actually bigger than 1 over n squared. 00:38:53.730 --> 00:38:57.510 align:middle line:84% This here is a convergent series. 00:38:57.510 --> 00:38:59.870 align:middle line:90% So this does not help you. 00:38:59.870 --> 00:39:03.570 align:middle line:84% This is not great for using the first test. 00:39:03.570 --> 00:39:05.170 align:middle line:90% You could have done it. 00:39:05.170 --> 00:39:07.110 align:middle line:84% You could still use the first test. 00:39:07.110 --> 00:39:09.830 align:middle line:84% But then you have to build a slightly more clever. 00:39:09.830 --> 00:39:16.910 align:middle line:84% So instead of that you can use this second test. 00:39:16.910 --> 00:39:22.430 align:middle line:84% And so in the second test you, set a n equal to 1 over n 00:39:22.430 --> 00:39:24.110 align:middle line:90% squared minus 1. 00:39:24.110 --> 00:39:28.390 align:middle line:84% And you set bn equal to 1 over n squared. 00:39:28.390 --> 00:39:33.190 align:middle line:84% And then you have that a n divided by bn. 00:39:33.190 --> 00:39:42.950 align:middle line:84% But this is just 1 over n squared minus 1 over n squared. 00:39:42.950 --> 00:39:44.390 align:middle line:90% Or 1 over n squared, sorry. 00:39:44.390 --> 00:39:47.270 align:middle line:90% 00:39:47.270 --> 00:39:56.470 align:middle line:84% And so this here is the same as n squared minus 1, like that. 00:39:56.470 --> 00:40:00.070 align:middle line:84% And now you can divide by n squared in both numerator 00:40:00.070 --> 00:40:01.130 align:middle line:90% and denominator. 00:40:01.130 --> 00:40:07.070 align:middle line:84% So you're getting 1 minus 1 over n squared. 00:40:07.070 --> 00:40:11.780 align:middle line:84% And now you just use the usual algebraic rules for limits. 00:40:11.780 --> 00:40:13.820 align:middle line:90% This here converts to 0. 00:40:13.820 --> 00:40:16.840 align:middle line:84% So it means that the denominator converts to 1. 00:40:16.840 --> 00:40:22.600 align:middle line:84% So this means that this ratio here converts to 1. 00:40:22.600 --> 00:40:27.360 align:middle line:84% And so you see that the ratios are converging to something that 00:40:27.360 --> 00:40:30.840 align:middle line:90% is neither zero nor infinity. 00:40:30.840 --> 00:40:35.160 align:middle line:84% And this means that this series here is convergent if 00:40:35.160 --> 00:40:37.020 align:middle line:90% and only if this convergent. 00:40:37.020 --> 00:40:39.420 align:middle line:84% You already know that this is convergent. 00:40:39.420 --> 00:40:41.800 align:middle line:84% So we conclude that this is convergent. 00:40:41.800 --> 00:40:44.530 align:middle line:84% So that's how it's typically used. 00:40:44.530 --> 00:40:53.300 align:middle line:90% 00:40:53.300 --> 00:41:01.620 align:middle line:84% So now the next test I want to talk about is the ratio test. 00:41:01.620 --> 00:41:04.560 align:middle line:90% And so this is-- 00:41:04.560 --> 00:41:08.900 align:middle line:84% I mean, the comparison tests are typically almost 00:41:08.900 --> 00:41:11.740 align:middle line:90% the easiest tests. 00:41:11.740 --> 00:41:14.430 align:middle line:84% But this one here is extremely useful. 00:41:14.430 --> 00:41:18.220 align:middle line:90% 00:41:18.220 --> 00:41:23.820 align:middle line:84% And so the next two tests, the ratio test-- 00:41:23.820 --> 00:41:40.140 align:middle line:84% both the ratio test and root test, 00:41:40.140 --> 00:41:55.300 align:middle line:84% are based on comparison with the geometric series. 00:41:55.300 --> 00:42:00.580 align:middle line:84% And so this is explaining why the geometric series 00:42:00.580 --> 00:42:02.620 align:middle line:90% is so crucial. 00:42:02.620 --> 00:42:05.230 align:middle line:90% So what is the ratio test? 00:42:05.230 --> 00:42:08.350 align:middle line:84% Well, the ratio test is simply the test 00:42:08.350 --> 00:42:12.670 align:middle line:90% that if you're looking at-- 00:42:12.670 --> 00:42:14.290 align:middle line:90% so if you have a series. 00:42:14.290 --> 00:42:16.930 align:middle line:84% You don't have to look at any other series. 00:42:16.930 --> 00:42:21.510 align:middle line:84% Like in the comparison test, you had to find another series. 00:42:21.510 --> 00:42:23.730 align:middle line:90% But in the ratio test-- 00:42:23.730 --> 00:42:25.080 align:middle line:90% so this is the ratio test. 00:42:25.080 --> 00:42:31.070 align:middle line:90% 00:42:31.070 --> 00:42:34.680 align:middle line:84% So you're looking at a series like this. 00:42:34.680 --> 00:42:38.870 align:middle line:90% 00:42:38.870 --> 00:42:48.190 align:middle line:84% And the ratio test is that you're looking at a n plus 1 00:42:48.190 --> 00:42:49.130 align:middle line:90% over n. 00:42:49.130 --> 00:42:55.310 align:middle line:84% So you are looking at the ratio between two consecutive elements 00:42:55.310 --> 00:42:57.510 align:middle line:90% of the a n's. 00:42:57.510 --> 00:43:01.710 align:middle line:84% So you're looking at this and you're looking 00:43:01.710 --> 00:43:04.690 align:middle line:90% at the absolute value of this. 00:43:04.690 --> 00:43:09.010 align:middle line:84% And you're asking if the limit here-- 00:43:09.010 --> 00:43:14.330 align:middle line:84% as n goes to infinity, suppose that the limit exists. 00:43:14.330 --> 00:43:17.770 align:middle line:84% And let's say that the limit is a. 00:43:17.770 --> 00:43:24.690 align:middle line:90% So it's asking you if this-- 00:43:24.690 --> 00:43:26.610 align:middle line:90% somehow the test here-- 00:43:26.610 --> 00:43:31.010 align:middle line:84% this form of the ratio test, it requires that the ratio here 00:43:31.010 --> 00:43:33.450 align:middle line:90% has a limit. 00:43:33.450 --> 00:43:40.450 align:middle line:84% So the ratio test then says that if a here is-- 00:43:40.450 --> 00:43:41.640 align:middle line:90% if a here is-- 00:43:41.640 --> 00:43:46.090 align:middle line:90% 00:43:46.090 --> 00:43:46.890 align:middle line:90% sorry. 00:43:46.890 --> 00:43:54.090 align:middle line:84% If a here is less than 1, then the series is-- so let 00:43:54.090 --> 00:43:55.190 align:middle line:90% me write it like this. 00:43:55.190 --> 00:43:57.970 align:middle line:90% 00:43:57.970 --> 00:44:04.190 align:middle line:84% So a less than 1, this here means 00:44:04.190 --> 00:44:05.970 align:middle line:90% that the series is convergent. 00:44:05.970 --> 00:44:13.270 align:middle line:90% 00:44:13.270 --> 00:44:17.230 align:middle line:84% Two, if this a here is bigger than 1, 00:44:17.230 --> 00:44:22.250 align:middle line:84% this implies that the series is divergent. 00:44:22.250 --> 00:44:26.150 align:middle line:90% 00:44:26.150 --> 00:44:30.950 align:middle line:84% And three is that if this a here is 1, 00:44:30.950 --> 00:44:36.256 align:middle line:90% then the test is inconclusive. 00:44:36.256 --> 00:44:40.150 align:middle line:90% 00:44:40.150 --> 00:44:44.470 align:middle line:84% But it's important to remember that it is possible 00:44:44.470 --> 00:44:46.610 align:middle line:84% that you don't get anything out of the test 00:44:46.610 --> 00:44:51.570 align:middle line:84% if you are in the unlucky case where this limit is 1. 00:44:51.570 --> 00:44:54.580 align:middle line:90% 00:44:54.580 --> 00:45:02.560 align:middle line:84% And I should also emphasize that it is crucial to have a limit. 00:45:02.560 --> 00:45:07.050 align:middle line:84% So the test here, in this form, it's crucial to have a limit. 00:45:07.050 --> 00:45:18.136 align:middle line:90% 00:45:18.136 --> 00:45:22.640 align:middle line:84% And we'll talk a little bit and we'll come back to that. 00:45:22.640 --> 00:45:23.900 align:middle line:90% We'll come back to this. 00:45:23.900 --> 00:45:27.760 align:middle line:90% 00:45:27.760 --> 00:45:37.440 align:middle line:84% So now let me try to explain why you have-- 00:45:37.440 --> 00:45:41.540 align:middle line:84% well, in conclusion, I don't really need to explain anything. 00:45:41.540 --> 00:45:44.740 align:middle line:90% But why in the case-- 00:45:44.740 --> 00:45:46.140 align:middle line:90% so proof of case one. 00:45:46.140 --> 00:45:52.320 align:middle line:90% 00:45:52.320 --> 00:45:55.680 align:middle line:84% So this means that I have to show-- 00:45:55.680 --> 00:46:08.940 align:middle line:84% so I need to show that if this limit here-- 00:46:08.940 --> 00:46:16.220 align:middle line:84% if this limit of a n plus 1 over a n, 00:46:16.220 --> 00:46:22.480 align:middle line:84% if this here is less than 1, then the series is convergent. 00:46:22.480 --> 00:46:31.540 align:middle line:90% 00:46:31.540 --> 00:46:37.260 align:middle line:84% And I'll just sketch the proof, because we're going to do-- 00:46:37.260 --> 00:46:37.780 align:middle line:90% yeah. 00:46:37.780 --> 00:46:39.760 align:middle line:90% I will just sketch the proof. 00:46:39.760 --> 00:46:41.980 align:middle line:84% We;ll come back, in a way, to the proof, 00:46:41.980 --> 00:46:45.620 align:middle line:84% because it's very close to the same proof that is used 00:46:45.620 --> 00:46:47.090 align:middle line:90% in the case of the root test. 00:46:47.090 --> 00:46:50.180 align:middle line:90% 00:46:50.180 --> 00:46:54.040 align:middle line:90% So that's what I need. 00:46:54.040 --> 00:46:56.570 align:middle line:90% So I have that there exists-- 00:46:56.570 --> 00:47:01.560 align:middle line:90% 00:47:01.560 --> 00:47:09.360 align:middle line:90% so given a0 less than 1. 00:47:09.360 --> 00:47:12.860 align:middle line:90% So given a0 with-- 00:47:12.860 --> 00:47:17.680 align:middle line:90% 00:47:17.680 --> 00:47:23.460 align:middle line:84% a0 bigger than this a and less than 1. 00:47:23.460 --> 00:47:26.600 align:middle line:84% So if you pick an a0 that is strictly 00:47:26.600 --> 00:47:30.480 align:middle line:84% bigger than a and still less than 1, 00:47:30.480 --> 00:47:40.400 align:middle line:84% then we know that there exists an n such 00:47:40.400 --> 00:47:51.040 align:middle line:84% that if little n is bigger than this capital N, 00:47:51.040 --> 00:47:58.850 align:middle line:90% then a n plus 1 over a n-- 00:47:58.850 --> 00:48:03.610 align:middle line:84% and really, I should have absolute value here on this. 00:48:03.610 --> 00:48:09.530 align:middle line:84% Then we have that this thing here is less than-- 00:48:09.530 --> 00:48:12.150 align:middle line:90% it's less than a0. 00:48:12.150 --> 00:48:16.370 align:middle line:84% Because if you're very far out, then this here 00:48:16.370 --> 00:48:19.490 align:middle line:84% is-- this ratio is going to be very close to this. 00:48:19.490 --> 00:48:21.830 align:middle line:84% But this is strictly less than that. 00:48:21.830 --> 00:48:24.090 align:middle line:90% So we have this. 00:48:24.090 --> 00:48:26.650 align:middle line:84% So this means that-- maybe I shouldn't 00:48:26.650 --> 00:48:30.890 align:middle line:84% call it a0 because there's too many a's now. 00:48:30.890 --> 00:48:32.710 align:middle line:90% Let's call it something else. 00:48:32.710 --> 00:48:36.380 align:middle line:84% Let's call it alpha to make it just look much more distinct. 00:48:36.380 --> 00:48:41.392 align:middle line:90% 00:48:41.392 --> 00:48:43.270 align:middle line:90% So this means that-- 00:48:43.270 --> 00:48:49.050 align:middle line:84% so you see that if-- so in other words, 00:48:49.050 --> 00:48:52.810 align:middle line:84% If n here is bigger than this capital N then 00:48:52.810 --> 00:48:57.830 align:middle line:84% you have-- just multiplying over, you have that a n plus 1 00:48:57.830 --> 00:49:01.620 align:middle line:84% is less than alpha times a n, like that. 00:49:01.620 --> 00:49:04.214 align:middle line:90% 00:49:04.214 --> 00:49:06.610 align:middle line:84% In the ratio test, I should have said that. 00:49:06.610 --> 00:49:10.790 align:middle line:84% But obviously, you are assuming that these a n's are not 0. 00:49:10.790 --> 00:49:12.450 align:middle line:90% So you can divide by them. 00:49:12.450 --> 00:49:17.230 align:middle line:90% 00:49:17.230 --> 00:49:19.150 align:middle line:90% So you have this. 00:49:19.150 --> 00:49:20.910 align:middle line:84% If you are far out, you have this, 00:49:20.910 --> 00:49:25.390 align:middle line:84% that the next one is a fraction of the previous one. 00:49:25.390 --> 00:49:30.530 align:middle line:84% But this means, of course, that if you take-- 00:49:30.530 --> 00:49:37.310 align:middle line:84% if you're looking at n is bigger or equal to this capital N, 00:49:37.310 --> 00:49:44.790 align:middle line:84% well, you have that a capital N plus 1. 00:49:44.790 --> 00:49:52.070 align:middle line:84% If you're using this a capital N as little n, 00:49:52.070 --> 00:49:57.682 align:middle line:84% then you have that this thing here, this is now the-- 00:49:57.682 --> 00:49:59.450 align:middle line:90% this here is now-- sorry. 00:49:59.450 --> 00:50:08.890 align:middle line:84% If you're using-- if you first pick n equal to capital N, then 00:50:08.890 --> 00:50:10.990 align:middle line:84% what does this inequality tell you? 00:50:10.990 --> 00:50:15.770 align:middle line:84% It tells you that a capital N plus 1 is less than or equal 00:50:15.770 --> 00:50:23.890 align:middle line:84% to alpha a capital N. Now apply this inequality here for n -- 00:50:23.890 --> 00:50:24.650 align:middle line:90% plus-- 00:50:24.650 --> 00:50:27.090 align:middle line:90% capital n plus 1. 00:50:27.090 --> 00:50:33.650 align:middle line:84% So then you have that a capital N plus true absolute value 00:50:33.650 --> 00:50:40.570 align:middle line:84% is less than or equal to alpha a n plus 1 like that. 00:50:40.570 --> 00:50:44.890 align:middle line:84% But you already proven that this thing here-- 00:50:44.890 --> 00:50:46.350 align:middle line:90% forget about the alpha here. 00:50:46.350 --> 00:50:47.830 align:middle line:84% But you've proven this inequality. 00:50:47.830 --> 00:50:51.650 align:middle line:84% So filling in that inequality, I have this alpha here. 00:50:51.650 --> 00:50:56.600 align:middle line:84% But then I have another alpha that is this and then like this. 00:50:56.600 --> 00:51:01.300 align:middle line:90% 00:51:01.300 --> 00:51:03.800 align:middle line:84% And so now, you can continue this way. 00:51:03.800 --> 00:51:09.540 align:middle line:84% So this here is like alpha squared times a capital N. 00:51:09.540 --> 00:51:12.300 align:middle line:84% And so now you see that you can continue. 00:51:12.300 --> 00:51:14.480 align:middle line:90% So if you're looking at-- 00:51:14.480 --> 00:51:20.740 align:middle line:84% so if you continue like this, then 00:51:20.740 --> 00:51:26.780 align:middle line:84% you see that a capital N plus some little n 00:51:26.780 --> 00:51:31.820 align:middle line:84% is less than or equal to alpha to the power m times 00:51:31.820 --> 00:51:35.540 align:middle line:90% a to the absolute value of a n. 00:51:35.540 --> 00:51:37.700 align:middle line:84% This here, the previous line, was just 00:51:37.700 --> 00:51:40.420 align:middle line:90% where m was equal to true. 00:51:40.420 --> 00:51:42.840 align:middle line:84% But you really get it for all integers. 00:51:42.840 --> 00:51:45.940 align:middle line:90% 00:51:45.940 --> 00:51:49.780 align:middle line:84% And so this says that at least the tail of this sequence here, 00:51:49.780 --> 00:51:53.160 align:middle line:84% the tail of the a n sequence, this is just 00:51:53.160 --> 00:51:54.540 align:middle line:90% some fixed constant. 00:51:54.540 --> 00:51:57.840 align:middle line:84% This here is a convergent geometric series. 00:51:57.840 --> 00:52:02.120 align:middle line:84% So the tail is bounded by a convergent geometric series. 00:52:02.120 --> 00:52:06.920 align:middle line:84% So you can just apply the first comparison test 00:52:06.920 --> 00:52:11.040 align:middle line:84% to prove that the sequence-- the series a n 00:52:11.040 --> 00:52:12.400 align:middle line:90% is actually convergent. 00:52:12.400 --> 00:52:16.640 align:middle line:90% 00:52:16.640 --> 00:52:23.120 align:middle line:84% So the ratio test, again, was to compare 00:52:23.120 --> 00:52:28.040 align:middle line:84% it, use the first comparison test where the comparison was 00:52:28.040 --> 00:52:29.090 align:middle line:90% the geometric series. 00:52:29.090 --> 00:52:36.520 align:middle line:90% 00:52:36.520 --> 00:52:39.560 align:middle line:90% And so of course, if the-- 00:52:39.560 --> 00:52:45.920 align:middle line:90% so this proves one, two here. 00:52:45.920 --> 00:52:49.080 align:middle line:84% But two is kind of really the same. 00:52:49.080 --> 00:52:53.140 align:middle line:90% Because if this alpha here-- 00:52:53.140 --> 00:52:56.980 align:middle line:90% so to show the second part. 00:52:56.980 --> 00:52:59.980 align:middle line:84% Again, the third one, to prove it's inconclusive, 00:52:59.980 --> 00:53:01.980 align:middle line:84% there's nothing to prove, because it 00:53:01.980 --> 00:53:05.220 align:middle line:90% leaves open any possibility. 00:53:05.220 --> 00:53:08.940 align:middle line:84% But you want to show that if this limit here-- so if this 00:53:08.940 --> 00:53:11.000 align:middle line:90% limit-- so proof of two. 00:53:11.000 --> 00:53:16.300 align:middle line:90% 00:53:16.300 --> 00:53:22.180 align:middle line:84% So you want-- so we're assuming that a here is bigger than 1. 00:53:22.180 --> 00:53:24.200 align:middle line:84% But this means that there exists-- 00:53:24.200 --> 00:53:26.820 align:middle line:90% so if you take alpha-- 00:53:26.820 --> 00:53:41.020 align:middle line:84% so this time, choose alpha such that alpha is bigger than 1, 00:53:41.020 --> 00:53:44.860 align:middle line:90% but it's smaller than a. 00:53:44.860 --> 00:53:48.080 align:middle line:90% But then you have that for-- 00:53:48.080 --> 00:53:53.030 align:middle line:84% since this ratio is converging for little n 00:53:53.030 --> 00:53:55.470 align:middle line:84% bigger than-- there exists some capital N, so 00:53:55.470 --> 00:53:58.990 align:middle line:84% that if this little n is bigger than capital N, then 00:53:58.990 --> 00:54:06.870 align:middle line:84% you have that a n plus 1 over a n, that this thing here, 00:54:06.870 --> 00:54:09.690 align:middle line:84% if it's far enough out, it's going to converge here. 00:54:09.690 --> 00:54:11.390 align:middle line:84% But at least far enough out, it's 00:54:11.390 --> 00:54:14.830 align:middle line:90% going to be bigger than alpha. 00:54:14.830 --> 00:54:22.150 align:middle line:84% Multiplying over again means that a n plus 1 00:54:22.150 --> 00:54:28.590 align:middle line:90% is bigger than alpha times a n. 00:54:28.590 --> 00:54:31.270 align:middle line:90% This here is now bigger than 1. 00:54:31.270 --> 00:54:33.410 align:middle line:84% And so in the same way as before, 00:54:33.410 --> 00:54:36.870 align:middle line:84% you see that if you're far enough out, 00:54:36.870 --> 00:54:42.070 align:middle line:84% then the next element is some multiple, 00:54:42.070 --> 00:54:48.850 align:middle line:84% some big number, at least bigger than 1, times the previous one. 00:54:48.850 --> 00:54:52.290 align:middle line:84% But this means that they're getting larger and larger. 00:54:52.290 --> 00:54:57.290 align:middle line:84% From some fixed stage on and outwards, the a n's are 00:54:57.290 --> 00:54:59.310 align:middle line:90% getting larger and larger. 00:54:59.310 --> 00:55:03.170 align:middle line:84% So they don't converge to 0 as they were supposed to. 00:55:03.170 --> 00:55:05.858 align:middle line:90% So this is proving true. 00:55:05.858 --> 00:55:07.650 align:middle line:84% And three, again, there's nothing to prove, 00:55:07.650 --> 00:55:10.190 align:middle line:84% because it leaves open both possibilities. 00:55:10.190 --> 00:55:18.067 align:middle line:90% 00:55:18.067 --> 00:55:20.150 align:middle line:84% AUDIENCE: I have a question about the first group. 00:55:20.150 --> 00:55:21.025 align:middle line:90% TOBIAS COLDING: Yeah. 00:55:21.025 --> 00:55:23.970 align:middle line:84% AUDIENCE: Does that assume that the series is monotonically 00:55:23.970 --> 00:55:26.810 align:middle line:90% decreasing? 00:55:26.810 --> 00:55:29.330 align:middle line:90% TOBIAS COLDING: Sorry? 00:55:29.330 --> 00:55:32.230 align:middle line:84% AUDIENCE: I guess just looking at it from a high level, 00:55:32.230 --> 00:55:34.625 align:middle line:84% we're saying that a of n plus 1 over a of n-- 00:55:34.625 --> 00:55:35.750 align:middle line:90% TOBIAS COLDING: Wait, wait. 00:55:35.750 --> 00:55:38.968 align:middle line:84% So you're asking-- what are you asking? 00:55:38.968 --> 00:55:41.510 align:middle line:84% AUDIENCE: So for part one, when we say that a is less than 1, 00:55:41.510 --> 00:55:45.690 align:middle line:84% that implies that it converges, is that only 00:55:45.690 --> 00:55:47.590 align:middle line:84% going to work for monotonically decreasing? 00:55:47.590 --> 00:55:48.715 align:middle line:90% TOBIAS COLDING: No, no, no. 00:55:48.715 --> 00:55:54.530 align:middle line:84% So there's no assumption here on whether the sequence is right. 00:55:54.530 --> 00:55:56.810 align:middle line:90% I mean, so 1-- 00:55:56.810 --> 00:55:57.310 align:middle line:90% yeah. 00:55:57.310 --> 00:56:04.950 align:middle line:84% So 1 implies that the absolute value is going down, as you say. 00:56:04.950 --> 00:56:07.850 align:middle line:84% But I could alternate signs and things like that. 00:56:07.850 --> 00:56:08.990 align:middle line:90% Yeah. 00:56:08.990 --> 00:56:10.430 align:middle line:90% But it is-- yeah. 00:56:10.430 --> 00:56:13.370 align:middle line:84% And so just because you're comparing it with-- 00:56:13.370 --> 00:56:17.030 align:middle line:84% you're going to compare it with a convergent geometric series. 00:56:17.030 --> 00:56:19.510 align:middle line:84% In a convergent geometric series, 00:56:19.510 --> 00:56:25.850 align:middle line:84% the next element is some fixed fraction of the previous one. 00:56:25.850 --> 00:56:26.350 align:middle line:90% Yeah. 00:56:26.350 --> 00:56:29.030 align:middle line:90% 00:56:29.030 --> 00:56:33.430 align:middle line:84% So you're right that in order for the ratio test 00:56:33.430 --> 00:56:37.750 align:middle line:84% to give you convergence, then it's 00:56:37.750 --> 00:56:41.350 align:middle line:84% going to require that sufficiently far out, 00:56:41.350 --> 00:56:46.960 align:middle line:84% the absolute value of the a n element has gone down. 00:56:46.960 --> 00:56:57.920 align:middle line:90% 00:56:57.920 --> 00:57:01.800 align:middle line:84% And this, in a way, addresses that point you're making there. 00:57:01.800 --> 00:57:08.760 align:middle line:84% So the other test that is also based on comparing it 00:57:08.760 --> 00:57:11.440 align:middle line:90% with the geometric series. 00:57:11.440 --> 00:57:15.080 align:middle line:84% But this does not compare consecutive elements. 00:57:15.080 --> 00:57:19.720 align:middle line:84% So it's leaving the possibility that the next element could 00:57:19.720 --> 00:57:22.250 align:middle line:90% be larger than the previous. 00:57:22.250 --> 00:57:25.760 align:middle line:90% 00:57:25.760 --> 00:57:27.880 align:middle line:90% And so this is the root test. 00:57:27.880 --> 00:57:31.800 align:middle line:84% But of course, it cannot be that-- 00:57:31.800 --> 00:57:35.420 align:middle line:84% you cannot keep having the next element larger than 00:57:35.420 --> 00:57:35.980 align:middle line:90% the previous. 00:57:35.980 --> 00:57:39.480 align:middle line:84% But you could have-- there could be somewhere where 00:57:39.480 --> 00:57:41.760 align:middle line:90% the next one actually went up. 00:57:41.760 --> 00:57:44.580 align:middle line:84% But then presumably, after that, they're 00:57:44.580 --> 00:57:46.403 align:middle line:84% going down, at least for a long time. 00:57:46.403 --> 00:57:47.820 align:middle line:84% And then maybe there's another one 00:57:47.820 --> 00:57:50.920 align:middle line:84% that jumps up, but not very much, et cetera. 00:57:50.920 --> 00:57:57.380 align:middle line:84% And the root test is OK with that, whereas that test wasn't 00:57:57.380 --> 00:57:58.380 align:middle line:90% OK with that. 00:57:58.380 --> 00:58:00.420 align:middle line:90% So the root test. 00:58:00.420 --> 00:58:05.020 align:middle line:84% So you have a series, again, like this. 00:58:05.020 --> 00:58:08.780 align:middle line:84% And the root test is that you're looking 00:58:08.780 --> 00:58:13.860 align:middle line:84% at-- you're taking the absolute value of this 00:58:13.860 --> 00:58:16.700 align:middle line:84% and then you're taking the n-th root. 00:58:16.700 --> 00:58:20.540 align:middle line:84% So this means-- this is the same as saying a n 00:58:20.540 --> 00:58:26.180 align:middle line:84% absolute value, if you want to write it like this, 1 over n. 00:58:26.180 --> 00:58:31.340 align:middle line:90% So it's dealing with this. 00:58:31.340 --> 00:58:43.360 align:middle line:84% And the root test is saying that suppose that when you form this 00:58:43.360 --> 00:58:46.360 align:middle line:90% here, that this has a limit. 00:58:46.360 --> 00:58:47.970 align:middle line:90% Suppose that this n-th root-- 00:58:47.970 --> 00:58:50.800 align:middle line:90% 00:58:50.800 --> 00:58:53.640 align:middle line:84% suppose that this here has a limit. 00:58:53.640 --> 00:58:56.720 align:middle line:90% And let's call this limit c. 00:58:56.720 --> 00:59:00.080 align:middle line:90% Suppose that you have this. 00:59:00.080 --> 00:59:06.950 align:middle line:84% Then it has the same three parts here as in the previous. 00:59:06.950 --> 00:59:15.600 align:middle line:90% 00:59:15.600 --> 00:59:19.320 align:middle line:90% And so this is-- 00:59:19.320 --> 00:59:21.640 align:middle line:90% so there's three possibilities. 00:59:21.640 --> 00:59:25.720 align:middle line:84% Where c here is less than 1, then this 00:59:25.720 --> 00:59:27.860 align:middle line:84% implies that the series is convergent. 00:59:27.860 --> 00:59:34.240 align:middle line:90% 00:59:34.240 --> 00:59:41.420 align:middle line:84% Two, c is bigger than 1 implies that the series is divergent. 00:59:41.420 --> 00:59:46.330 align:middle line:90% 00:59:46.330 --> 00:59:52.530 align:middle line:84% And again, the case where c here is equal to 1, in conclusion. 00:59:52.530 --> 00:59:53.950 align:middle line:90% Test is in conclusion. 00:59:53.950 --> 01:00:01.690 align:middle line:90% 01:00:01.690 --> 01:00:04.330 align:middle line:84% And again, we're going to compare 01:00:04.330 --> 01:00:05.750 align:middle line:90% with the geometric series. 01:00:05.750 --> 01:00:07.160 align:middle line:84% So let's see how the proof works. 01:00:07.160 --> 01:00:20.990 align:middle line:90% 01:00:20.990 --> 01:00:21.490 align:middle line:90% OK. 01:00:21.490 --> 01:00:23.300 align:middle line:90% So now let's prove one. 01:00:23.300 --> 01:00:26.050 align:middle line:90% 01:00:26.050 --> 01:00:28.530 align:middle line:90% Proof of one. 01:00:28.530 --> 01:00:30.570 align:middle line:84% So again, we are in a situation where 01:00:30.570 --> 01:00:37.290 align:middle line:84% we have that the n-th root of the absolute value here 01:00:37.290 --> 01:00:43.070 align:middle line:84% as n goes to infinity is equal to some c, 01:00:43.070 --> 01:00:47.270 align:middle line:84% and we're assuming that the c is less than 1. 01:00:47.270 --> 01:00:53.390 align:middle line:84% So now I'm again choosing something 01:00:53.390 --> 01:00:57.490 align:middle line:90% that is between c and 1. 01:00:57.490 --> 01:01:08.140 align:middle line:84% So choose c-- let's call it c0 with a-- 01:01:08.140 --> 01:01:12.270 align:middle line:90% 01:01:12.270 --> 01:01:16.790 align:middle line:90% c0 less than 1. 01:01:16.790 --> 01:01:20.390 align:middle line:84% And now we know that-- because we have this limit, 01:01:20.390 --> 01:01:28.790 align:middle line:84% we know that there exists capital N such 01:01:28.790 --> 01:01:38.730 align:middle line:84% that if little n is bigger than capital N, 01:01:38.730 --> 01:01:43.330 align:middle line:84% then the n-th root of this absolute value sign 01:01:43.330 --> 01:01:47.930 align:middle line:84% is actually less than or equal to c0. 01:01:47.930 --> 01:01:53.650 align:middle line:84% This is just because c0 was strictly bigger than c. 01:01:53.650 --> 01:01:54.850 align:middle line:90% So we have that. 01:01:54.850 --> 01:01:58.790 align:middle line:84% So now we just check the n-th power on both sides. 01:01:58.790 --> 01:02:12.570 align:middle line:84% So this implies by taking the n-th power, 01:02:12.570 --> 01:02:22.450 align:middle line:84% we get that absolute value of a n is less or equal to c0 01:02:22.450 --> 01:02:23.410 align:middle line:90% to the power n. 01:02:23.410 --> 01:02:27.850 align:middle line:90% 01:02:27.850 --> 01:02:32.970 align:middle line:84% But that's already great, because this means that you have 01:02:32.970 --> 01:02:37.290 align:middle line:84% compared-- at least from a certain stage and an outward, 01:02:37.290 --> 01:02:40.320 align:middle line:90% you have compared the a n with-- 01:02:40.320 --> 01:02:46.220 align:middle line:84% if you're thinking that this thing here is like the b n, 01:02:46.220 --> 01:02:50.020 align:middle line:84% then you have compared these guys here 01:02:50.020 --> 01:02:53.900 align:middle line:84% with this series here based on these, at least 01:02:53.900 --> 01:02:57.060 align:middle line:84% from this capital N and outwards. 01:02:57.060 --> 01:03:00.300 align:middle line:84% And this is a geometric series because it's 01:03:00.300 --> 01:03:03.220 align:middle line:84% just some number strictly smaller than 1 01:03:03.220 --> 01:03:05.800 align:middle line:90% multiplied by itself n times. 01:03:05.800 --> 01:03:09.290 align:middle line:84% So again, by the first comparison test-- 01:03:09.290 --> 01:03:20.140 align:middle line:90% 01:03:20.140 --> 01:03:39.140 align:middle line:84% so by the first comparison test, the a n is convergent. 01:03:39.140 --> 01:03:42.760 align:middle line:84% And note here that what you're proving is-- 01:03:42.760 --> 01:03:47.680 align:middle line:84% you're actually proving that this series is convergent. 01:03:47.680 --> 01:03:50.280 align:middle line:84% Because you're comparing this series here 01:03:50.280 --> 01:03:52.200 align:middle line:90% with the geometric series. 01:03:52.200 --> 01:03:53.800 align:middle line:90% But this here is just-- 01:03:53.800 --> 01:04:01.520 align:middle line:84% this just means-- sorry, means that the original series here 01:04:01.520 --> 01:04:03.300 align:middle line:90% was absolutely convergent. 01:04:03.300 --> 01:04:09.440 align:middle line:90% 01:04:09.440 --> 01:04:13.960 align:middle line:84% But absolute convergence implies convergence. 01:04:13.960 --> 01:04:18.860 align:middle line:84% So you're actually proving absolute convergence. 01:04:18.860 --> 01:04:22.826 align:middle line:90% 01:04:22.826 --> 01:04:27.020 align:middle line:84% Now let's try to-- again, there's three parts to it. 01:04:27.020 --> 01:04:30.900 align:middle line:84% But the last one leaves open any possibility, 01:04:30.900 --> 01:04:32.460 align:middle line:90% so there's nothing to prove. 01:04:32.460 --> 01:04:35.900 align:middle line:84% So let's just prove the second part here. 01:04:35.900 --> 01:04:39.220 align:middle line:90% And that just goes similarly. 01:04:39.220 --> 01:04:43.900 align:middle line:84% And again, it fails dramatically to be convergent, 01:04:43.900 --> 01:04:47.260 align:middle line:84% because the a n, as you will see, 01:04:47.260 --> 01:04:49.820 align:middle line:90% does not actually goes to 0. 01:04:49.820 --> 01:04:52.160 align:middle line:90% So proof of two. 01:04:52.160 --> 01:04:56.220 align:middle line:90% 01:04:56.220 --> 01:04:58.820 align:middle line:90% So we have that this-- 01:04:58.820 --> 01:05:04.380 align:middle line:84% so we're assuming that this thing here, this 01:05:04.380 --> 01:05:11.180 align:middle line:84% should converge to some c here, which is now bigger than 1. 01:05:11.180 --> 01:05:13.460 align:middle line:84% So this means that there exists-- 01:05:13.460 --> 01:05:23.660 align:middle line:84% so this means that there exists capital N such that 01:05:23.660 --> 01:05:28.080 align:middle line:84% if little n is bigger than this capital N-- sorry. 01:05:28.080 --> 01:05:31.260 align:middle line:90% So we have this thing here. 01:05:31.260 --> 01:05:35.710 align:middle line:84% And what I should before I do this, I should choose my-- 01:05:35.710 --> 01:05:44.910 align:middle line:84% just like before, I should choose my c0 such 01:05:44.910 --> 01:05:53.330 align:middle line:84% that c0 is strictly bigger than 1, 01:05:53.330 --> 01:05:55.463 align:middle line:84% but yet is strictly smaller than c. 01:05:55.463 --> 01:05:58.790 align:middle line:90% 01:05:58.790 --> 01:06:00.050 align:middle line:90% So you choose this. 01:06:00.050 --> 01:06:03.390 align:middle line:84% And now using this, we have-- because we 01:06:03.390 --> 01:06:06.130 align:middle line:84% have that this limit here, it converts to c. 01:06:06.130 --> 01:06:09.790 align:middle line:84% So it converts to something that is strictly above this c0. 01:06:09.790 --> 01:06:16.950 align:middle line:84% This means that there exists capital N such 01:06:16.950 --> 01:06:22.630 align:middle line:84% that if little n is bigger than capital N, 01:06:22.630 --> 01:06:35.115 align:middle line:84% then the n-th root here is bigger than c0. 01:06:35.115 --> 01:06:38.370 align:middle line:90% 01:06:38.370 --> 01:06:41.770 align:middle line:84% But now, again, taking the n-th power on both sides, 01:06:41.770 --> 01:06:47.530 align:middle line:84% then you have that a n is bigger than-- 01:06:47.530 --> 01:06:53.410 align:middle line:84% strictly bigger than, in fact, c0 to the n-th power. 01:06:53.410 --> 01:06:56.710 align:middle line:90% But you see these guys here? 01:06:56.710 --> 01:06:58.950 align:middle line:90% c0 is strictly bigger than 1. 01:06:58.950 --> 01:07:02.470 align:middle line:84% So this here actually even goes to infinity. 01:07:02.470 --> 01:07:04.100 align:middle line:90% So this here does not go to 0. 01:07:04.100 --> 01:07:08.714 align:middle line:90% 01:07:08.714 --> 01:07:13.130 align:middle line:84% So this proves, again, that it's divergent. 01:07:13.130 --> 01:07:14.170 align:middle line:90% So this proved true. 01:07:14.170 --> 01:07:17.210 align:middle line:90% 01:07:17.210 --> 01:07:19.570 align:middle line:90% Now the next thing-- 01:07:19.570 --> 01:07:22.400 align:middle line:84% the next thing we want to talk about is power series. 01:07:22.400 --> 01:07:35.550 align:middle line:90% 01:07:35.550 --> 01:07:40.550 align:middle line:84% So remember that-- so the first example of a power series 01:07:40.550 --> 01:07:44.510 align:middle line:84% is something we already looked at, more or less. 01:07:44.510 --> 01:07:47.230 align:middle line:90% So suppose you take-- 01:07:47.230 --> 01:07:51.790 align:middle line:84% suppose that you're looking at the series, some number, 01:07:51.790 --> 01:07:57.590 align:middle line:84% I'll call it x now instead of c, to the power n, and n equal to 0 01:07:57.590 --> 01:07:59.390 align:middle line:90% to infinity. 01:07:59.390 --> 01:08:00.644 align:middle line:90% Yeah? 01:08:00.644 --> 01:08:03.488 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:08:03.488 --> 01:08:05.390 align:middle line:90% 01:08:05.390 --> 01:08:07.550 align:middle line:84% TOBIAS COLDING: So what we proved-- 01:08:07.550 --> 01:08:12.430 align:middle line:84% so in the root test here, we proved that the series 01:08:12.430 --> 01:08:15.490 align:middle line:90% is absolutely convergent. 01:08:15.490 --> 01:08:19.510 align:middle line:90% 01:08:19.510 --> 01:08:23.069 align:middle line:90% Here we proved it's divergent. 01:08:23.069 --> 01:08:27.229 align:middle line:84% And what we actually proved is we proved that a n-- 01:08:27.229 --> 01:08:33.040 align:middle line:84% in fact, this here a n that is here 01:08:33.040 --> 01:08:40.000 align:middle line:84% allows me to write that it goes to infinity like this. 01:08:40.000 --> 01:08:43.840 align:middle line:84% They were supposed to go to 0, but we actually write-- 01:08:43.840 --> 01:08:48.680 align:middle line:84% if the series was converging, the most elementary thing 01:08:48.680 --> 01:08:53.200 align:middle line:84% is, the most basic thing is that the a n has to go to 0. 01:08:53.200 --> 01:08:55.319 align:middle line:84% Otherwise, it's not going to converge. 01:08:55.319 --> 01:08:58.560 align:middle line:84% But in fact, not only do they not go to 0, 01:08:58.560 --> 01:09:01.580 align:middle line:84% they actually, in absolute value, go to infinity. 01:09:01.580 --> 01:09:04.479 align:middle line:84% So it's very, very far from being convergent. 01:09:04.479 --> 01:09:12.720 align:middle line:90% 01:09:12.720 --> 01:09:16.479 align:middle line:84% So suppose we looked at the geometric series. 01:09:16.479 --> 01:09:20.040 align:middle line:84% If you think about x just representing any number, 01:09:20.040 --> 01:09:21.819 align:middle line:84% you can look at this series here. 01:09:21.819 --> 01:09:25.240 align:middle line:90% This is the geometric series. 01:09:25.240 --> 01:09:27.520 align:middle line:84% And we know that it's convergent. 01:09:27.520 --> 01:09:29.840 align:middle line:84% So we know that this has convergence. 01:09:29.840 --> 01:09:34.180 align:middle line:90% 01:09:34.180 --> 01:09:39.200 align:middle line:84% It's convergent if the absolute value is less than 1. 01:09:39.200 --> 01:09:45.979 align:middle line:84% And in that case, the sum here is just this familiar function. 01:09:45.979 --> 01:09:54.560 align:middle line:84% But in general, you may have a sequence. 01:09:54.560 --> 01:10:04.460 align:middle line:84% So in general, if a n is a sequence, 01:10:04.460 --> 01:10:07.860 align:middle line:84% then you can form a power series. 01:10:07.860 --> 01:10:11.500 align:middle line:84% And the power series is like that, 01:10:11.500 --> 01:10:13.740 align:middle line:90% where this is the coefficient. 01:10:13.740 --> 01:10:19.980 align:middle line:84% The a n is now the coefficient to the x to the power n. 01:10:19.980 --> 01:10:26.120 align:middle line:84% So if you will, this is like an infinite polynomial, 01:10:26.120 --> 01:10:29.680 align:middle line:84% a polynomial of infinite degree because the sum 01:10:29.680 --> 01:10:33.500 align:middle line:90% of all these polynomials. 01:10:33.500 --> 01:10:34.940 align:middle line:84% And this is called a power series. 01:10:34.940 --> 01:10:37.047 align:middle line:90% So this is a power series. 01:10:37.047 --> 01:10:38.130 align:middle line:90% So this is a power series. 01:10:38.130 --> 01:10:41.600 align:middle line:90% 01:10:41.600 --> 01:10:44.960 align:middle line:90% And the question is then-- 01:10:44.960 --> 01:10:50.240 align:middle line:84% the question, which turned out to be quite subtle, is-- 01:10:50.240 --> 01:10:53.360 align:middle line:84% so a lot of the basic functions can 01:10:53.360 --> 01:10:56.000 align:middle line:90% be represented by power series. 01:10:56.000 --> 01:10:59.480 align:middle line:90% This here was one example. 01:10:59.480 --> 01:11:04.640 align:middle line:84% Another example is-- so another example of-- 01:11:04.640 --> 01:11:07.200 align:middle line:84% if you're looking at the exponential function-- and we'll 01:11:07.200 --> 01:11:09.320 align:middle line:90% come back and discuss this more. 01:11:09.320 --> 01:11:12.640 align:middle line:84% But the exponential function has a nice power series 01:11:12.640 --> 01:11:14.520 align:middle line:90% representation. 01:11:14.520 --> 01:11:17.860 align:middle line:84% And this is the exponential function, 01:11:17.860 --> 01:11:22.760 align:middle line:84% the power series representation of the exponential function. 01:11:22.760 --> 01:11:26.170 align:middle line:90% And also the sine and cosine. 01:11:26.170 --> 01:11:33.010 align:middle line:84% So cosine x, this also has a power series representation. 01:11:33.010 --> 01:11:38.210 align:middle line:84% And the power series is minus 1 to the power n x to the power 2n 01:11:38.210 --> 01:11:45.210 align:middle line:84% divided by 2n factorial, where you're summing from 0 01:11:45.210 --> 01:11:46.930 align:middle line:90% to infinity. 01:11:46.930 --> 01:11:50.170 align:middle line:90% And sine has a similar one. 01:11:50.170 --> 01:11:54.730 align:middle line:84% So sine x has a similar one where you're 01:11:54.730 --> 01:11:57.170 align:middle line:90% summing from 0 to infinity. 01:11:57.170 --> 01:12:05.810 align:middle line:84% And then it's minus 1 to power n x to the power 2n plus 1 over 2n 01:12:05.810 --> 01:12:08.270 align:middle line:90% plus 1 factorial. 01:12:08.270 --> 01:12:11.202 align:middle line:90% 01:12:11.202 --> 01:12:16.970 align:middle line:84% And lots of functions have a power series representation. 01:12:16.970 --> 01:12:19.470 align:middle line:90% But it's not every function. 01:12:19.470 --> 01:12:23.250 align:middle line:84% If I write down a function, then-- 01:12:23.250 --> 01:12:26.990 align:middle line:84% so in fact, it's easy to write a function that doesn't have. 01:12:26.990 --> 01:12:30.230 align:middle line:84% So if you have-- but I'm not going to prove to you that it 01:12:30.230 --> 01:12:30.870 align:middle line:90% doesn't. 01:12:30.870 --> 01:12:33.910 align:middle line:84% So if I take the function that starts off 01:12:33.910 --> 01:12:39.170 align:middle line:84% being dead zero and now it's becoming interesting. 01:12:39.170 --> 01:12:43.050 align:middle line:84% So this function here, this one does not have a power series. 01:12:43.050 --> 01:12:45.990 align:middle line:84% Even locally does not have a power series. 01:12:45.990 --> 01:12:51.190 align:middle line:84% And the thing is that a power series 01:12:51.190 --> 01:12:52.870 align:middle line:84% is an example of something that's 01:12:52.870 --> 01:12:55.030 align:middle line:90% called an analytic function. 01:12:55.030 --> 01:12:57.850 align:middle line:84% And this here is not an analytic function. 01:12:57.850 --> 01:13:00.070 align:middle line:84% So that's why I know that it definitely 01:13:00.070 --> 01:13:01.350 align:middle line:90% doesn't have a power series. 01:13:01.350 --> 01:13:01.850 align:middle line:90% But anyway. 01:13:01.850 --> 01:13:11.390 align:middle line:90% 01:13:11.390 --> 01:13:16.510 align:middle line:84% So the question is if you take a power series-- 01:13:16.510 --> 01:13:23.550 align:middle line:84% so suppose you take a sequence, a n, and you write it down. 01:13:23.550 --> 01:13:28.370 align:middle line:84% Suppose you take a sequence a n and then you write down 01:13:28.370 --> 01:13:32.730 align:middle line:90% a power series based on it. 01:13:32.730 --> 01:13:35.130 align:middle line:84% Then the first question that comes to mind 01:13:35.130 --> 01:13:41.770 align:middle line:84% is, when does this-- for which x does this series converge? 01:13:41.770 --> 01:13:51.140 align:middle line:84% So for which x does the series converge? 01:13:51.140 --> 01:13:57.810 align:middle line:90% 01:13:57.810 --> 01:14:01.250 align:middle line:84% So this is like the most basic thing. 01:14:01.250 --> 01:14:04.360 align:middle line:90% And let's look at some examples. 01:14:04.360 --> 01:14:13.370 align:middle line:90% 01:14:13.370 --> 01:14:21.250 align:middle line:84% Of course, if you look at the-- so 01:14:21.250 --> 01:14:27.840 align:middle line:84% the first example is what we already talked about. 01:14:27.840 --> 01:14:31.540 align:middle line:84% Suppose I'm looking at this thing here. 01:14:31.540 --> 01:14:33.900 align:middle line:90% Well, this here is x. 01:14:33.900 --> 01:14:35.960 align:middle line:84% If you think about x as being fixed, 01:14:35.960 --> 01:14:38.060 align:middle line:90% this is a geometric series. 01:14:38.060 --> 01:14:39.740 align:middle line:90% And we already know-- 01:14:39.740 --> 01:14:52.050 align:middle line:84% so this is for each fixed x, this is the geometric series. 01:14:52.050 --> 01:14:58.340 align:middle line:90% 01:14:58.340 --> 01:15:01.160 align:middle line:84% So we already know that this thing here converges. 01:15:01.160 --> 01:15:09.480 align:middle line:84% So converges as long as the absolute value sign-- 01:15:09.480 --> 01:15:12.300 align:middle line:84% absolute value of x is less than 1. 01:15:12.300 --> 01:15:21.320 align:middle line:84% And we also know that n diverges otherwise, because we've already 01:15:21.320 --> 01:15:22.260 align:middle line:90% determined that. 01:15:22.260 --> 01:15:26.560 align:middle line:90% 01:15:26.560 --> 01:15:33.440 align:middle line:84% Now you see-- the next example I want to look at is the one power 01:15:33.440 --> 01:15:36.520 align:middle line:90% series that represents-- 01:15:36.520 --> 01:15:38.180 align:middle line:90% but we haven't proven that yet. 01:15:38.180 --> 01:15:40.840 align:middle line:84% But we'll talk about that next time. 01:15:40.840 --> 01:15:42.820 align:middle line:90% It's the exponential function. 01:15:42.820 --> 01:15:48.880 align:middle line:84% If I'm looking at this power series here, then I have-- 01:15:48.880 --> 01:15:54.200 align:middle line:84% then I'm interested in for what x-- 01:15:54.200 --> 01:16:01.570 align:middle line:90% for what x does this converge? 01:16:01.570 --> 01:16:09.400 align:middle line:90% 01:16:09.400 --> 01:16:13.620 align:middle line:84% And so it's natural here to try to apply one of the tests. 01:16:13.620 --> 01:16:16.320 align:middle line:84% So you just think about x as being fixed 01:16:16.320 --> 01:16:19.100 align:middle line:84% and you just apply one of the tests. 01:16:19.100 --> 01:16:23.180 align:middle line:84% And the natural one is either the root test or the ratio test. 01:16:23.180 --> 01:16:24.550 align:middle line:90% Let's do the ratio test. 01:16:24.550 --> 01:16:27.180 align:middle line:90% 01:16:27.180 --> 01:16:32.080 align:middle line:84% So x is fixed, and I'm not interested in x equal to 0. 01:16:32.080 --> 01:16:36.840 align:middle line:84% I'm assuming that x is not equal to 0, because when x is 0, 01:16:36.840 --> 01:16:38.420 align:middle line:90% this here is just-- 01:16:38.420 --> 01:16:44.780 align:middle line:84% it's only for-- x to the power of 0 is by definition one. 01:16:44.780 --> 01:16:50.540 align:middle line:84% And all the other things, all the other terms, are 0. 01:16:50.540 --> 01:16:53.380 align:middle line:84% So this here is-- it's definitely converging 01:16:53.380 --> 01:16:55.360 align:middle line:90% when x is equal to 0. 01:16:55.360 --> 01:16:58.425 align:middle line:90% So let's not look at that. 01:16:58.425 --> 01:17:00.300 align:middle line:84% And then-- because if you don't look at that, 01:17:00.300 --> 01:17:02.900 align:middle line:84% you can look at-- you can think about a n. 01:17:02.900 --> 01:17:08.660 align:middle line:84% You can now think about, sorry, as the series 01:17:08.660 --> 01:17:14.780 align:middle line:84% where bn is x to the power n over n factorial. 01:17:14.780 --> 01:17:17.180 align:middle line:84% If you think about this thing here, 01:17:17.180 --> 01:17:19.950 align:middle line:84% and you think about the series as now bn, 01:17:19.950 --> 01:17:23.590 align:middle line:84% then the ratio test tells you that if you're 01:17:23.590 --> 01:17:29.280 align:middle line:84% looking at the ratio of bn plus 1 over bn-- 01:17:29.280 --> 01:17:35.270 align:middle line:90% 01:17:35.270 --> 01:17:40.510 align:middle line:84% so if you're looking at bn plus 1 over bn, 01:17:40.510 --> 01:17:43.070 align:middle line:84% you take the absolute value of this here. 01:17:43.070 --> 01:17:47.630 align:middle line:84% Well, this is just x to the power n 01:17:47.630 --> 01:17:52.650 align:middle line:90% plus 1 over n plus 1 factorial. 01:17:52.650 --> 01:17:57.190 align:middle line:84% And then you have to divide it by x to the power 01:17:57.190 --> 01:18:00.070 align:middle line:90% n over n factorial. 01:18:00.070 --> 01:18:05.630 align:middle line:84% But if you do that, this here flips upside down, 01:18:05.630 --> 01:18:10.510 align:middle line:84% and you're getting x to the power n plus 1 times 01:18:10.510 --> 01:18:12.430 align:middle line:90% n factorial. 01:18:12.430 --> 01:18:18.170 align:middle line:84% And then you're getting x to the n times n plus 1 factorial. 01:18:18.170 --> 01:18:21.330 align:middle line:90% So this is what it's equal to. 01:18:21.330 --> 01:18:24.410 align:middle line:84% And now, of course, here, there are 01:18:24.410 --> 01:18:27.930 align:middle line:90% x times itself n plus 1 times. 01:18:27.930 --> 01:18:31.750 align:middle line:84% The n of those is canceling out with this. 01:18:31.750 --> 01:18:34.890 align:middle line:90% So this is x. 01:18:34.890 --> 01:18:41.170 align:middle line:84% And here, you see here the n plus 1 factorial. 01:18:41.170 --> 01:18:45.890 align:middle line:84% You can write as n plus 1 and then times 01:18:45.890 --> 01:18:48.130 align:middle line:90% n times n minus 1 cetera. 01:18:48.130 --> 01:18:50.610 align:middle line:90% But that's just n factorial. 01:18:50.610 --> 01:18:52.250 align:middle line:90% So you can write it like that. 01:18:52.250 --> 01:18:56.050 align:middle line:84% And so you see that this here cancels that. 01:18:56.050 --> 01:18:59.250 align:middle line:90% So you're left with n plus 1. 01:18:59.250 --> 01:19:06.130 align:middle line:84% Remember, if x is fixed as n goes to infinity, this here-- 01:19:06.130 --> 01:19:10.390 align:middle line:84% for each fixed x, this just goes to 0. 01:19:10.390 --> 01:19:15.430 align:middle line:84% So now you have, from the ratio test, if that limit-- 01:19:15.430 --> 01:19:18.270 align:middle line:84% if it has a limit-- it does have a limit. 01:19:18.270 --> 01:19:23.890 align:middle line:84% If that limit is strictly less than 1, but it's actually 0. 01:19:23.890 --> 01:19:26.530 align:middle line:84% So it's definitely fallen under 1. 01:19:26.530 --> 01:19:28.670 align:middle line:90% So it's convergent. 01:19:28.670 --> 01:19:32.990 align:middle line:90% So the ratio test tells you-- 01:19:32.990 --> 01:19:39.890 align:middle line:84% so by the ratio test, this is convergent. 01:19:39.890 --> 01:19:46.694 align:middle line:90% 01:19:46.694 --> 01:19:49.350 align:middle line:90% And so we'll finish here. 01:19:49.350 --> 01:19:50.550 align:middle line:90% There's a little bit more. 01:19:50.550 --> 01:19:54.110 align:middle line:84% I posted already my lecture notes. 01:19:54.110 --> 01:19:56.270 align:middle line:90% There's a little bit more. 01:19:56.270 --> 01:19:59.750 align:middle line:84% And I'll probably talk about that next time. 01:19:59.750 --> 01:20:02.070 align:middle line:84% I'll just start with just talking about the stuff I 01:20:02.070 --> 01:20:02.910 align:middle line:90% didn't get to. 01:20:02.910 --> 01:20:05.390 align:middle line:90% The stuff I didn't get to was-- 01:20:05.390 --> 01:20:08.190 align:middle line:84% and we touched upon that earlier-- 01:20:08.190 --> 01:20:10.534 align:middle line:90% is lim sup and lim inf. 01:20:10.534 --> 01:20:13.150 align:middle line:90% 01:20:13.150 --> 01:20:16.760 align:middle line:84% And the reason why we're covering it here 01:20:16.760 --> 01:20:19.840 align:middle line:90% is that what if the ratio-- 01:20:19.840 --> 01:20:22.800 align:middle line:84% what if these things-- what if the n-th roots-- so you're 01:20:22.800 --> 01:20:26.240 align:middle line:84% asking for either the ratio or the n-th root 01:20:26.240 --> 01:20:28.760 align:middle line:90% of something to have a limit. 01:20:28.760 --> 01:20:31.520 align:middle line:84% But what if it doesn't have a limit? 01:20:31.520 --> 01:20:33.620 align:middle line:84% And so it's dealing with that case. 01:20:33.620 --> 01:20:35.560 align:middle line:90% OK. 01:20:35.560 --> 01:20:36.320 align:middle line:90% All right. 01:20:36.320 --> 01:20:37.520 align:middle line:90% Any questions? 01:20:37.520 --> 01:20:39.793 align:middle line:84% AUDIENCE: That means for all x, right, you're saying? 01:20:39.793 --> 01:20:41.210 align:middle line:84% TOBIAS COLDING: This is for all x. 01:20:41.210 --> 01:20:44.240 align:middle line:84% But think about x as being fixed. 01:20:44.240 --> 01:20:46.320 align:middle line:90% Think about x. 01:20:46.320 --> 01:20:48.860 align:middle line:84% You're just asking for one particular x, 01:20:48.860 --> 01:20:50.300 align:middle line:90% but it holds for all x. 01:20:50.300 --> 01:20:54.400 align:middle line:84% But don't think about x also as being variable right. 01:20:54.400 --> 01:20:58.240 align:middle line:84% Because you think about x as being fixed, and then you let n 01:20:58.240 --> 01:21:00.033 align:middle line:90% go to infinity. 01:21:00.033 --> 01:21:00.700 align:middle line:90% AUDIENCE: Right. 01:21:00.700 --> 01:21:01.960 align:middle line:90% And then after, you generalize. 01:21:01.960 --> 01:21:02.480 align:middle line:90% TOBIAS COLDING: Yeah, yeah. 01:21:02.480 --> 01:21:03.022 align:middle line:90% That's right. 01:21:03.022 --> 01:21:07.280 align:middle line:84% And so you have for all x, this makes sense. 01:21:07.280 --> 01:21:09.460 align:middle line:84% Then you can ask, how does it vary in x? 01:21:09.460 --> 01:21:11.710 align:middle line:90% But that's a different story. 01:21:11.710 --> 01:21:14.000 align:middle line:90%