WEBVTT 00:00:00.000 --> 00:00:01.948 align:middle line:90% [SQUEAKING] 00:00:01.948 --> 00:00:04.383 align:middle line:90% [RUSTLING] 00:00:04.383 --> 00:00:07.305 align:middle line:90% [CLICKING] 00:00:07.305 --> 00:00:10.720 align:middle line:90% 00:00:10.720 --> 00:00:17.830 align:middle line:84% TOBIAS COLDING: OK, so last time we talked about, 00:00:17.830 --> 00:00:19.580 align:middle line:84% in particular, we talked about two things. 00:00:19.580 --> 00:00:24.580 align:middle line:84% We talked about the Cauchy Convergence Theorem. 00:00:24.580 --> 00:00:30.840 align:middle line:90% 00:00:30.840 --> 00:00:32.840 align:middle line:84% So the Cauchy Convergence Theorem 00:00:32.840 --> 00:00:47.220 align:middle line:84% says that any Cauchy sequence is convergent. 00:00:47.220 --> 00:00:51.280 align:middle line:90% 00:00:51.280 --> 00:00:56.200 align:middle line:84% So this is one thing we talked quite a bit about last time. 00:00:56.200 --> 00:00:57.600 align:middle line:90% And we proved it. 00:00:57.600 --> 00:01:00.300 align:middle line:84% But we proved it assuming another theorem, 00:01:00.300 --> 00:01:12.420 align:middle line:84% so we showed it, the proof used the Bolzano-Weierstrass Theorem. 00:01:12.420 --> 00:01:21.040 align:middle line:90% 00:01:21.040 --> 00:01:26.160 align:middle line:84% Now, the Bolzano-Weierstrass Theorem-- 00:01:26.160 --> 00:01:27.900 align:middle line:90% so that's the following-- 00:01:27.900 --> 00:01:39.600 align:middle line:90% 00:01:39.600 --> 00:01:42.880 align:middle line:84% so this is the following theorem. 00:01:42.880 --> 00:01:44.900 align:middle line:90% Let's say that any bounded-- 00:01:44.900 --> 00:01:49.600 align:middle line:90% 00:01:49.600 --> 00:02:01.460 align:middle line:84% sorry, any bounded sequence, an, has a convergent subsequence. 00:02:01.460 --> 00:02:12.920 align:middle line:90% 00:02:12.920 --> 00:02:15.320 align:middle line:90% And let's call it ank. 00:02:15.320 --> 00:02:18.080 align:middle line:84% That was the Bolzano-Weierstrass Theorem. 00:02:18.080 --> 00:02:21.620 align:middle line:84% So using this-- remember, the Cauchy Convergence Theorem, 00:02:21.620 --> 00:02:25.080 align:middle line:84% the proof then went like this, that if you take a Cauchy 00:02:25.080 --> 00:02:29.560 align:middle line:84% sequence, then it was easy to prove that it was bounded, 00:02:29.560 --> 00:02:33.320 align:middle line:84% then you use the Bolzano-Weierstrass Theorem 00:02:33.320 --> 00:02:36.500 align:middle line:84% to say that now the sequence is bounded-- 00:02:36.500 --> 00:02:39.000 align:middle line:84% it's a Cauchy sequence-- the sequence is bounded, 00:02:39.000 --> 00:02:44.000 align:middle line:84% so it has a convergent subsequence, and the limit-- 00:02:44.000 --> 00:02:47.320 align:middle line:84% you would then want to prove that the limit is actually 00:02:47.320 --> 00:02:48.920 align:middle line:90% a limit for the whole sequence. 00:02:48.920 --> 00:02:52.640 align:middle line:90% That's what we did, right? 00:02:52.640 --> 00:02:55.540 align:middle line:84% But we didn't prove the Bolzano-Weierstrass Theorem. 00:02:55.540 --> 00:03:01.760 align:middle line:84% So let's try to prove it now, so proof 00:03:01.760 --> 00:03:03.826 align:middle line:90% of the Bolzano-Weierstrass. 00:03:03.826 --> 00:03:13.560 align:middle line:90% 00:03:13.560 --> 00:03:17.520 align:middle line:84% So let me try to explain the idea. 00:03:17.520 --> 00:03:24.240 align:middle line:84% So you have a sequence, a n, and a n is bounded. 00:03:24.240 --> 00:03:26.320 align:middle line:84% Let's just say it doesn't matter. 00:03:26.320 --> 00:03:31.480 align:middle line:84% But let's say that a n lie in the interval from 0 to 1. 00:03:31.480 --> 00:03:34.320 align:middle line:84% It could be any other bounded interval. 00:03:34.320 --> 00:03:38.520 align:middle line:90% But let's just say 0 to 1. 00:03:38.520 --> 00:03:44.840 align:middle line:84% Then the idea is that you want to prove that an-- 00:03:44.840 --> 00:04:04.570 align:middle line:84% so the idea is that we want to find a subsequence, ank, 00:04:04.570 --> 00:04:22.490 align:middle line:84% and two other sequences, a-- sorry, bk and ck. 00:04:22.490 --> 00:04:24.410 align:middle line:84% And these two other sequences have 00:04:24.410 --> 00:04:31.450 align:middle line:84% the property that if you have here ank, then 00:04:31.450 --> 00:04:35.350 align:middle line:90% the b's lie below-- 00:04:35.350 --> 00:04:38.050 align:middle line:90% they lie below the a and k. 00:04:38.050 --> 00:04:42.150 align:middle line:90% And the c's lies above it. 00:04:42.150 --> 00:04:44.730 align:middle line:90% 00:04:44.730 --> 00:04:50.370 align:middle line:84% So the sequence, the subsequence that we will find 00:04:50.370 --> 00:04:55.750 align:middle line:84% is going to be squeezed between two other sequences. 00:04:55.750 --> 00:05:06.910 align:middle line:84% So a and k is squeezed between two other sequences. 00:05:06.910 --> 00:05:12.850 align:middle line:90% 00:05:12.850 --> 00:05:15.450 align:middle line:84% And the thing about the other sequences 00:05:15.450 --> 00:05:19.270 align:middle line:90% is that they are much nicer. 00:05:19.270 --> 00:05:21.370 align:middle line:90% So why are they much nicer? 00:05:21.370 --> 00:05:28.130 align:middle line:84% The bk's are monotone increasing. 00:05:28.130 --> 00:05:33.310 align:middle line:84% And the ck are monotone decreasing. 00:05:33.310 --> 00:05:35.890 align:middle line:90% 00:05:35.890 --> 00:05:40.270 align:middle line:90% So you have that. 00:05:40.270 --> 00:05:47.090 align:middle line:84% So the picture is that the b's are always 00:05:47.090 --> 00:05:48.970 align:middle line:90% moving in this direction. 00:05:48.970 --> 00:05:50.310 align:middle line:90% The c's here. 00:05:50.310 --> 00:05:51.930 align:middle line:90% This is the b's. 00:05:51.930 --> 00:05:57.410 align:middle line:84% As k gets larger, they're moving to the right. 00:05:57.410 --> 00:06:00.810 align:middle line:84% And the ck's are moving to the left. 00:06:00.810 --> 00:06:06.850 align:middle line:84% And these guys, a and k, they will 00:06:06.850 --> 00:06:08.770 align:middle line:90% be squeezed between these two. 00:06:08.770 --> 00:06:15.170 align:middle line:84% So if you're looking at a later one, so this is like one. 00:06:15.170 --> 00:06:22.290 align:middle line:84% And now imagine that you take a later in the subsequence-- 00:06:22.290 --> 00:06:26.530 align:middle line:84% sorry, let me indicate it here, an, say, 00:06:26.530 --> 00:06:33.130 align:middle line:84% l, and l is now further out in the subsequence-- 00:06:33.130 --> 00:06:42.330 align:middle line:84% then this is going to be lying between bl and cl. 00:06:42.330 --> 00:06:48.410 align:middle line:84% But these here, these here will lie-- the bl, because it's 00:06:48.410 --> 00:06:50.370 align:middle line:84% further out in subsequence, it will 00:06:50.370 --> 00:06:54.210 align:middle line:84% lie to the right of the-- so here will be bk. 00:06:54.210 --> 00:06:57.090 align:middle line:84% It would be bk would be to the left, 00:06:57.090 --> 00:07:01.850 align:middle line:84% and ck would be to the right of this, right? 00:07:01.850 --> 00:07:05.030 align:middle line:84% So you see that these here are coming together. 00:07:05.030 --> 00:07:07.610 align:middle line:84% They're coming together like this. 00:07:07.610 --> 00:07:12.530 align:middle line:84% And the a, the subsequence is going 00:07:12.530 --> 00:07:15.850 align:middle line:84% to be squeezed in smaller and smaller interval. 00:07:15.850 --> 00:07:18.730 align:middle line:84% And that's going to force it to converge. 00:07:18.730 --> 00:07:24.810 align:middle line:84% And the point here is that because this here is increasing, 00:07:24.810 --> 00:07:28.410 align:middle line:84% and they always lie between the ones that go below, 00:07:28.410 --> 00:07:32.450 align:middle line:84% this automatically means that the bk's will always 00:07:32.450 --> 00:07:37.530 align:middle line:90% be less than the first c. 00:07:37.530 --> 00:07:46.970 align:middle line:84% And these ck's will always be bigger than the first b. 00:07:46.970 --> 00:07:51.730 align:middle line:84% And so this is a monotone sequence that go up, 00:07:51.730 --> 00:07:54.650 align:middle line:84% but it's bounded, so it has a limit. 00:07:54.650 --> 00:07:59.770 align:middle line:84% So bk is converging to something. 00:07:59.770 --> 00:08:03.010 align:middle line:84% The ck's are going down, but they're 00:08:03.010 --> 00:08:05.090 align:middle line:90% uniformly bounded from below. 00:08:05.090 --> 00:08:09.490 align:middle line:84% So they will also be converging, right? 00:08:09.490 --> 00:08:12.210 align:middle line:84% But now they're coming closer together. 00:08:12.210 --> 00:08:14.310 align:middle line:84% They're coming closer and closer together. 00:08:14.310 --> 00:08:15.590 align:middle line:90% This is converging to that. 00:08:15.590 --> 00:08:17.210 align:middle line:90% And this is converging to this. 00:08:17.210 --> 00:08:22.630 align:middle line:84% So that's going to imply that b is equal to c. 00:08:22.630 --> 00:08:23.390 align:middle line:90% That's the idea. 00:08:23.390 --> 00:08:24.530 align:middle line:90% Yes, sir? 00:08:24.530 --> 00:08:26.750 align:middle line:84% AUDIENCE: Do they need to be strictly monotonic? 00:08:26.750 --> 00:08:34.650 align:middle line:84% TOBIAS COLDING: They will have-- sorry, 00:08:34.650 --> 00:08:40.809 align:middle line:84% the bk will be less or equal to bk plus 1. 00:08:40.809 --> 00:08:51.250 align:middle line:84% And the ck will be bigger or equal to ck plus 1. 00:08:51.250 --> 00:08:52.430 align:middle line:90% So it's not strict. 00:08:52.430 --> 00:08:54.870 align:middle line:90% It's inequality, like this. 00:08:54.870 --> 00:08:57.510 align:middle line:84% AUDIENCE: [INAUDIBLE] you had a constant [INAUDIBLE] constant? 00:08:57.510 --> 00:08:58.927 align:middle line:84% TOBIAS COLDING: Yeah that's right. 00:08:58.927 --> 00:09:02.730 align:middle line:84% But the thing is also that they are going to, 00:09:02.730 --> 00:09:04.250 align:middle line:90% that the difference-- 00:09:04.250 --> 00:09:10.610 align:middle line:90% so the ck is lying above the bk. 00:09:10.610 --> 00:09:12.370 align:middle line:90% So this here is positive. 00:09:12.370 --> 00:09:17.990 align:middle line:84% And we will prove that this here is bounded by 2. 00:09:17.990 --> 00:09:21.170 align:middle line:84% It was actually going to be equal to. 00:09:21.170 --> 00:09:27.530 align:middle line:84% We would actually prove that this here is equal to 1 minus, 00:09:27.530 --> 00:09:28.730 align:middle line:90% like this. 00:09:28.730 --> 00:09:31.010 align:middle line:90% And you see this here goes to 0. 00:09:31.010 --> 00:09:35.250 align:middle line:84% So you're right that if they didn't come together, 00:09:35.250 --> 00:09:37.310 align:middle line:90% we couldn't conclude things. 00:09:37.310 --> 00:09:38.670 align:middle line:90% But they will come together. 00:09:38.670 --> 00:09:40.050 align:middle line:90% This I will guarantee. 00:09:40.050 --> 00:09:41.350 align:middle line:90% So how do we do this? 00:09:41.350 --> 00:09:42.070 align:middle line:90% That's the idea. 00:09:42.070 --> 00:09:45.530 align:middle line:90% How do we do this? 00:09:45.530 --> 00:09:51.840 align:middle line:84% Well, and again, I'm just doing 0 to 1. 00:09:51.840 --> 00:09:55.030 align:middle line:84% But the same argument works for any other interval. 00:09:55.030 --> 00:09:58.840 align:middle line:90% 00:09:58.840 --> 00:10:02.420 align:middle line:84% It will look simpler if I do it from 0 to 1, 00:10:02.420 --> 00:10:06.780 align:middle line:84% but it's not-- there's nothing essential I'm using about it. 00:10:06.780 --> 00:10:13.380 align:middle line:84% So here is your starting interval from 0 to 1. 00:10:13.380 --> 00:10:16.060 align:middle line:84% We want to construct a subsequence. 00:10:16.060 --> 00:10:20.500 align:middle line:84% And not only do we want to construct a subsequence, 00:10:20.500 --> 00:10:24.940 align:middle line:84% but we also want to construct the sequence bk and the sequence 00:10:24.940 --> 00:10:27.180 align:middle line:90% ck, right? 00:10:27.180 --> 00:10:34.300 align:middle line:90% So let me define b1. 00:10:34.300 --> 00:10:37.540 align:middle line:90% b1 is just going to be 0. 00:10:37.540 --> 00:10:41.780 align:middle line:90% c1 is going to be 1. 00:10:41.780 --> 00:10:49.620 align:middle line:84% And then an1-- that's the subsequence, the first element 00:10:49.620 --> 00:10:50.780 align:middle line:90% is subsequence. 00:10:50.780 --> 00:10:54.260 align:middle line:84% I just take it to be the first element in sequence. 00:10:54.260 --> 00:10:58.340 align:middle line:84% So now of course, we have trivially 00:10:58.340 --> 00:11:02.900 align:middle line:84% that we have that b1, which is 0, 00:11:02.900 --> 00:11:05.460 align:middle line:84% this is less than or equal to a1, 00:11:05.460 --> 00:11:09.900 align:middle line:84% because it lie in this interval, less than or equal to c1, 00:11:09.900 --> 00:11:11.060 align:middle line:90% which is 1. 00:11:11.060 --> 00:11:18.360 align:middle line:84% Now, then you divide it into two intervals here. 00:11:18.360 --> 00:11:27.500 align:middle line:84% So the second step is that you take the interval from 0 to 1. 00:11:27.500 --> 00:11:31.500 align:middle line:90% You divide it into two halves. 00:11:31.500 --> 00:11:35.860 align:middle line:84% So you have between 0 and 1/2 and 1/2 and 1. 00:11:35.860 --> 00:11:39.900 align:middle line:84% The whole sequence lie in this interval. 00:11:39.900 --> 00:11:41.700 align:middle line:84% So this means that there are either 00:11:41.700 --> 00:11:46.380 align:middle line:84% infinitely many here or infinitely many there, right? 00:11:46.380 --> 00:11:51.980 align:middle line:84% Let's say that infinitely many of the a n lie in this interval. 00:11:51.980 --> 00:11:56.920 align:middle line:90% So then we're going to forward. 00:11:56.920 --> 00:11:59.820 align:middle line:84% We were going to just focus on this interval 00:11:59.820 --> 00:12:01.780 align:middle line:90% and on subintervals of this. 00:12:01.780 --> 00:12:08.100 align:middle line:90% So we have defined a1. 00:12:08.100 --> 00:12:09.860 align:middle line:90% Now we want to define-- 00:12:09.860 --> 00:12:12.480 align:middle line:90% and b1 and c1. 00:12:12.480 --> 00:12:29.140 align:middle line:84% So now I can assume that infinitely many a n's lie 00:12:29.140 --> 00:12:31.380 align:middle line:90% in this interval, right? 00:12:31.380 --> 00:12:36.620 align:middle line:90% So then we take an2-- 00:12:36.620 --> 00:12:39.580 align:middle line:84% the second element in a subsequence, 00:12:39.580 --> 00:12:43.660 align:middle line:84% we let it be the next element after a1 that 00:12:43.660 --> 00:12:45.940 align:middle line:90% is in this smaller interval. 00:12:45.940 --> 00:13:04.340 align:middle line:84% So this here is the second element in the original sequence 00:13:04.340 --> 00:13:14.740 align:middle line:84% that lies in the interval from 0 to 1/2. 00:13:14.740 --> 00:13:16.340 align:middle line:90% OK? 00:13:16.340 --> 00:13:18.400 align:middle line:90% That's an2, right? 00:13:18.400 --> 00:13:20.480 align:middle line:90% We don't want to have a1 again. 00:13:20.480 --> 00:13:23.030 align:middle line:84% We need another element, because otherwise it's 00:13:23.030 --> 00:13:24.280 align:middle line:90% not going to be a subsequence. 00:13:24.280 --> 00:13:27.222 align:middle line:90% You can't pick the A1. 00:13:27.222 --> 00:13:28.680 align:middle line:84% I mean, it could be the same value. 00:13:28.680 --> 00:13:32.580 align:middle line:84% But it will have to come from a different subscript. 00:13:32.580 --> 00:13:38.900 align:middle line:84% Now, b2 is then just going to be the left endpoint 00:13:38.900 --> 00:13:40.460 align:middle line:90% of this interval. 00:13:40.460 --> 00:13:42.460 align:middle line:90% So this is again 0. 00:13:42.460 --> 00:13:47.620 align:middle line:84% And c2 is going to be the right endpoint of this interval we're 00:13:47.620 --> 00:13:48.940 align:middle line:90% now focusing on. 00:13:48.940 --> 00:13:49.640 align:middle line:90% So that's 1/2. 00:13:49.640 --> 00:13:53.140 align:middle line:90% 00:13:53.140 --> 00:13:58.000 align:middle line:90% Now, so now to the next-- 00:13:58.000 --> 00:14:05.420 align:middle line:84% so now, how do we construct the third element in the subsequence 00:14:05.420 --> 00:14:11.620 align:middle line:84% and the corresponding b's and c's? 00:14:11.620 --> 00:14:15.820 align:middle line:84% The thing is that-- and here you have to be a little bit careful, 00:14:15.820 --> 00:14:21.820 align:middle line:84% because now, we assumed that there was infinitely 00:14:21.820 --> 00:14:24.460 align:middle line:84% many elements in the sequence that 00:14:24.460 --> 00:14:26.980 align:middle line:90% lie in this smaller interval. 00:14:26.980 --> 00:14:29.520 align:middle line:84% You are going to only-- from here on out, 00:14:29.520 --> 00:14:33.540 align:middle line:84% you're going to only look at those infinitely many. 00:14:33.540 --> 00:14:35.900 align:middle line:84% You're not going to care about, you know, 00:14:35.900 --> 00:14:38.160 align:middle line:84% so this may be the fifth, the seventh, 00:14:38.160 --> 00:14:40.840 align:middle line:84% the 15th element in the original sequence. 00:14:40.840 --> 00:14:44.220 align:middle line:84% You're not going to worry about what lies here. 00:14:44.220 --> 00:14:48.980 align:middle line:84% And so now you divide this one here into two halves. 00:14:48.980 --> 00:14:54.860 align:middle line:84% This interval between 0 and 1/2, you divide it into halves. 00:14:54.860 --> 00:14:58.960 align:middle line:84% And so now, imagine you have then the picture here. 00:14:58.960 --> 00:15:00.460 align:middle line:90% Here you have 0. 00:15:00.460 --> 00:15:04.900 align:middle line:90% Here you have a 1/2. 00:15:04.900 --> 00:15:08.880 align:middle line:84% And then you divide it into this integral from 0 to 1/2. 00:15:08.880 --> 00:15:14.980 align:middle line:84% You divide it into two equally large intervals. 00:15:14.980 --> 00:15:22.740 align:middle line:84% So now, we're just focusing on the infinitely many 00:15:22.740 --> 00:15:25.980 align:middle line:84% of the original sequences that lie here. 00:15:25.980 --> 00:15:29.500 align:middle line:84% But of those infinitely many, either infinitely many 00:15:29.500 --> 00:15:33.380 align:middle line:84% lie here, or infinitely many lie there. 00:15:33.380 --> 00:15:37.160 align:middle line:84% Let's say that there's only finitely many here, say, 00:15:37.160 --> 00:15:40.310 align:middle line:84% but there are infinitely many here. 00:15:40.310 --> 00:15:43.270 align:middle line:84% So if there's infinitely many in this interval 00:15:43.270 --> 00:15:46.990 align:middle line:84% here, then from here on out, you would only 00:15:46.990 --> 00:15:49.230 align:middle line:90% focus on those infinitely many. 00:15:49.230 --> 00:15:52.990 align:middle line:90% And the next element would be-- 00:15:52.990 --> 00:15:56.390 align:middle line:84% we know that by the next element would 00:15:56.390 --> 00:16:00.630 align:middle line:84% be the next element after the one 00:16:00.630 --> 00:16:05.790 align:middle line:84% you pick to be the second element in the subsequence that 00:16:05.790 --> 00:16:09.870 align:middle line:90% would lie in this subsequence. 00:16:09.870 --> 00:16:15.670 align:middle line:84% So we define a3 is now the next element. 00:16:15.670 --> 00:16:25.630 align:middle line:84% So an3, this is now this element that lies in this interval here. 00:16:25.630 --> 00:16:33.830 align:middle line:84% And then b3 is then going to be 1 over 4. 00:16:33.830 --> 00:16:41.430 align:middle line:84% And b-- sorry, and c3 is going to be the right endpoint of this 00:16:41.430 --> 00:16:44.310 align:middle line:90% interval, 1/4-- 00:16:44.310 --> 00:16:46.550 align:middle line:90% 1/2, sorry. 00:16:46.550 --> 00:16:52.110 align:middle line:90% And then you continue. 00:16:52.110 --> 00:16:54.050 align:middle line:90% Then you continue this way. 00:16:54.050 --> 00:16:56.650 align:middle line:84% So let me just make sure that it's totally clear. 00:16:56.650 --> 00:17:00.310 align:middle line:84% Let me just go over how we did it, 00:17:00.310 --> 00:17:07.310 align:middle line:84% so just because I want to be careful that once you-- so we 00:17:07.310 --> 00:17:10.030 align:middle line:84% started with the interval from 0 to 1. 00:17:10.030 --> 00:17:13.190 align:middle line:84% The first element you just pick in the subsequence, 00:17:13.190 --> 00:17:16.710 align:middle line:84% that was just the first element in the original sequence. 00:17:16.710 --> 00:17:22.109 align:middle line:84% And the b1 was just the left endpoint here, so that was 0. 00:17:22.109 --> 00:17:23.930 align:middle line:90% c1 was the right endpoint. 00:17:23.930 --> 00:17:26.670 align:middle line:90% That was 1. 00:17:26.670 --> 00:17:29.270 align:middle line:90% Now you divide it into 2. 00:17:29.270 --> 00:17:31.590 align:middle line:90% And one of the two halves-- 00:17:31.590 --> 00:17:33.790 align:middle line:84% potentially, of course, both of the two halves 00:17:33.790 --> 00:17:35.910 align:middle line:84% could have infinitely many elements 00:17:35.910 --> 00:17:37.910 align:middle line:90% in the original sequence. 00:17:37.910 --> 00:17:41.390 align:middle line:84% You just focus on one of the two halves 00:17:41.390 --> 00:17:43.710 align:middle line:90% that have infinite elements. 00:17:43.710 --> 00:17:47.950 align:middle line:84% If they're both, fine, but you just take one of them. 00:17:47.950 --> 00:17:50.310 align:middle line:84% And so from here on out, we're just 00:17:50.310 --> 00:17:54.070 align:middle line:84% going to focus on those infinitely many. 00:17:54.070 --> 00:17:57.510 align:middle line:84% And then you're going to pick the next one 00:17:57.510 --> 00:18:03.650 align:middle line:84% after the a1 that lies in that interval you're focusing on. 00:18:03.650 --> 00:18:06.630 align:middle line:84% That's the second element in the subsequence. 00:18:06.630 --> 00:18:11.230 align:middle line:84% And the b's will always be the left endpoint of the interval. 00:18:11.230 --> 00:18:15.030 align:middle line:84% The c's will be the right endpoint of the interval. 00:18:15.030 --> 00:18:17.830 align:middle line:84% Then when you do the third element, 00:18:17.830 --> 00:18:20.630 align:middle line:84% then you have to remember that you were 00:18:20.630 --> 00:18:22.950 align:middle line:90% looking at infinitely many. 00:18:22.950 --> 00:18:25.350 align:middle line:84% And so now it's like you're looking at-- you 00:18:25.350 --> 00:18:28.630 align:middle line:90% divide this interval into two. 00:18:28.630 --> 00:18:32.070 align:middle line:84% And now either this one here or that one 00:18:32.070 --> 00:18:35.190 align:middle line:90% here has infinitely many. 00:18:35.190 --> 00:18:37.090 align:middle line:90% And you pick whatever it is. 00:18:37.090 --> 00:18:40.590 align:middle line:84% I mean, I guess we assume it was this interval. 00:18:40.590 --> 00:18:44.950 align:middle line:84% And then the next element after the two first you 00:18:44.950 --> 00:18:48.190 align:middle line:84% picked in the original sequence, the next element that 00:18:48.190 --> 00:18:51.110 align:middle line:84% lies in this, that's going to be your third element 00:18:51.110 --> 00:18:52.630 align:middle line:90% of the sequence. 00:18:52.630 --> 00:18:56.470 align:middle line:84% And then still, the b's will be then 00:18:56.470 --> 00:18:58.030 align:middle line:84% of this interval you're focusing on 00:18:58.030 --> 00:19:01.790 align:middle line:84% would be the left endpoint and the c, the right endpoint. 00:19:01.790 --> 00:19:06.430 align:middle line:84% If you do that, you see each time you're halving it. 00:19:06.430 --> 00:19:08.650 align:middle line:84% So it means-- well, first of all, 00:19:08.650 --> 00:19:13.870 align:middle line:84% it means that the b's are going to be increasing, 00:19:13.870 --> 00:19:20.430 align:middle line:84% because it's always by each time you divide into a half, 00:19:20.430 --> 00:19:21.590 align:middle line:90% and it's always-- 00:19:21.590 --> 00:19:24.590 align:middle line:84% but the interval you focus on previously 00:19:24.590 --> 00:19:26.710 align:middle line:90% you divide into half, like this. 00:19:26.710 --> 00:19:30.850 align:middle line:84% And then either you pick either this point or that point. 00:19:30.850 --> 00:19:33.510 align:middle line:90% And this was the previous one. 00:19:33.510 --> 00:19:36.470 align:middle line:84% So the b's will always be increasing. 00:19:36.470 --> 00:19:41.430 align:middle line:84% Likewise, the c's will be a decreasing sequence. 00:19:41.430 --> 00:19:47.190 align:middle line:84% And you also see that the difference between ck-- 00:19:47.190 --> 00:19:52.870 align:middle line:90% say, ck plus 1 minus bk plus 1-- 00:19:52.870 --> 00:19:56.390 align:middle line:84% that the difference here, but the difference 00:19:56.390 --> 00:20:00.910 align:middle line:84% is going to be like half of the length of the previous interval. 00:20:00.910 --> 00:20:04.870 align:middle line:84% So the difference here is going to be 1/2-- 00:20:04.870 --> 00:20:14.910 align:middle line:84% it's actually going to be 1/2 and then bck minus bk. 00:20:14.910 --> 00:20:17.950 align:middle line:84% So you're getting this form here. 00:20:17.950 --> 00:20:22.670 align:middle line:84% And now if you just use that you're starting with-- 00:20:22.670 --> 00:20:25.790 align:middle line:84% if you're just using that there are [INAUDIBLE], 00:20:25.790 --> 00:20:30.090 align:middle line:84% so you get this recursive formula here. 00:20:30.090 --> 00:20:36.790 align:middle line:84% And you also have that c1 minus b1. 00:20:36.790 --> 00:20:40.430 align:middle line:84% This is the length of the original interval, 00:20:40.430 --> 00:20:44.630 align:middle line:90% because b1 was 0, c1 was 1. 00:20:44.630 --> 00:20:47.030 align:middle line:90% So this is 1. 00:20:47.030 --> 00:20:49.150 align:middle line:84% And so you're just putting this in. 00:20:49.150 --> 00:20:57.870 align:middle line:84% And so you get the formula that ck minus bk is equal to 2 00:20:57.870 --> 00:21:01.830 align:middle line:90% to the 1 minus k. 00:21:01.830 --> 00:21:05.470 align:middle line:84% So you're getting this formula here. 00:21:05.470 --> 00:21:09.430 align:middle line:84% This just comes from this recursive formula 00:21:09.430 --> 00:21:11.710 align:middle line:90% and how this starts. 00:21:11.710 --> 00:21:13.070 align:middle line:90% So this gives you this. 00:21:13.070 --> 00:21:16.430 align:middle line:84% And so you see that the length of this interval 00:21:16.430 --> 00:21:18.390 align:middle line:90% becomes smaller and smaller. 00:21:18.390 --> 00:21:22.190 align:middle line:84% Again, the sequence bk is increasing. 00:21:22.190 --> 00:21:25.230 align:middle line:90% The ck is decreasing. 00:21:25.230 --> 00:21:29.090 align:middle line:84% So this means that this here is converging to some b. 00:21:29.090 --> 00:21:31.480 align:middle line:90% This is converging to some c. 00:21:31.480 --> 00:21:35.080 align:middle line:84% And because the difference between these goes to 0, 00:21:35.080 --> 00:21:37.644 align:middle line:90% then the b must be equal to c. 00:21:37.644 --> 00:21:40.920 align:middle line:90% 00:21:40.920 --> 00:21:45.920 align:middle line:84% And so this will prove that the limit that you construct, 00:21:45.920 --> 00:21:54.240 align:middle line:84% because this is going to be squeezed, 00:21:54.240 --> 00:22:00.520 align:middle line:84% you will have this property that if you're looking at ak 00:22:00.520 --> 00:22:02.880 align:middle line:90% and even-- 00:22:02.880 --> 00:22:12.800 align:middle line:84% sorry, ank and even anl, the addition here, 00:22:12.800 --> 00:22:19.200 align:middle line:84% this here will, by construction, will lie between these two. 00:22:19.200 --> 00:22:23.720 align:middle line:84% But all the later ones will also lie in this interval. 00:22:23.720 --> 00:22:28.240 align:middle line:84% So this here will also lie in this interval, 00:22:28.240 --> 00:22:32.060 align:middle line:84% ck, as long as l here is bigger or equal to k. 00:22:32.060 --> 00:22:34.760 align:middle line:90% 00:22:34.760 --> 00:22:38.760 align:middle line:84% And so you see that this thing here, this thing here 00:22:38.760 --> 00:22:42.780 align:middle line:84% will-- because this here is converging to b, 00:22:42.780 --> 00:22:45.000 align:middle line:84% and actually now, we prove that this converges 00:22:45.000 --> 00:22:48.160 align:middle line:84% to the same number, b, so this means that these guys here 00:22:48.160 --> 00:22:50.840 align:middle line:84% that is squeezed between these two other sequence 00:22:50.840 --> 00:22:54.740 align:middle line:90% will also converge to b. 00:22:54.740 --> 00:22:57.300 align:middle line:84% So that's a Bolzano-Weierstrass Theorem. 00:22:57.300 --> 00:23:00.400 align:middle line:90% 00:23:00.400 --> 00:23:05.840 align:middle line:84% OK, another key concept in this class 00:23:05.840 --> 00:23:07.970 align:middle line:90% is the concept of a series. 00:23:07.970 --> 00:23:17.720 align:middle line:90% 00:23:17.720 --> 00:23:28.706 align:middle line:84% And so this is the concept of a series, oops. 00:23:28.706 --> 00:23:32.280 align:middle line:90% 00:23:32.280 --> 00:23:35.600 align:middle line:84% And so a series is like as follows, 00:23:35.600 --> 00:23:40.770 align:middle line:84% that suppose you have-- so let an be a sequence. 00:23:40.770 --> 00:23:45.480 align:middle line:90% 00:23:45.480 --> 00:23:49.120 align:middle line:84% From a sequence, you can form another sequence. 00:23:49.120 --> 00:23:51.400 align:middle line:84% And you can form another sequence 00:23:51.400 --> 00:23:57.490 align:middle line:84% in the following way, that s-- so we form another sequence. 00:23:57.490 --> 00:24:03.320 align:middle line:90% 00:24:03.320 --> 00:24:07.280 align:middle line:90% And that is that s1. 00:24:07.280 --> 00:24:13.520 align:middle line:84% And we call this other sequence sn, another sequence, sn. 00:24:13.520 --> 00:24:20.640 align:middle line:84% And the sequence is defined as follows, that s1 is a1, 00:24:20.640 --> 00:24:32.450 align:middle line:84% s2 is a1 plus a2, s3 is a1 plus a2 plus a3. 00:24:32.450 --> 00:24:36.040 align:middle line:90% 00:24:36.040 --> 00:24:40.280 align:middle line:84% s4 is where you're summing the first four elements 00:24:40.280 --> 00:24:42.930 align:middle line:90% in the original sequence. 00:24:42.930 --> 00:24:45.720 align:middle line:90% 00:24:45.720 --> 00:24:47.680 align:middle line:90% And then you write-- 00:24:47.680 --> 00:24:49.860 align:middle line:84% sorry, this is the first four element. 00:24:49.860 --> 00:24:52.480 align:middle line:84% And then you write the n-th element. 00:24:52.480 --> 00:24:59.590 align:middle line:84% You write it just as a sum where i equals to 1 to n of the ai's. 00:24:59.590 --> 00:25:03.200 align:middle line:90% 00:25:03.200 --> 00:25:08.680 align:middle line:84% So if you have a sequence, then you can form another sequence. 00:25:08.680 --> 00:25:11.840 align:middle line:90% And this we denote by s. 00:25:11.840 --> 00:25:15.760 align:middle line:90% And this here is-- 00:25:15.760 --> 00:25:21.180 align:middle line:84% so this here, other sequence, if a sequence is of this form, 00:25:21.180 --> 00:25:28.640 align:middle line:84% then you call it a series, so sn here. 00:25:28.640 --> 00:25:29.260 align:middle line:90% Sorry. 00:25:29.260 --> 00:25:31.240 align:middle line:90% But sn is just a sequence. 00:25:31.240 --> 00:25:39.530 align:middle line:84% This here is said to be a series. 00:25:39.530 --> 00:25:42.440 align:middle line:90% 00:25:42.440 --> 00:25:53.200 align:middle line:84% Now, let's go to the most-- so I'm sure probably everybody have 00:25:53.200 --> 00:25:57.560 align:middle line:84% heard about this kind of one of the famous Greek paradoxes 00:25:57.560 --> 00:25:58.360 align:middle line:90% is this. 00:25:58.360 --> 00:26:00.680 align:middle line:84% I mean, they have various different forms. 00:26:00.680 --> 00:26:07.120 align:middle line:84% This is one form of it is with Achilles and the turtle, 00:26:07.120 --> 00:26:10.640 align:middle line:84% where the turtle is getting a head start of Achilles. 00:26:10.640 --> 00:26:15.340 align:middle line:84% And the paradox is that because the turtle have a head start, 00:26:15.340 --> 00:26:18.180 align:middle line:84% then Achilles can never catch up with the turtle, 00:26:18.180 --> 00:26:21.680 align:middle line:84% even though the turtle is moving incredibly slowly, 00:26:21.680 --> 00:26:24.780 align:middle line:84% because in order to catch up with the turtle, 00:26:24.780 --> 00:26:30.760 align:middle line:84% then Achilles first have to go half of the distance 00:26:30.760 --> 00:26:33.280 align:middle line:90% that the turtle have gone. 00:26:33.280 --> 00:26:34.920 align:middle line:84% And then once you've done that, you 00:26:34.920 --> 00:26:38.800 align:middle line:84% have to go half of the distance that the turtle now 00:26:38.800 --> 00:26:41.080 align:middle line:90% has traveled, et cetera. 00:26:41.080 --> 00:26:43.940 align:middle line:84% And so this goes on infinitely long, 00:26:43.940 --> 00:26:46.300 align:middle line:84% and therefore, he can never catch up with the turtle. 00:26:46.300 --> 00:26:49.820 align:middle line:84% There's various other version of this. 00:26:49.820 --> 00:26:52.080 align:middle line:90% And that's exactly a series. 00:26:52.080 --> 00:26:56.680 align:middle line:84% And this is also like the most important series 00:26:56.680 --> 00:27:01.230 align:middle line:84% is what's called the geometric series, so the geometric series. 00:27:01.230 --> 00:27:03.960 align:middle line:90% 00:27:03.960 --> 00:27:08.860 align:middle line:84% So this is the most important series. 00:27:08.860 --> 00:27:10.800 align:middle line:84% And it's the most important because it's 00:27:10.800 --> 00:27:13.160 align:middle line:84% kind of a non-trivial series that 00:27:13.160 --> 00:27:17.680 align:middle line:84% can be used to determine whether other series 00:27:17.680 --> 00:27:25.410 align:middle line:84% or whether for other sums, whether this sequence sn, 00:27:25.410 --> 00:27:27.690 align:middle line:90% whether it converges or not. 00:27:27.690 --> 00:27:31.690 align:middle line:84% So the geometric series is the following, that c-- 00:27:31.690 --> 00:27:33.730 align:middle line:90% you take some number, c-- 00:27:33.730 --> 00:27:39.730 align:middle line:84% and then you are summing this number, but to a power. 00:27:39.730 --> 00:27:41.370 align:middle line:84% So you're looking at where you have 00:27:41.370 --> 00:27:45.730 align:middle line:90% some power, say, from 0 to n. 00:27:45.730 --> 00:27:47.930 align:middle line:90% So this means that-- 00:27:47.930 --> 00:27:49.530 align:middle line:90% so this is sn. 00:27:49.530 --> 00:28:00.800 align:middle line:84% So this means that s serum, this is just where you have a-- 00:28:00.800 --> 00:28:05.050 align:middle line:84% so c to the power 0 by definition is 1. 00:28:05.050 --> 00:28:10.450 align:middle line:84% And s1 is then when you're summing c to the power 0 00:28:10.450 --> 00:28:13.410 align:middle line:90% plus c to the power 1. 00:28:13.410 --> 00:28:17.250 align:middle line:90% So that's just 1 plus c. 00:28:17.250 --> 00:28:32.770 align:middle line:84% And s2 is then c0 plus c1 plus c squared. 00:28:32.770 --> 00:28:39.090 align:middle line:84% So that's just 1 plus c plus c squared et cetera. 00:28:39.090 --> 00:28:43.850 align:middle line:84% So that's a geometric series where it's just a number. 00:28:43.850 --> 00:28:51.010 align:middle line:84% And each time you multiply that number by itself the n time, 00:28:51.010 --> 00:28:53.890 align:middle line:84% and then you're adding it to the previous. 00:28:53.890 --> 00:28:55.930 align:middle line:90% So that's a geometric series. 00:28:55.930 --> 00:29:01.570 align:middle line:84% And the reason why this is important 00:29:01.570 --> 00:29:05.290 align:middle line:84% is that this is very useful to determine 00:29:05.290 --> 00:29:08.290 align:middle line:84% whether other series are convergent or not 00:29:08.290 --> 00:29:11.970 align:middle line:84% because you could compare it, you can compare other series 00:29:11.970 --> 00:29:15.730 align:middle line:90% with the geometric series. 00:29:15.730 --> 00:29:17.970 align:middle line:84% And the geometric series is easier 00:29:17.970 --> 00:29:21.150 align:middle line:84% to determine whether it is convergent or not. 00:29:21.150 --> 00:29:24.810 align:middle line:84% And so this is because of the following little calculation 00:29:24.810 --> 00:29:25.930 align:middle line:90% here. 00:29:25.930 --> 00:29:31.690 align:middle line:84% Let's suppose I'm looking at some summing here from 0 to n 00:29:31.690 --> 00:29:36.210 align:middle line:90% of c to the power i. 00:29:36.210 --> 00:29:41.970 align:middle line:90% And now I multiply this with c. 00:29:41.970 --> 00:29:44.690 align:middle line:84% So then this here would just be where 00:29:44.690 --> 00:29:47.970 align:middle line:84% I'm summing here c-- if I multiply by c, 00:29:47.970 --> 00:29:50.490 align:middle line:90% I can move that c inside. 00:29:50.490 --> 00:29:54.850 align:middle line:84% And then the first time it's really right-- so you 00:29:54.850 --> 00:29:56.190 align:middle line:90% can write this thing here. 00:29:56.190 --> 00:29:57.390 align:middle line:90% So let me just write it out. 00:29:57.390 --> 00:29:58.850 align:middle line:90% So this is c. 00:29:58.850 --> 00:30:05.090 align:middle line:84% And then it's 1 plus c plus c squared plus c cubed, et cetera, 00:30:05.090 --> 00:30:06.650 align:middle line:90% like that, up to cn. 00:30:06.650 --> 00:30:09.410 align:middle line:90% 00:30:09.410 --> 00:30:12.230 align:middle line:84% And if you multiply by c, then, of course, 00:30:12.230 --> 00:30:17.170 align:middle line:84% this is just c plus c squared plus c cubed up 00:30:17.170 --> 00:30:20.970 align:middle line:90% to c to the n plus 1. 00:30:20.970 --> 00:30:26.150 align:middle line:84% And so this means that if you take this thing here, 00:30:26.150 --> 00:30:31.690 align:middle line:84% so if you're looking at the sum from 0 to n of ci, 00:30:31.690 --> 00:30:36.730 align:middle line:84% and now you multiply by 1 minus c, 00:30:36.730 --> 00:30:44.190 align:middle line:84% then this is the same as just 1 times this. 00:30:44.190 --> 00:30:45.670 align:middle line:90% So let me just write it out. 00:30:45.670 --> 00:30:49.690 align:middle line:84% This just gives you the original series. 00:30:49.690 --> 00:30:53.050 align:middle line:90% So this gives you this up to cn. 00:30:53.050 --> 00:30:55.970 align:middle line:84% That's when we multiply the one n. 00:30:55.970 --> 00:30:59.810 align:middle line:84% When we multiply it by minus c, then we get this thing 00:30:59.810 --> 00:31:01.630 align:middle line:90% here with a minus sign. 00:31:01.630 --> 00:31:05.090 align:middle line:90% So we're getting minus c. 00:31:05.090 --> 00:31:06.830 align:middle line:90% So I can write it like this. 00:31:06.830 --> 00:31:11.650 align:middle line:90% 00:31:11.650 --> 00:31:13.690 align:middle line:90% Like that, right? 00:31:13.690 --> 00:31:17.290 align:middle line:84% But now you see that there's a lot of cancelation. 00:31:17.290 --> 00:31:21.170 align:middle line:84% This thing here cancel with this here. 00:31:21.170 --> 00:31:23.410 align:middle line:84% The next one would cancel with this. 00:31:23.410 --> 00:31:26.810 align:middle line:84% This here would cancel with the second last. 00:31:26.810 --> 00:31:32.310 align:middle line:84% And so you are only left with this one here and that one here. 00:31:32.310 --> 00:31:35.530 align:middle line:84% But this one here comes in with a minus sign. 00:31:35.530 --> 00:31:40.650 align:middle line:84% So this is 1 minus c to the n plus 1. 00:31:40.650 --> 00:31:42.660 align:middle line:90% So we have this little formula. 00:31:42.660 --> 00:31:49.130 align:middle line:90% 00:31:49.130 --> 00:31:58.610 align:middle line:84% And this is a formula, say, that if you take 1 minus c and you 00:31:58.610 --> 00:32:03.570 align:middle line:84% sum from i equal to 0 to n of ci-- 00:32:03.570 --> 00:32:06.290 align:middle line:90% 00:32:06.290 --> 00:32:08.130 align:middle line:84% this is what we did here-- then you're 00:32:08.130 --> 00:32:14.450 align:middle line:84% getting 1 minus c to the n plus 1. 00:32:14.450 --> 00:32:16.050 align:middle line:90% OK. 00:32:16.050 --> 00:32:19.530 align:middle line:84% So now, of course, if c is 1-- let's 00:32:19.530 --> 00:32:21.710 align:middle line:90% wait with figuring that out. 00:32:21.710 --> 00:32:27.010 align:middle line:84% But assume that c here is not equal to 1. 00:32:27.010 --> 00:32:33.450 align:middle line:84% Then I can write this here as that this sum here, i equal to 0 00:32:33.450 --> 00:32:39.770 align:middle line:84% to n ci, I can just divide here by this here on both sides. 00:32:39.770 --> 00:32:43.890 align:middle line:84% So I get this thing here, 1 minus c. 00:32:43.890 --> 00:32:49.290 align:middle line:84% So I have this formula here, assuming 00:32:49.290 --> 00:32:54.250 align:middle line:84% that c is not equal to 1, so I can divide by 1 minus c. 00:32:54.250 --> 00:32:57.770 align:middle line:90% And now you see that-- 00:32:57.770 --> 00:33:02.275 align:middle line:84% so now from this here, we can determine-- so we 00:33:02.275 --> 00:33:03.400 align:middle line:90% can prove a little theorem. 00:33:03.400 --> 00:33:07.530 align:middle line:90% 00:33:07.530 --> 00:33:10.330 align:middle line:90% So that the geometric series-- 00:33:10.330 --> 00:33:12.670 align:middle line:84% so this is geometric series convergence. 00:33:12.670 --> 00:33:22.320 align:middle line:90% 00:33:22.320 --> 00:33:27.060 align:middle line:84% Remember that this thing here, so a series here-- 00:33:27.060 --> 00:33:29.560 align:middle line:84% so maybe I should just emphasize that over here. 00:33:29.560 --> 00:33:32.460 align:middle line:90% 00:33:32.460 --> 00:33:36.860 align:middle line:84% So if you write down a series, so again, a series 00:33:36.860 --> 00:33:41.260 align:middle line:84% is always that you're starting with a sequence like this. 00:33:41.260 --> 00:33:45.340 align:middle line:84% And then you form this other sequence, sn, where 00:33:45.340 --> 00:33:50.406 align:middle line:90% you're summing the first n-- 00:33:50.406 --> 00:33:53.720 align:middle line:84% I guess in this case, I'm starting from 0. 00:33:53.720 --> 00:34:00.340 align:middle line:84% So then it would be n plus 1 elements, like this. 00:34:00.340 --> 00:34:05.220 align:middle line:84% And sometimes one also just write for the series, 00:34:05.220 --> 00:34:14.260 align:middle line:84% one write it as this, where one write like this. 00:34:14.260 --> 00:34:19.500 align:middle line:84% But really, what you're thinking about is this sequence here. 00:34:19.500 --> 00:34:23.960 align:middle line:84% But you, with a slight abuse of notation, 00:34:23.960 --> 00:34:27.199 align:middle line:90% you write this here for series. 00:34:27.199 --> 00:34:30.460 align:middle line:84% But what you're thinking about is this sequence. 00:34:30.460 --> 00:34:32.806 align:middle line:90% And so a series is convergent. 00:34:32.806 --> 00:34:42.460 align:middle line:90% 00:34:42.460 --> 00:34:53.639 align:middle line:84% It just means that the sequence sn is convergent. 00:34:53.639 --> 00:34:58.740 align:middle line:90% 00:34:58.740 --> 00:35:05.300 align:middle line:84% So it just means that this sequence, sn, is convergent. 00:35:05.300 --> 00:35:13.430 align:middle line:84% And the sum and the limit of the series-- 00:35:13.430 --> 00:35:16.020 align:middle line:90% 00:35:16.020 --> 00:35:25.420 align:middle line:84% this is just the limit, is the limit of the sn's. 00:35:25.420 --> 00:35:30.740 align:middle line:90% 00:35:30.740 --> 00:35:47.280 align:middle line:84% That is usually denoted by this, by this infinite sum, a. 00:35:47.280 --> 00:35:52.260 align:middle line:84% Sorry, this was i, ai, like that. 00:35:52.260 --> 00:35:52.760 align:middle line:90% Right. 00:35:52.760 --> 00:35:56.820 align:middle line:84% So you see there's a little bit of abuse of notation, 00:35:56.820 --> 00:36:00.300 align:middle line:84% that one talk about a series and one 00:36:00.300 --> 00:36:02.500 align:middle line:90% typically write it like this. 00:36:02.500 --> 00:36:06.620 align:middle line:84% What one really means is that one thing about this sequence 00:36:06.620 --> 00:36:07.860 align:middle line:90% here. 00:36:07.860 --> 00:36:10.260 align:middle line:84% And then if it's convergent, this just 00:36:10.260 --> 00:36:13.940 align:middle line:90% means that these sn converge. 00:36:13.940 --> 00:36:19.020 align:middle line:84% And the limit, but the limit you would also write as this. 00:36:19.020 --> 00:36:23.200 align:middle line:84% So in a way, this symbol here is used for two different things. 00:36:23.200 --> 00:36:25.140 align:middle line:90% It's used for the series. 00:36:25.140 --> 00:36:30.620 align:middle line:84% And if the series is convergent, it's also used for the limit. 00:36:30.620 --> 00:36:31.850 align:middle line:90% This is like standard. 00:36:31.850 --> 00:36:34.420 align:middle line:90% 00:36:34.420 --> 00:36:38.420 align:middle line:84% OK, so now let's come back to the geometric series. 00:36:38.420 --> 00:36:43.300 align:middle line:84% So again, the geometric series were that you take some c, 00:36:43.300 --> 00:36:47.340 align:middle line:84% and then you form this series here, 00:36:47.340 --> 00:36:51.520 align:middle line:90% ci, i equal to 0 to infinity. 00:36:51.520 --> 00:36:54.480 align:middle line:84% It's, of course, important where you start. 00:36:54.480 --> 00:36:57.460 align:middle line:84% I mean, the convergence is not important. 00:36:57.460 --> 00:37:01.420 align:middle line:84% But what the limit is depends on where you start. 00:37:01.420 --> 00:37:05.460 align:middle line:90% So this is a geometric series. 00:37:05.460 --> 00:37:10.540 align:middle line:90% And we have that if c here-- 00:37:10.540 --> 00:37:23.220 align:middle line:84% so we have now the following, that the geometric series 00:37:23.220 --> 00:37:35.900 align:middle line:84% is convergent if c here in absolute value is less than 1 00:37:35.900 --> 00:37:49.700 align:middle line:84% and divergent otherwise, if the absolute value of c 00:37:49.700 --> 00:37:52.140 align:middle line:90% is bigger than 1. 00:37:52.140 --> 00:37:54.970 align:middle line:90% So this is the claim that-- 00:37:54.970 --> 00:37:57.580 align:middle line:90% 00:37:57.580 --> 00:38:00.180 align:middle line:90% and so this is a claim. 00:38:00.180 --> 00:38:10.696 align:middle line:84% And in fact, so one more thing is that-- 00:38:10.696 --> 00:38:14.700 align:middle line:90% 00:38:14.700 --> 00:38:29.460 align:middle line:84% so moreover, if c is less than 1, then the limit-- 00:38:29.460 --> 00:38:36.660 align:middle line:84% so then the claim is that it's convergent-- then the limit 00:38:36.660 --> 00:38:41.700 align:middle line:90% is 1 minus c. 00:38:41.700 --> 00:38:42.880 align:middle line:90% This is the limit. 00:38:42.880 --> 00:38:48.210 align:middle line:84% And so we write this here as this infinite sum. 00:38:48.210 --> 00:38:53.260 align:middle line:90% 00:38:53.260 --> 00:38:56.940 align:middle line:84% Now, let's just try to prove this. 00:38:56.940 --> 00:39:02.660 align:middle line:84% So let me first do-- and we're almost there already. 00:39:02.660 --> 00:39:05.760 align:middle line:90% So the thing is that-- 00:39:05.760 --> 00:39:14.430 align:middle line:90% 00:39:14.430 --> 00:39:16.300 align:middle line:90% so let's first look at the case. 00:39:16.300 --> 00:39:21.310 align:middle line:90% 00:39:21.310 --> 00:39:29.070 align:middle line:84% Assume c in absolute value is strictly less than 1. 00:39:29.070 --> 00:39:36.030 align:middle line:84% But then we have already proven that sn is-- so sn, 00:39:36.030 --> 00:39:41.310 align:middle line:84% this here was the sum from i equal to 0 to n of ci. 00:39:41.310 --> 00:39:46.470 align:middle line:84% And we have already proven that this is 1 minus cn plus 1 over 1 00:39:46.470 --> 00:39:48.990 align:middle line:90% minus c. 00:39:48.990 --> 00:39:51.350 align:middle line:90% And this makes sense. 00:39:51.350 --> 00:39:53.130 align:middle line:90% Norm is strictly less than 1. 00:39:53.130 --> 00:39:55.290 align:middle line:84% So in particular, c is not 1, right? 00:39:55.290 --> 00:39:57.710 align:middle line:90% So this here is fine. 00:39:57.710 --> 00:39:59.670 align:middle line:84% But now you see that if you take something 00:39:59.670 --> 00:40:04.430 align:middle line:84% with norm strictly less than 1, then just because cn 00:40:04.430 --> 00:40:10.190 align:middle line:84% plus 1, the norm of this here, but if you take something 00:40:10.190 --> 00:40:14.630 align:middle line:84% and you multiply it by another-- so if you take two numbers, a 00:40:14.630 --> 00:40:17.630 align:middle line:84% and b, and you take the absolute value 00:40:17.630 --> 00:40:20.430 align:middle line:84% of the product, that's the same as the product 00:40:20.430 --> 00:40:23.310 align:middle line:90% of the absolute values. 00:40:23.310 --> 00:40:27.990 align:middle line:84% So this means that if you take c to the power n plus 1-- 00:40:27.990 --> 00:40:29.950 align:middle line:84% so if you take c and you multiply it 00:40:29.950 --> 00:40:33.670 align:middle line:84% by itself n plus 1 time, this is the same 00:40:33.670 --> 00:40:37.030 align:middle line:84% as the absolute value times [INAUDIBLE] raised 00:40:37.030 --> 00:40:39.650 align:middle line:90% to the power n plus 1 right. 00:40:39.650 --> 00:40:42.710 align:middle line:84% And so now you see that because this c has 00:40:42.710 --> 00:40:46.950 align:middle line:84% absolute value less than 1, strictly less than 1, 00:40:46.950 --> 00:40:52.350 align:middle line:84% then as n goes to infinity, this here converges to 0. 00:40:52.350 --> 00:40:55.910 align:middle line:84% So this means that as n goes to infinity, 00:40:55.910 --> 00:40:59.150 align:middle line:90% this thing here might converge. 00:40:59.150 --> 00:41:00.870 align:middle line:90% This here goes to 0. 00:41:00.870 --> 00:41:05.270 align:middle line:84% So just by the algebraic rules for limits, 00:41:05.270 --> 00:41:10.670 align:middle line:84% this thing here is going to converge to 1 minus c over 1, 00:41:10.670 --> 00:41:11.430 align:middle line:90% like that. 00:41:11.430 --> 00:41:14.710 align:middle line:90% 00:41:14.710 --> 00:41:20.830 align:middle line:84% So you see that in the case where c in absolute value 00:41:20.830 --> 00:41:23.790 align:middle line:84% is strictly less than 1, then we have now 00:41:23.790 --> 00:41:26.810 align:middle line:84% proven that the geometric series is convergent. 00:41:26.810 --> 00:41:30.230 align:middle line:84% And in fact, we have even, which is really nice, 00:41:30.230 --> 00:41:34.090 align:middle line:90% and unusual for a series. 00:41:34.090 --> 00:41:37.190 align:middle line:84% I mean, usually you cannot actually determine what 00:41:37.190 --> 00:41:38.270 align:middle line:90% the limit is. 00:41:38.270 --> 00:41:39.790 align:middle line:90% But in this case, we could. 00:41:39.790 --> 00:41:43.070 align:middle line:84% But we still need to figure out why 00:41:43.070 --> 00:41:51.790 align:middle line:84% is it not convergent if the absolute value of c 00:41:51.790 --> 00:41:54.710 align:middle line:90% is bigger or equal to 1. 00:41:54.710 --> 00:41:58.350 align:middle line:90% And so here's an easy thing. 00:41:58.350 --> 00:42:00.950 align:middle line:90% I mean, but you could also-- 00:42:00.950 --> 00:42:04.710 align:middle line:84% yeah, so here's an easy thing, that if you're looking at c-- 00:42:04.710 --> 00:42:14.590 align:middle line:84% sorry, sn plus 1 minus sn, where sn plus 1 or sn say, 00:42:14.590 --> 00:42:23.850 align:middle line:84% this here was just c to the power 0 up to c to the power n. 00:42:23.850 --> 00:42:28.950 align:middle line:84% Sn plus 1 is where you do the same stuff. 00:42:28.950 --> 00:42:32.190 align:middle line:90% We're adding these guys. 00:42:32.190 --> 00:42:36.630 align:middle line:84% But then you have to add one more, right? 00:42:36.630 --> 00:42:40.230 align:middle line:84% So this means that this difference here is really 00:42:40.230 --> 00:42:45.190 align:middle line:90% c to the n plus 1, right? 00:42:45.190 --> 00:42:47.990 align:middle line:84% But now I could look at the absolute value 00:42:47.990 --> 00:42:50.670 align:middle line:90% of this difference. 00:42:50.670 --> 00:42:53.790 align:middle line:90% This is this. 00:42:53.790 --> 00:43:00.470 align:middle line:84% But this is the same, again, as c to the power n plus 1. 00:43:00.470 --> 00:43:05.470 align:middle line:84% But you see, if c here is the absolute value is bigger 00:43:05.470 --> 00:43:09.070 align:middle line:84% or equal to 1, then this thing here-- oops-- 00:43:09.070 --> 00:43:12.630 align:middle line:84% then this thing here, so if the absolute value here 00:43:12.630 --> 00:43:16.070 align:middle line:84% is bigger or equal to 1, then this thing here 00:43:16.070 --> 00:43:19.630 align:middle line:90% must be bigger or equal to 1. 00:43:19.630 --> 00:43:23.290 align:middle line:84% Remember that if a sequence is convergent, 00:43:23.290 --> 00:43:27.830 align:middle line:84% if a sequence is convergent, then it is a Cauchy sequence. 00:43:27.830 --> 00:43:31.430 align:middle line:84% It's a Cauchy sequence that if you go sufficiently far 00:43:31.430 --> 00:43:36.710 align:middle line:84% out, then from there on and out, everything bunch very, very 00:43:36.710 --> 00:43:37.890 align:middle line:90% close together. 00:43:37.890 --> 00:43:41.230 align:middle line:84% But you see here, not even two consecutive 00:43:41.230 --> 00:43:44.990 align:middle line:84% is bunching together, because independent on how far, 00:43:44.990 --> 00:43:49.110 align:middle line:84% how large n is, the difference is always at least 1. 00:43:49.110 --> 00:43:51.110 align:middle line:90% So this is not even a Cauchy-- 00:43:51.110 --> 00:43:53.850 align:middle line:84% I mean, well, it's actually equivalent, of course, 00:43:53.850 --> 00:43:55.370 align:middle line:90% but it's not a Cauchy sequence. 00:43:55.370 --> 00:43:56.910 align:middle line:90% It's failed. 00:43:56.910 --> 00:44:00.470 align:middle line:84% Like the most basic thing with consecutive ones 00:44:00.470 --> 00:44:03.870 align:middle line:84% does not even lie close together. 00:44:03.870 --> 00:44:09.550 align:middle line:84% So this proves this theorem about the geometric convergence, 00:44:09.550 --> 00:44:12.630 align:middle line:84% the geometric series convergence. 00:44:12.630 --> 00:44:23.750 align:middle line:84% So this is the most important example of a sequence. 00:44:23.750 --> 00:44:28.270 align:middle line:84% Maybe the second most important is 00:44:28.270 --> 00:44:35.920 align:middle line:84% what's called the harmonic series, so harmonic series. 00:44:35.920 --> 00:44:44.750 align:middle line:90% 00:44:44.750 --> 00:44:48.830 align:middle line:84% But the harmonic series is unfortunately not a series 00:44:48.830 --> 00:44:50.670 align:middle line:90% that is convergent. 00:44:50.670 --> 00:44:52.770 align:middle line:84% So the harmonic series is the following. 00:44:52.770 --> 00:44:54.150 align:middle line:90% So again, I'm writing it-- 00:44:54.150 --> 00:44:59.440 align:middle line:84% I'm using this abusive notation that I'm writing the series as 00:44:59.440 --> 00:45:01.280 align:middle line:90% like an infinite sum. 00:45:01.280 --> 00:45:06.540 align:middle line:84% But this infinite sum, but it's not a convergent series, 00:45:06.540 --> 00:45:08.920 align:middle line:84% so the infinite sum doesn't really make sense. 00:45:08.920 --> 00:45:14.460 align:middle line:84% So here I have to start because I'm dividing by n. 00:45:14.460 --> 00:45:16.560 align:middle line:90% So I'm starting by 1 with 1. 00:45:16.560 --> 00:45:19.760 align:middle line:84% So this is called the harmonic series. 00:45:19.760 --> 00:45:20.300 align:middle line:90% OK. 00:45:20.300 --> 00:45:31.280 align:middle line:84% And so for series, there's a bunch 00:45:31.280 --> 00:45:34.120 align:middle line:84% of things, a bunch of tests that are 00:45:34.120 --> 00:45:37.420 align:middle line:84% useful to determine whether a series is convergent or not. 00:45:37.420 --> 00:45:41.720 align:middle line:90% 00:45:41.720 --> 00:45:43.640 align:middle line:84% And we will talk about these tests. 00:45:43.640 --> 00:45:45.980 align:middle line:84% I'll talk a little bit about later in this lecture. 00:45:45.980 --> 00:45:48.820 align:middle line:84% I'll talk about some of the tests, but not all of them. 00:45:48.820 --> 00:45:51.760 align:middle line:84% And we will later talk about more. 00:45:51.760 --> 00:45:55.020 align:middle line:84% The test that is particularly useful in this case, 00:45:55.020 --> 00:45:56.580 align:middle line:84% it's what's called the integral test. 00:45:56.580 --> 00:45:59.760 align:middle line:84% But I won't talk about that today. 00:45:59.760 --> 00:46:05.560 align:middle line:84% But instead, I will show you a very simple proof of why 00:46:05.560 --> 00:46:09.020 align:middle line:90% this here is not convergent. 00:46:09.020 --> 00:46:12.000 align:middle line:90% 00:46:12.000 --> 00:46:15.750 align:middle line:90% So here's a theorem. 00:46:15.750 --> 00:46:19.920 align:middle line:90% 00:46:19.920 --> 00:46:30.162 align:middle line:84% The harmonic series is a divergence. 00:46:30.162 --> 00:46:31.120 align:middle line:90% So it's not convergent. 00:46:31.120 --> 00:46:32.100 align:middle line:90% It's divergent. 00:46:32.100 --> 00:46:34.680 align:middle line:90% 00:46:34.680 --> 00:46:37.000 align:middle line:84% And so in fact, what we will prove 00:46:37.000 --> 00:46:44.080 align:middle line:84% is the following, that we will show that if you're looking at-- 00:46:44.080 --> 00:46:46.240 align:middle line:90% so n is an integer. 00:46:46.240 --> 00:46:48.820 align:middle line:90% If you're looking at-- 00:46:48.820 --> 00:46:51.200 align:middle line:90% so this is like-- 00:46:51.200 --> 00:46:57.920 align:middle line:84% so this is only some of the-- you only hit, for this here, 00:46:57.920 --> 00:47:03.200 align:middle line:84% you only hit some of the elements in the sequence. 00:47:03.200 --> 00:47:11.120 align:middle line:84% So this is where I'm summing again from i equal to 1 to 2 00:47:11.120 --> 00:47:16.760 align:middle line:84% to the n minus 1, n is an integer, of 1 over i. 00:47:16.760 --> 00:47:21.280 align:middle line:84% So I want to prove that these guys here, 00:47:21.280 --> 00:47:24.900 align:middle line:84% if the original sequence had been convergent, 00:47:24.900 --> 00:47:28.400 align:middle line:84% then of course, any subsequence would also be convergent. 00:47:28.400 --> 00:47:32.440 align:middle line:84% And we're going to prove that this thing here 00:47:32.440 --> 00:47:36.000 align:middle line:90% is bigger or equal to n over 2. 00:47:36.000 --> 00:47:37.820 align:middle line:90% So I claim that we have this. 00:47:37.820 --> 00:47:44.840 align:middle line:84% So this is a claim that you have this. 00:47:44.840 --> 00:47:47.440 align:middle line:84% But if you have this thing here, you see, 00:47:47.440 --> 00:47:51.340 align:middle line:84% that then this subsequence here is unbounded. 00:47:51.340 --> 00:47:54.640 align:middle line:84% So of course, the original harmonic series 00:47:54.640 --> 00:47:57.280 align:middle line:90% is not convergent. 00:47:57.280 --> 00:47:59.440 align:middle line:90% So let's try to prove this. 00:47:59.440 --> 00:48:06.320 align:middle line:84% Again, this is just like a simple proof of that-- 00:48:06.320 --> 00:48:09.280 align:middle line:84% very simple-minded proof-- that the harmonic series 00:48:09.280 --> 00:48:11.040 align:middle line:90% is divergent. 00:48:11.040 --> 00:48:12.953 align:middle line:90% But there are more. 00:48:12.953 --> 00:48:15.960 align:middle line:90% 00:48:15.960 --> 00:48:18.700 align:middle line:84% Once we learn about the integral test and so on, 00:48:18.700 --> 00:48:24.520 align:middle line:84% there are other way better ways of doing it. 00:48:24.520 --> 00:48:26.220 align:middle line:90% So let me try to prove this. 00:48:26.220 --> 00:48:31.310 align:middle line:84% So I want to prove the claim, proof of the claim. 00:48:31.310 --> 00:48:37.080 align:middle line:90% 00:48:37.080 --> 00:48:38.860 align:middle line:90% So I'm going to prove it, right? 00:48:38.860 --> 00:48:40.440 align:middle line:90% So I just proved. 00:48:40.440 --> 00:48:43.440 align:middle line:84% So I want to establish this for all n. 00:48:43.440 --> 00:48:51.960 align:middle line:84% And I'm going to prove it first when n is equal to 1. 00:48:51.960 --> 00:48:55.920 align:middle line:84% So let's show this for n equal to 1. 00:48:55.920 --> 00:49:05.100 align:middle line:84% So for n equal to 1, well then, the claim here means that-- 00:49:05.100 --> 00:49:07.600 align:middle line:90% so 2 to the 1, that's true. 00:49:07.600 --> 00:49:16.020 align:middle line:84% So this is s1, right, because 2 to the 1 is 2 minus 1. 00:49:16.020 --> 00:49:17.400 align:middle line:90% So this is 1. 00:49:17.400 --> 00:49:19.720 align:middle line:90% So I claim that. 00:49:19.720 --> 00:49:25.920 align:middle line:84% So I want to prove that s1 is bigger or equal to 1/2. 00:49:25.920 --> 00:49:26.860 align:middle line:90% [INAUDIBLE] to show. 00:49:26.860 --> 00:49:30.000 align:middle line:90% 00:49:30.000 --> 00:49:36.280 align:middle line:84% But of course, s1 here, s1 is just 1 over 1. 00:49:36.280 --> 00:49:40.040 align:middle line:90% So that's definitely fine. 00:49:40.040 --> 00:49:47.240 align:middle line:84% So this is-- so we want to prove the claim. 00:49:47.240 --> 00:49:49.640 align:middle line:90% And the claim is OK. 00:49:49.640 --> 00:49:56.120 align:middle line:90% So claim is OK for n equal to 1. 00:49:56.120 --> 00:49:58.740 align:middle line:90% So now I want to prove it. 00:49:58.740 --> 00:50:15.160 align:middle line:84% So I want to assume the claim is correct for n. 00:50:15.160 --> 00:50:22.440 align:middle line:84% And we want to prove it for, want to show 00:50:22.440 --> 00:50:30.060 align:middle line:90% the claim for the next one. 00:50:30.060 --> 00:50:33.300 align:middle line:90% And so I have that. 00:50:33.300 --> 00:50:43.880 align:middle line:84% So we assume that s2n minus 1, that this 00:50:43.880 --> 00:50:47.840 align:middle line:90% is bigger than n over 2. 00:50:47.840 --> 00:50:55.270 align:middle line:84% And then I want to look at s2n plus 1 minus 1. 00:50:55.270 --> 00:50:58.450 align:middle line:84% OK, how can I write this thing here? 00:50:58.450 --> 00:51:00.990 align:middle line:90% So I'm summing, right? 00:51:00.990 --> 00:51:04.290 align:middle line:84% This here is a sum of the 1 over i, where 00:51:04.290 --> 00:51:06.690 align:middle line:90% I do it up to this thing here. 00:51:06.690 --> 00:51:10.330 align:middle line:84% But I can think about that as the sum of the one 00:51:10.330 --> 00:51:14.530 align:middle line:90% where I do up to this here. 00:51:14.530 --> 00:51:20.210 align:middle line:84% And then I have to sum the rest also. 00:51:20.210 --> 00:51:34.370 align:middle line:84% So this is i equal to 2n, to 2n plus 1, 2n plus 1 minus 1, 00:51:34.370 --> 00:51:38.090 align:middle line:90% like this, of 1 over i. 00:51:38.090 --> 00:51:47.000 align:middle line:84% But now you see, I want a bounded from below. 00:51:47.000 --> 00:51:47.500 align:middle line:90% Fine. 00:51:47.500 --> 00:51:50.090 align:middle line:90% 00:51:50.090 --> 00:51:54.450 align:middle line:84% So if you continue this line, then by assumption, 00:51:54.450 --> 00:51:58.730 align:middle line:84% this thing here is bounded by n over 2. 00:51:58.730 --> 00:52:05.870 align:middle line:84% This thing here, I'm summing 2n numbers. 00:52:05.870 --> 00:52:10.770 align:middle line:84% From here to here, there are 2n numbers. 00:52:10.770 --> 00:52:16.570 align:middle line:90% And so it's 2n numbers. 00:52:16.570 --> 00:52:20.210 align:middle line:84% And if I take the smallest of those two numbers, 00:52:20.210 --> 00:52:24.650 align:middle line:84% if I take the smallest of these guys and I multiply it by 2n, 00:52:24.650 --> 00:52:26.510 align:middle line:84% then that's definitely a lower bound. 00:52:26.510 --> 00:52:27.710 align:middle line:90% So let's try to do this. 00:52:27.710 --> 00:52:30.010 align:middle line:90% So I say 2n, 2 to the n-- 00:52:30.010 --> 00:52:33.690 align:middle line:84% and then the smallest, but the smallest is the last one. 00:52:33.690 --> 00:52:40.570 align:middle line:84% So the smallest is 1 over 2 to the n plus 1 minus 1. 00:52:40.570 --> 00:52:47.490 align:middle line:84% But this thing here is, of course, obviously bigger 00:52:47.490 --> 00:52:51.590 align:middle line:84% than-- so n over 2, this thing here-- well, this is true, then. 00:52:51.590 --> 00:52:55.650 align:middle line:84% But this thing here, if I divide by a larger 00:52:55.650 --> 00:52:59.170 align:middle line:84% number and the larger number is just this here, 00:52:59.170 --> 00:53:01.610 align:middle line:84% then I get something even smaller. 00:53:01.610 --> 00:53:05.970 align:middle line:84% So I would have 1 over 2 to the n plus 1. 00:53:05.970 --> 00:53:11.250 align:middle line:84% But now you see that this thing here cancel with these guys 00:53:11.250 --> 00:53:12.290 align:middle line:90% here. 00:53:12.290 --> 00:53:17.370 align:middle line:84% And so you're getting that this is n over 2 plus 1/2. 00:53:17.370 --> 00:53:20.190 align:middle line:90% And so this is n plus 1 over 2. 00:53:20.190 --> 00:53:25.170 align:middle line:84% And that's exactly the claim for n plus 1. 00:53:25.170 --> 00:53:28.490 align:middle line:84% So we're proven that if it was true for n, then 00:53:28.490 --> 00:53:30.810 align:middle line:90% it's also true for n plus 1. 00:53:30.810 --> 00:53:37.330 align:middle line:84% And we remember to check that the beginning was also OK. 00:53:37.330 --> 00:53:38.950 align:middle line:90% So that's a harmonic series. 00:53:38.950 --> 00:53:43.530 align:middle line:84% So you see that the harmonic series is divergent. 00:53:43.530 --> 00:53:48.150 align:middle line:84% Now, the harmonic series is, of course, 00:53:48.150 --> 00:53:54.810 align:middle line:84% an example of something else, namely that if you're looking 00:53:54.810 --> 00:53:58.970 align:middle line:84% at-- so when you're looking at a series, 00:53:58.970 --> 00:54:01.430 align:middle line:84% the first thing you look at typically is-- 00:54:01.430 --> 00:54:07.650 align:middle line:90% 00:54:07.650 --> 00:54:12.310 align:middle line:84% OK, so let me just have a side here. 00:54:12.310 --> 00:54:17.490 align:middle line:90% So if you give me a series-- 00:54:17.490 --> 00:54:19.370 align:middle line:90% it doesn't really matter where. 00:54:19.370 --> 00:54:24.290 align:middle line:84% I mean, it doesn't matter very much where it's-- 00:54:24.290 --> 00:54:27.190 align:middle line:84% for whether a series is convergent, 00:54:27.190 --> 00:54:29.010 align:middle line:84% it doesn't matter where you start, 00:54:29.010 --> 00:54:32.610 align:middle line:84% because this is just adding something to it. 00:54:32.610 --> 00:54:34.910 align:middle line:84% It will matter where you start for the limit, 00:54:34.910 --> 00:54:38.650 align:middle line:84% but not whether or not the series is convergent. 00:54:38.650 --> 00:54:42.450 align:middle line:84% But if you give me a limit, the first thing I always check 00:54:42.450 --> 00:54:44.230 align:middle line:90% is that this-- 00:54:44.230 --> 00:54:59.420 align:middle line:84% so first, so for a series, always check if an goes to 0 00:54:59.420 --> 00:54:59.920 align:middle line:90% or not. 00:54:59.920 --> 00:55:02.930 align:middle line:90% 00:55:02.930 --> 00:55:04.130 align:middle line:90% And why? 00:55:04.130 --> 00:55:09.210 align:middle line:84% Because if you give me a series, the question about a series 00:55:09.210 --> 00:55:11.530 align:middle line:84% is always whether it's converging or not. 00:55:11.530 --> 00:55:18.450 align:middle line:84% And again, just like we did, if you're looking at sn, say, 00:55:18.450 --> 00:55:21.890 align:middle line:84% minus sn minus 1, this is where I'm looking 00:55:21.890 --> 00:55:23.770 align:middle line:90% at two consecutive things. 00:55:23.770 --> 00:55:27.330 align:middle line:84% But if I'm looking at this difference here, 00:55:27.330 --> 00:55:31.350 align:middle line:84% the difference here, this is where I'm summing up to the a, 00:55:31.350 --> 00:55:32.930 align:middle line:90% n minus 1. 00:55:32.930 --> 00:55:35.050 align:middle line:90% Here I'm summing up to the an. 00:55:35.050 --> 00:55:38.290 align:middle line:90% So the difference here is an. 00:55:38.290 --> 00:55:40.170 align:middle line:84% And if the sequence-- just like we 00:55:40.170 --> 00:55:43.990 align:middle line:84% saw, if the sequence of the series is convergent, 00:55:43.990 --> 00:55:47.150 align:middle line:84% it means that this sequence here is convergent. 00:55:47.150 --> 00:55:50.050 align:middle line:84% But it means that it has to be a Cauchy sequence, that things 00:55:50.050 --> 00:55:55.570 align:middle line:84% have to bunch together, but that two consecutive bunch together 00:55:55.570 --> 00:55:58.630 align:middle line:84% is determined whether an goes to 0 or not. 00:55:58.630 --> 00:56:03.350 align:middle line:84% So if an does not go to 0, you're already done, right? 00:56:03.350 --> 00:56:05.210 align:middle line:90% There's no reason to continue. 00:56:05.210 --> 00:56:08.730 align:middle line:84% It's already clear that if an does not go to 0, 00:56:08.730 --> 00:56:10.930 align:middle line:84% you can just say it doesn't converge, 00:56:10.930 --> 00:56:12.590 align:middle line:90% because an doesn't go to 0. 00:56:12.590 --> 00:56:14.770 align:middle line:84% So there's no reason to start with any kind of test 00:56:14.770 --> 00:56:18.640 align:middle line:84% or anything, any sort of good ideas. 00:56:18.640 --> 00:56:22.410 align:middle line:90% 00:56:22.410 --> 00:56:25.610 align:middle line:90% So that was just a side remark. 00:56:25.610 --> 00:56:29.730 align:middle line:84% The other thing is that for the harmonic series 00:56:29.730 --> 00:56:33.420 align:middle line:84% is-- so this, of course, failed this test. 00:56:33.420 --> 00:56:35.930 align:middle line:90% 00:56:35.930 --> 00:56:42.340 align:middle line:84% But it also is an example of something where you're taking-- 00:56:42.340 --> 00:56:43.820 align:middle line:90% so suppose you take a-- 00:56:43.820 --> 00:56:47.640 align:middle line:90% so you have a sequence, an. 00:56:47.640 --> 00:56:55.860 align:middle line:84% And then you're forming the series, this series here. 00:56:55.860 --> 00:56:57.450 align:middle line:90% Now, if all of a n's-- 00:56:57.450 --> 00:57:00.190 align:middle line:90% 00:57:00.190 --> 00:57:04.220 align:middle line:84% you post that all of the a n's are non-negative, 00:57:04.220 --> 00:57:11.140 align:middle line:84% then if you're looking at sn plus 1, you can think about sn 00:57:11.140 --> 00:57:18.500 align:middle line:84% plus 1 as where you're summing the first n elements, 00:57:18.500 --> 00:57:22.680 align:middle line:84% and then you're adding the n plus 1 element. 00:57:22.680 --> 00:57:30.260 align:middle line:84% So this here you can think about as sn plus an plus 1. 00:57:30.260 --> 00:57:34.980 align:middle line:84% And so of course, if all of these elements is non-negative, 00:57:34.980 --> 00:57:36.760 align:middle line:84% this means that this has non-negative. 00:57:36.760 --> 00:57:39.700 align:middle line:84% So this thing here is bigger than sn. 00:57:39.700 --> 00:57:43.340 align:middle line:84% So you see that if all of the element, an, is non-negative, 00:57:43.340 --> 00:57:47.640 align:middle line:84% then this sequence, sn, is monotone. 00:57:47.640 --> 00:57:49.500 align:middle line:84% So in this case here, if you have this, 00:57:49.500 --> 00:57:56.180 align:middle line:84% this imply that this sn is monotone increasing. 00:57:56.180 --> 00:58:01.140 align:middle line:84% And so now to say whether this, if you have this, 00:58:01.140 --> 00:58:05.580 align:middle line:84% to determine whether this series here is convergent or not, 00:58:05.580 --> 00:58:10.180 align:middle line:84% it's just a matter of proving whether to determine 00:58:10.180 --> 00:58:15.155 align:middle line:84% whether this sequence is bounded or not. 00:58:15.155 --> 00:58:16.530 align:middle line:84% So you just need to find a bound. 00:58:16.530 --> 00:58:19.620 align:middle line:90% 00:58:19.620 --> 00:58:24.180 align:middle line:84% So this naturally leads to looking at what's 00:58:24.180 --> 00:58:27.420 align:middle line:90% called absolute convergence. 00:58:27.420 --> 00:58:30.080 align:middle line:84% So this is something stronger than convergence. 00:58:30.080 --> 00:58:33.180 align:middle line:90% 00:58:33.180 --> 00:58:35.500 align:middle line:90% And for many things-- 00:58:35.500 --> 00:58:38.540 align:middle line:90% for many, although not all-- 00:58:38.540 --> 00:58:40.340 align:middle line:84% there is a particular test that deals 00:58:40.340 --> 00:58:46.780 align:middle line:84% with it where you don't have both things or you [INAUDIBLE] 00:58:46.780 --> 00:58:48.060 align:middle line:90% have a stronger property. 00:58:48.060 --> 00:58:54.500 align:middle line:84% So for many things you have absolute convergence. 00:58:54.500 --> 00:59:00.680 align:middle line:84% And so let me try to-- so absolute convergence. 00:59:00.680 --> 00:59:09.100 align:middle line:90% 00:59:09.100 --> 00:59:13.080 align:middle line:84% So again, you have a series here. 00:59:13.080 --> 00:59:17.220 align:middle line:90% 00:59:17.220 --> 00:59:21.400 align:middle line:84% This series here is said to be absolute convergence or converge 00:59:21.400 --> 00:59:27.300 align:middle line:84% absolutely-- so converge absolutely-- 00:59:27.300 --> 00:59:31.620 align:middle line:90% 00:59:31.620 --> 00:59:38.340 align:middle line:84% if the series where you slap absolute values on them, 00:59:38.340 --> 00:59:41.410 align:middle line:90% if this thing here, converge. 00:59:41.410 --> 00:59:51.400 align:middle line:90% 00:59:51.400 --> 00:59:54.020 align:middle line:84% So we say that this here converge absolutely 00:59:54.020 --> 00:59:55.700 align:middle line:90% if this here converge. 00:59:55.700 --> 01:00:04.220 align:middle line:84% Now, so I claim that absolute convergence 01:00:04.220 --> 01:00:06.000 align:middle line:90% is stronger than convergence. 01:00:06.000 --> 01:00:16.740 align:middle line:90% 01:00:16.740 --> 01:00:32.288 align:middle line:84% So absolute convergence implies convergence. 01:00:32.288 --> 01:00:34.100 align:middle line:90% So this is a theorem. 01:00:34.100 --> 01:00:38.420 align:middle line:90% 01:00:38.420 --> 01:00:40.860 align:middle line:84% But it's not the other way around. 01:00:40.860 --> 01:00:44.140 align:middle line:84% So if you have what's called the-- 01:00:44.140 --> 01:00:46.980 align:middle line:90% so here, an example. 01:00:46.980 --> 01:00:49.900 align:middle line:84% And we haven't yet proved that it's converging, 01:00:49.900 --> 01:00:52.050 align:middle line:84% but we'll do that in some later lecture. 01:00:52.050 --> 01:00:59.180 align:middle line:90% 01:00:59.180 --> 01:01:03.460 align:middle line:90% So here's an example. 01:01:03.460 --> 01:01:07.140 align:middle line:84% If you're looking at the series minus 1 01:01:07.140 --> 01:01:11.300 align:middle line:84% to the power n divided by n, n equal to 1 01:01:11.300 --> 01:01:17.340 align:middle line:84% to infinity, let's say you see if this had just had been 1, 01:01:17.340 --> 01:01:19.860 align:middle line:84% then it had been the harmonic series. 01:01:19.860 --> 01:01:25.280 align:middle line:84% This here series is also kind of like a famous series. 01:01:25.280 --> 01:01:29.590 align:middle line:84% It's called the alternating harmonic series. 01:01:29.590 --> 01:01:34.780 align:middle line:90% 01:01:34.780 --> 01:01:35.960 align:middle line:90% And it's alternating. 01:01:35.960 --> 01:01:39.560 align:middle line:84% Alternating means that the sign of it is alternating. 01:01:39.560 --> 01:01:43.020 align:middle line:84% So the sign of this an element, it 01:01:43.020 --> 01:01:47.540 align:middle line:84% keeps changing from a-- in this case, the first one is negative. 01:01:47.540 --> 01:01:49.140 align:middle line:90% The next one is positive. 01:01:49.140 --> 01:01:59.020 align:middle line:84% So go back and forth between around 0, so harmonic series. 01:01:59.020 --> 01:02:10.740 align:middle line:84% And this series is convergent, but not absolutely convergent, 01:02:10.740 --> 01:02:16.880 align:middle line:84% because if you slap absolute value sign, 01:02:16.880 --> 01:02:18.160 align:middle line:90% you get the harmonic series. 01:02:18.160 --> 01:02:21.260 align:middle line:84% And we already know that this is not convergent, but not 01:02:21.260 --> 01:02:22.520 align:middle line:90% absolutely convergent. 01:02:22.520 --> 01:02:30.340 align:middle line:90% 01:02:30.340 --> 01:02:36.130 align:middle line:84% So this is the alternating harmonic series. 01:02:36.130 --> 01:02:41.710 align:middle line:84% Now, let's try to prove that absolute convergence implies 01:02:41.710 --> 01:02:42.690 align:middle line:90% convergence. 01:02:42.690 --> 01:02:45.710 align:middle line:90% 01:02:45.710 --> 01:02:49.010 align:middle line:84% And so I'm going to look at-- so I take two series. 01:02:49.010 --> 01:02:50.790 align:middle line:90% So I have sn. 01:02:50.790 --> 01:02:58.670 align:middle line:84% That's my original series, ai, i equal to 0, to n, like that. 01:02:58.670 --> 01:03:02.150 align:middle line:84% And then I take the one where I snap absolute value sign 01:03:02.150 --> 01:03:11.110 align:middle line:84% and I denote it with a bar, so i equal to 0 to n of ai. 01:03:11.110 --> 01:03:15.350 align:middle line:84% And then I make a little observation. 01:03:15.350 --> 01:03:21.550 align:middle line:84% And the observation is that if I take-- 01:03:21.550 --> 01:03:28.790 align:middle line:84% if I'm looking at sn and sm, so just 01:03:28.790 --> 01:03:35.670 align:middle line:84% think about m here is larger than n, 01:03:35.670 --> 01:03:38.510 align:middle line:84% then you can think about this thing here, 01:03:38.510 --> 01:03:42.350 align:middle line:84% as this thing here is just where you're summing 01:03:42.350 --> 01:03:53.830 align:middle line:90% from n plus 1 n up to an. 01:03:53.830 --> 01:03:57.070 align:middle line:90% That's this difference, right? 01:03:57.070 --> 01:04:03.710 align:middle line:84% But if you take absolute value on this, absolute value on that, 01:04:03.710 --> 01:04:08.150 align:middle line:84% well, that's of course lesser or equal to where 01:04:08.150 --> 01:04:23.050 align:middle line:84% you're taking absolute value on each individual one, like that. 01:04:23.050 --> 01:04:29.110 align:middle line:90% But this thing here is just-- 01:04:29.110 --> 01:04:34.510 align:middle line:84% this thing here is just for the other series, this series. 01:04:34.510 --> 01:04:36.570 align:middle line:90% It's the same difference. 01:04:36.570 --> 01:04:41.510 align:middle line:90% 01:04:41.510 --> 01:04:44.237 align:middle line:84% And here, actually, I don't even need to take absolute value, 01:04:44.237 --> 01:04:45.570 align:middle line:90% but I could take absolute value. 01:04:45.570 --> 01:04:49.270 align:middle line:90% But this is always [INAUDIBLE]. 01:04:49.270 --> 01:04:51.830 align:middle line:90% And so you see that-- 01:04:51.830 --> 01:04:54.370 align:middle line:84% so you make this little observation. 01:04:54.370 --> 01:04:59.430 align:middle line:84% But now we know that if it is absolutely convergent, 01:04:59.430 --> 01:05:01.870 align:middle line:84% then this sequence here converge. 01:05:01.870 --> 01:05:04.310 align:middle line:84% So this means that in particular, it's 01:05:04.310 --> 01:05:07.630 align:middle line:84% equivalent to saying that it's a Cauchy sequence. 01:05:07.630 --> 01:05:11.510 align:middle line:84% And so this means that since if you are sufficiently far out, 01:05:11.510 --> 01:05:13.550 align:middle line:84% these things have bunched together, 01:05:13.550 --> 01:05:17.350 align:middle line:84% epsilon together if you are bigger than some capital N. 01:05:17.350 --> 01:05:20.070 align:middle line:84% But now, this inequality say that this thing here 01:05:20.070 --> 01:05:21.190 align:middle line:90% bunch together. 01:05:21.190 --> 01:05:25.430 align:middle line:84% So you see that if this is convergent, 01:05:25.430 --> 01:05:28.270 align:middle line:90% then it's a Cauchy sequence. 01:05:28.270 --> 01:05:31.050 align:middle line:84% But this means that this here is a Cauchy sequence. 01:05:31.050 --> 01:05:34.250 align:middle line:84% So it means that the original series was convergent. 01:05:34.250 --> 01:05:40.030 align:middle line:90% 01:05:40.030 --> 01:05:47.710 align:middle line:84% And so now, let's look at one more example. 01:05:47.710 --> 01:05:50.240 align:middle line:84% And then we'll talk about some of the tests. 01:05:50.240 --> 01:05:54.150 align:middle line:90% 01:05:54.150 --> 01:06:00.550 align:middle line:84% So let's look at-- so we saw that the harmonic series is 01:06:00.550 --> 01:06:01.490 align:middle line:90% divergent. 01:06:01.490 --> 01:06:10.230 align:middle line:90% 01:06:10.230 --> 01:06:14.030 align:middle line:84% So we saw that the harmonic series is divergent. 01:06:14.030 --> 01:06:15.720 align:middle line:90% But what about-- 01:06:15.720 --> 01:06:20.350 align:middle line:90% 01:06:20.350 --> 01:06:22.090 align:middle line:90% I mean, what about-- 01:06:22.090 --> 01:06:25.450 align:middle line:84% so there's another thing you could do. 01:06:25.450 --> 01:06:30.670 align:middle line:84% So maybe the most important series is the geometric series. 01:06:30.670 --> 01:06:33.950 align:middle line:84% After that, you can debate a little bit which series 01:06:33.950 --> 01:06:35.790 align:middle line:90% is the most important. 01:06:35.790 --> 01:06:40.270 align:middle line:84% But this here is some other candidates for that. 01:06:40.270 --> 01:06:49.350 align:middle line:84% So I'm looking at now the sum here, n equal to 0, to infinity. 01:06:49.350 --> 01:06:55.590 align:middle line:84% And I'm looking at 1 over n squared. 01:06:55.590 --> 01:06:57.950 align:middle line:84% But you could do it more generally where 01:06:57.950 --> 01:06:59.790 align:middle line:90% you raise it to some power. 01:06:59.790 --> 01:07:03.510 align:middle line:84% If that power was 1, you got the alternating. 01:07:03.510 --> 01:07:05.050 align:middle line:90% You got the harmonic series. 01:07:05.050 --> 01:07:08.950 align:middle line:84% But now I'm just looking at a specific power, 2. 01:07:08.950 --> 01:07:11.350 align:middle line:90% OK. 01:07:11.350 --> 01:07:16.510 align:middle line:84% And so I claim that this series here, this here is-- 01:07:16.510 --> 01:07:20.137 align:middle line:90% 01:07:20.137 --> 01:07:21.970 align:middle line:84% I mean, this one doesn't really have a name, 01:07:21.970 --> 01:07:24.430 align:middle line:90% but it is a key series. 01:07:24.430 --> 01:07:27.850 align:middle line:84% And so I claim that this here is convergent. 01:07:27.850 --> 01:07:37.510 align:middle line:90% 01:07:37.510 --> 01:07:44.510 align:middle line:84% Now, you observe that the an, in this case, this is just 1 01:07:44.510 --> 01:07:46.470 align:middle line:90% over n squared. 01:07:46.470 --> 01:07:48.750 align:middle line:90% And these here are non-negative. 01:07:48.750 --> 01:07:52.050 align:middle line:84% So all we need to do is-- so we just need to prove. 01:07:52.050 --> 01:07:55.170 align:middle line:84% So when the a n's are non-negative, 01:07:55.170 --> 01:07:59.750 align:middle line:84% all you need to prove is that the sequence-- 01:07:59.750 --> 01:08:02.830 align:middle line:84% you just need to prove, need to show 01:08:02.830 --> 01:08:12.950 align:middle line:84% that the sn, which is where you're summing here, 01:08:12.950 --> 01:08:15.230 align:middle line:90% i equal to-- 01:08:15.230 --> 01:08:19.430 align:middle line:84% or you can't, you shouldn't sum from 0, sum from 1, 01:08:19.430 --> 01:08:23.560 align:middle line:90% i equal to 1 to n-- 01:08:23.560 --> 01:08:27.920 align:middle line:84% that we will need to prove that these here are bounded. 01:08:27.920 --> 01:08:31.899 align:middle line:84% This is because the a n's are non-negative. 01:08:31.899 --> 01:08:35.240 align:middle line:90% 01:08:35.240 --> 01:08:40.800 align:middle line:90% So now the idea here is that-- 01:08:40.800 --> 01:08:48.000 align:middle line:84% I mean, again, this is not necessarily how I would do it. 01:08:48.000 --> 01:08:53.120 align:middle line:84% But the advantage with this here is that it 01:08:53.120 --> 01:08:55.460 align:middle line:90% doesn't use anything fancy. 01:08:55.460 --> 01:09:00.240 align:middle line:84% I mean, it's not like the way later on you do it, 01:09:00.240 --> 01:09:01.899 align:middle line:90% you will use the integral test. 01:09:01.899 --> 01:09:04.520 align:middle line:84% But we haven't talked about this integral test. 01:09:04.520 --> 01:09:08.000 align:middle line:84% So you can do it very simple minded. 01:09:08.000 --> 01:09:17.040 align:middle line:84% And so the claim here is that you want to compare. 01:09:17.040 --> 01:09:29.319 align:middle line:84% So we want to make a foreign claim that if you're looking 01:09:29.319 --> 01:09:33.600 align:middle line:90% at s to the 2n plus 1-- 01:09:33.600 --> 01:09:38.840 align:middle line:84% sorry, to the 2n, so it's just similar to what we did before. 01:09:38.840 --> 01:09:44.880 align:middle line:84% If we want to prove that this thing here for this series, 01:09:44.880 --> 01:09:54.800 align:middle line:84% that this is less than the sum here from i equal to 0 to n 01:09:54.800 --> 01:10:02.920 align:middle line:84% of 1/2 to the power, i, so we'll put n to n minus 1. 01:10:02.920 --> 01:10:06.320 align:middle line:90% So we claim this thing here. 01:10:06.320 --> 01:10:08.640 align:middle line:90% Now, you see that-- 01:10:08.640 --> 01:10:13.680 align:middle line:84% so we claim that we have this inequality for all n. 01:10:13.680 --> 01:10:18.640 align:middle line:84% But this thing here, we already know that this here is 01:10:18.640 --> 01:10:29.380 align:middle line:84% a geometric series with c equal to 1/2, so it's convergent. 01:10:29.380 --> 01:10:33.440 align:middle line:90% 01:10:33.440 --> 01:10:37.600 align:middle line:84% And you see this thing here is a subsequence. 01:10:37.600 --> 01:10:41.080 align:middle line:84% But it's a subsequence because this is convergent. 01:10:41.080 --> 01:10:47.640 align:middle line:84% This here is bounded by something fixed. 01:10:47.640 --> 01:10:52.280 align:middle line:84% So this means that this subsequent here is bounded. 01:10:52.280 --> 01:10:57.700 align:middle line:84% But the original series was a monotone increasing sequence. 01:10:57.700 --> 01:11:01.960 align:middle line:84% So if a subsequence is bounded, then the original sequence 01:11:01.960 --> 01:11:04.360 align:middle line:90% is bounded and so convergent. 01:11:04.360 --> 01:11:06.120 align:middle line:90% So we just need to prove-- 01:11:06.120 --> 01:11:09.080 align:middle line:84% we just need to give a bound, like this. 01:11:09.080 --> 01:11:11.920 align:middle line:84% And the bound here you get is by we already determined 01:11:11.920 --> 01:11:18.160 align:middle line:84% that this here is less than or equal to 1 over 1 minus c. 01:11:18.160 --> 01:11:19.920 align:middle line:90% c is 1/2. 01:11:19.920 --> 01:11:21.260 align:middle line:90% So this is like this. 01:11:21.260 --> 01:11:23.680 align:middle line:90% So this is 1/2. 01:11:23.680 --> 01:11:27.200 align:middle line:90% So this whole thing here is 2. 01:11:27.200 --> 01:11:31.560 align:middle line:84% So this here would be bounded by 2, right? 01:11:31.560 --> 01:11:35.040 align:middle line:90% So now we want to prove. 01:11:35.040 --> 01:11:37.180 align:middle line:84% So we want to prove this claim here. 01:11:37.180 --> 01:11:42.260 align:middle line:84% So if you can prove this claim, then we have that the series, 01:11:42.260 --> 01:11:46.200 align:middle line:90% this series here is convergent. 01:11:46.200 --> 01:11:47.720 align:middle line:90% OK. 01:11:47.720 --> 01:11:51.760 align:middle line:90% So now let's try to do that. 01:11:51.760 --> 01:11:57.320 align:middle line:84% So when n is equal to 1, so we do it again 01:11:57.320 --> 01:12:01.820 align:middle line:84% in the same way, which is called by induction. 01:12:01.820 --> 01:12:07.600 align:middle line:84% We do it for, first of all, for n equal to 1. 01:12:07.600 --> 01:12:13.660 align:middle line:84% For n equal to 1, it is a claim that s-- 01:12:13.660 --> 01:12:16.860 align:middle line:84% if n is equal to 1, this is true, minus 1. 01:12:16.860 --> 01:12:17.840 align:middle line:90% That's 1. 01:12:17.840 --> 01:12:23.740 align:middle line:84% So it's a claim that this thing here is bigger equals-- 01:12:23.740 --> 01:12:26.160 align:middle line:90% it's less than or equal to. 01:12:26.160 --> 01:12:30.480 align:middle line:84% And now when n is equal to 1, you're 01:12:30.480 --> 01:12:31.820 align:middle line:90% just summing over one thing. 01:12:31.820 --> 01:12:33.040 align:middle line:90% It's just one thing. 01:12:33.040 --> 01:12:36.340 align:middle line:90% And it is 1/2 to the power 0. 01:12:36.340 --> 01:12:37.480 align:middle line:90% And that's 1. 01:12:37.480 --> 01:12:41.300 align:middle line:84% So for n equal to 1, this is what we need to show. 01:12:41.300 --> 01:12:46.000 align:middle line:90% 01:12:46.000 --> 01:12:53.600 align:middle line:84% But now, let's see for n equal to 1, 01:12:53.600 --> 01:12:55.820 align:middle line:90% well, this here is indeed 1. 01:12:55.820 --> 01:12:57.200 align:middle line:90% So that's OK. 01:12:57.200 --> 01:12:59.900 align:middle line:90% So this is OK. 01:12:59.900 --> 01:13:03.960 align:middle line:90% 01:13:03.960 --> 01:13:14.760 align:middle line:84% So now assume that s to the power 2n minus 1 is bounded 01:13:14.760 --> 01:13:23.880 align:middle line:84% by this i equal to 0 to n minus 1 of 1/2 to the power i. 01:13:23.880 --> 01:13:26.040 align:middle line:90% So assume this. 01:13:26.040 --> 01:13:31.500 align:middle line:84% And we want to prove the same thing, but for n plus 1, 01:13:31.500 --> 01:13:46.290 align:middle line:84% so want to show the same inequality for n plus 1. 01:13:46.290 --> 01:13:46.790 align:middle line:90% OK. 01:13:46.790 --> 01:13:50.480 align:middle line:90% 01:13:50.480 --> 01:13:51.860 align:middle line:90% So let me do that. 01:13:51.860 --> 01:14:01.280 align:middle line:90% 01:14:01.280 --> 01:14:08.160 align:middle line:84% So I'm looking at s2n plus 1 minus 1. 01:14:08.160 --> 01:14:14.200 align:middle line:84% So s2n plus 1 minus 1 I can write as where I'm summing 01:14:14.200 --> 01:14:18.850 align:middle line:90% the first 2n minus 1. 01:14:18.850 --> 01:14:24.890 align:middle line:84% And then I'm summing, and then plus, 01:14:24.890 --> 01:14:38.090 align:middle line:84% and then where I'm summing from i equal to 2n to up to 2n plus 1 01:14:38.090 --> 01:14:40.130 align:middle line:90% minus 1. 01:14:40.130 --> 01:14:44.650 align:middle line:90% And then it's 1 over i squared. 01:14:44.650 --> 01:14:49.170 align:middle line:84% Now, in this case, remember before we 01:14:49.170 --> 01:14:52.010 align:middle line:84% wanted to prove that something was divergent. 01:14:52.010 --> 01:14:54.110 align:middle line:84% So we wanted to give a lower bound. 01:14:54.110 --> 01:14:56.570 align:middle line:84% In this case, we want to give an upper bound. 01:14:56.570 --> 01:15:01.712 align:middle line:84% So when we do this, we think about what is the-- 01:15:01.712 --> 01:15:03.170 align:middle line:84% instead of what is the smallest, we 01:15:03.170 --> 01:15:06.290 align:middle line:84% think about what is the largest of these numbers. 01:15:06.290 --> 01:15:09.210 align:middle line:90% But that's the first one. 01:15:09.210 --> 01:15:11.790 align:middle line:84% And there are, again, just like before, 01:15:11.790 --> 01:15:14.170 align:middle line:90% there are 2n many of them. 01:15:14.170 --> 01:15:20.210 align:middle line:84% So this thing here is less than or equal to s2n minus 1. 01:15:20.210 --> 01:15:22.610 align:middle line:90% Sorry, like this. 01:15:22.610 --> 01:15:24.950 align:middle line:90% And then there are 2n many. 01:15:24.950 --> 01:15:26.070 align:middle line:90% They're all non-negative. 01:15:26.070 --> 01:15:34.490 align:middle line:84% And the first one is the largest, so 1 over 2n squared, 01:15:34.490 --> 01:15:38.530 align:middle line:90% like this, right. 01:15:38.530 --> 01:15:41.870 align:middle line:84% And so you see that this thing here is-- 01:15:41.870 --> 01:15:45.490 align:middle line:84% so now we can fill in the assumption. 01:15:45.490 --> 01:15:50.770 align:middle line:84% We are assuming that this here was less than this. 01:15:50.770 --> 01:16:00.110 align:middle line:84% This is sum from i equal to 0 to n minus 1 of 1/2 to the power i. 01:16:00.110 --> 01:16:05.210 align:middle line:90% 01:16:05.210 --> 01:16:11.210 align:middle line:84% And this thing here, but the square kills that one 01:16:11.210 --> 01:16:15.050 align:middle line:84% and you're still left with 1 over 2n. 01:16:15.050 --> 01:16:19.250 align:middle line:84% But you see, this is, of course, exactly the next one 01:16:19.250 --> 01:16:20.150 align:middle line:90% in the series. 01:16:20.150 --> 01:16:29.970 align:middle line:84% So this is i equal to 0 to n of 1/2 to the power i, again. 01:16:29.970 --> 01:16:34.242 align:middle line:84% And that's, indeed, this is like the induction step, right? 01:16:34.242 --> 01:16:35.950 align:middle line:84% This is what's called the induction step. 01:16:35.950 --> 01:16:38.530 align:middle line:90% So it's true also. 01:16:38.530 --> 01:16:41.110 align:middle line:84% So if it's true for n, the claim, 01:16:41.110 --> 01:16:43.150 align:middle line:84% then it's also true for n plus 1. 01:16:43.150 --> 01:16:52.570 align:middle line:90% 01:16:52.570 --> 01:16:54.450 align:middle line:90% So now let me-- 01:16:54.450 --> 01:16:55.750 align:middle line:90% I have just four minutes left. 01:16:55.750 --> 01:16:58.370 align:middle line:84% There's a little bit more, and I already 01:16:58.370 --> 01:17:01.010 align:middle line:90% posted the lecture notes. 01:17:01.010 --> 01:17:05.340 align:middle line:84% And there's a bit more here I want to talk about. 01:17:05.340 --> 01:17:07.970 align:middle line:90% 01:17:07.970 --> 01:17:10.710 align:middle line:84% So I want to talk about some of the tests. 01:17:10.710 --> 01:17:14.650 align:middle line:84% But we'll talk about that more next time, also. 01:17:14.650 --> 01:17:33.130 align:middle line:84% So to determine whether or not a series is convergent, converge, 01:17:33.130 --> 01:17:37.140 align:middle line:84% then there are a number of tests. 01:17:37.140 --> 01:17:43.330 align:middle line:90% 01:17:43.330 --> 01:17:50.250 align:middle line:84% And then of course, the key point is to use the right test. 01:17:50.250 --> 01:17:52.910 align:middle line:90% So I will just talk about-- 01:17:52.910 --> 01:17:54.590 align:middle line:84% so I just mentioned some of the tests. 01:17:54.590 --> 01:17:57.550 align:middle line:84% But we won't anyway go through all of them now. 01:17:57.550 --> 01:17:59.910 align:middle line:84% So maybe I can't go through any of them. 01:17:59.910 --> 01:18:01.120 align:middle line:90% So there's a comparison test. 01:18:01.120 --> 01:18:05.170 align:middle line:90% 01:18:05.170 --> 01:18:06.840 align:middle line:90% There's a ratio test. 01:18:06.840 --> 01:18:10.730 align:middle line:90% 01:18:10.730 --> 01:18:11.540 align:middle line:90% There's root test. 01:18:11.540 --> 01:18:14.530 align:middle line:90% 01:18:14.530 --> 01:18:17.370 align:middle line:84% And there's also the one I talked about several times, 01:18:17.370 --> 01:18:18.220 align:middle line:90% the integral test. 01:18:18.220 --> 01:18:21.370 align:middle line:90% 01:18:21.370 --> 01:18:24.740 align:middle line:84% And then there's even one called the alternate series test. 01:18:24.740 --> 01:18:34.450 align:middle line:90% 01:18:34.450 --> 01:18:39.310 align:middle line:84% Now, maybe I can just talk about the comparison test. 01:18:39.310 --> 01:18:41.110 align:middle line:84% That's, in a way, the simplest of them. 01:18:41.110 --> 01:18:44.210 align:middle line:90% 01:18:44.210 --> 01:18:50.730 align:middle line:84% And so the comparison test is that you're looking at two. 01:18:50.730 --> 01:18:52.630 align:middle line:84% So you have these two series here. 01:18:52.630 --> 01:18:55.250 align:middle line:90% 01:18:55.250 --> 01:18:57.070 align:middle line:84% And you have this other series here. 01:18:57.070 --> 01:19:03.290 align:middle line:90% 01:19:03.290 --> 01:19:10.070 align:middle line:84% And the comparison tests say that if you have an equal to 0, 01:19:10.070 --> 01:19:12.290 align:middle line:90% lesser equal to bn-- 01:19:12.290 --> 01:19:22.330 align:middle line:84% so if you have this end, and this here is here, 01:19:22.330 --> 01:19:32.770 align:middle line:84% converge, then the corresponding thing with an 01:19:32.770 --> 01:19:33.760 align:middle line:90% will also converge. 01:19:33.760 --> 01:19:40.810 align:middle line:90% 01:19:40.810 --> 01:19:46.010 align:middle line:84% And this is the simplest version of the comparison test. 01:19:46.010 --> 01:19:49.930 align:middle line:84% And this is, in a way, used in two ways. 01:19:49.930 --> 01:19:52.650 align:middle line:90% So often you have a series. 01:19:52.650 --> 01:19:56.530 align:middle line:84% And maybe it could be a super complicated series. 01:19:56.530 --> 01:20:00.010 align:middle line:84% And but you see some kind of path. 01:20:00.010 --> 01:20:04.250 align:middle line:84% And so you can compare it with a much simpler series 01:20:04.250 --> 01:20:07.280 align:middle line:84% that perhaps you already know whether it's converging. 01:20:07.280 --> 01:20:11.660 align:middle line:84% And so if you can compare it with something else 01:20:11.660 --> 01:20:15.100 align:middle line:84% and you know that that converge, then you 01:20:15.100 --> 01:20:19.020 align:middle line:84% conclude that the an series is convergent. 01:20:19.020 --> 01:20:22.100 align:middle line:84% It's also sometimes used that you 01:20:22.100 --> 01:20:25.620 align:middle line:84% want to determine whether this series is convergent. 01:20:25.620 --> 01:20:29.180 align:middle line:84% But actually, you can easily compare it 01:20:29.180 --> 01:20:32.960 align:middle line:84% with some other series where you have this relationship here, 01:20:32.960 --> 01:20:36.340 align:middle line:84% and this one you already know is divergent. 01:20:36.340 --> 01:20:38.540 align:middle line:90% So it's used in both ways. 01:20:38.540 --> 01:20:42.320 align:middle line:84% It's used to both determine that a series is convergent. 01:20:42.320 --> 01:20:45.320 align:middle line:84% And it's also used to determine that the series is divergent. 01:20:45.320 --> 01:20:48.620 align:middle line:84% So there's a little bit more about tests 01:20:48.620 --> 01:20:51.660 align:middle line:90% in the lecture notes. 01:20:51.660 --> 01:20:53.800 align:middle line:84% But we will, anyway, talk about it more next time. 01:20:53.800 --> 01:20:54.220 align:middle line:90% Yeah? 01:20:54.220 --> 01:20:56.178 align:middle line:84% AUDIENCE: I have a question about the intuition 01:20:56.178 --> 01:20:58.108 align:middle line:90% of this [INAUDIBLE]. 01:20:58.108 --> 01:21:02.523 align:middle line:90% 01:21:02.523 --> 01:21:03.440 align:middle line:90% TOBIAS COLDING: Right. 01:21:03.440 --> 01:21:05.240 align:middle line:90% So that's right. 01:21:05.240 --> 01:21:12.380 align:middle line:90% So that sort of thing is-- 01:21:12.380 --> 01:21:17.820 align:middle line:84% I mean, this last one shows you that the geometric series 01:21:17.820 --> 01:21:23.640 align:middle line:84% is one that come up often as a kind of comparison, in a way. 01:21:23.640 --> 01:21:28.320 align:middle line:84% How you get it is different for different people. 01:21:28.320 --> 01:21:30.300 align:middle line:84% Obviously you don't start with the claim. 01:21:30.300 --> 01:21:32.660 align:middle line:84% You're looking at the first few things 01:21:32.660 --> 01:21:34.600 align:middle line:84% and you see a pattern, basically. 01:21:34.600 --> 01:21:35.780 align:middle line:90% But it's not-- 01:21:35.780 --> 01:21:39.220 align:middle line:84% I mean, this is not the way you want to really-- 01:21:39.220 --> 01:21:44.540 align:middle line:84% later on, I would never prove any kind of convergence 01:21:44.540 --> 01:21:45.700 align:middle line:90% like this. 01:21:45.700 --> 01:21:51.460 align:middle line:84% There's a way of comparing it with the-- 01:21:51.460 --> 01:21:53.620 align:middle line:84% so the idea is, let me just tell you. 01:21:53.620 --> 01:21:57.540 align:middle line:84% But of course, the lecture is over. 01:21:57.540 --> 01:22:01.880 align:middle line:84% So the integral test is that you're looking at-- 01:22:01.880 --> 01:22:03.720 align:middle line:90% so you're looking at a series. 01:22:03.720 --> 01:22:06.940 align:middle line:84% And if you have a series of non-negative numbers, 01:22:06.940 --> 01:22:12.900 align:middle line:84% you can think about that the series is like the area below. 01:22:12.900 --> 01:22:19.660 align:middle line:84% If you think about this here as a1, and this is maybe a2, 01:22:19.660 --> 01:22:24.060 align:middle line:84% then you can think about the sum as the sum of the areas, 01:22:24.060 --> 01:22:25.300 align:middle line:90% like this. 01:22:25.300 --> 01:22:28.980 align:middle line:84% And so the integral of testers that you're comparing it 01:22:28.980 --> 01:22:34.020 align:middle line:84% with some curve and the area that lies below it, then it's 01:22:34.020 --> 01:22:35.540 align:middle line:90% much more clear. 01:22:35.540 --> 01:22:39.380 align:middle line:84% The other one is like where you have to do it case by case. 01:22:39.380 --> 01:22:42.640 align:middle line:84% Here there's a kind of general idea. 01:22:42.640 --> 01:22:45.958 align:middle line:84% So it's nice, in this particular case. 01:22:45.958 --> 01:22:46.500 align:middle line:90% I'm not sure. 01:22:46.500 --> 01:22:47.360 align:middle line:90% You know what I mean? 01:22:47.360 --> 01:22:49.360 align:middle line:84% I mean, you have to look at the first few things 01:22:49.360 --> 01:22:50.840 align:middle line:90% and see if you see a pattern. 01:22:50.840 --> 01:22:53.240 align:middle line:84% And of course, you don't start with a claim. 01:22:53.240 --> 01:22:56.900 align:middle line:84% That's like the end, when you actually have a claim, 01:22:56.900 --> 01:23:00.090 align:middle line:84% then you already know what's going on. 01:23:00.090 --> 01:23:03.000 align:middle line:90%