WEBVTT 00:00:00.000 --> 00:00:02.495 align:middle line:90% [SQUEAKING] 00:00:02.495 --> 00:00:03.992 align:middle line:90% [RUSTLING] 00:00:03.992 --> 00:00:05.489 align:middle line:90% [CLICKING] 00:00:05.489 --> 00:00:13.990 align:middle line:90% 00:00:13.990 --> 00:00:17.163 align:middle line:84% CASEY RODRIGUEZ: OK, so let's continue our study of sequences 00:00:17.163 --> 00:00:17.830 align:middle line:90% of real numbers. 00:00:17.830 --> 00:00:24.240 align:middle line:90% 00:00:24.240 --> 00:00:28.140 align:middle line:84% So we've seen special types of sequences, monotone sequences, 00:00:28.140 --> 00:00:28.650 align:middle line:90% before. 00:00:28.650 --> 00:00:32.770 align:middle line:84% And then in the previous lecture, 00:00:32.770 --> 00:00:39.220 align:middle line:84% we looked at sequences obtained from sequences, 00:00:39.220 --> 00:00:42.178 align:middle line:84% namely the sequences that give you the lim sup and lim inf. 00:00:42.178 --> 00:00:43.720 align:middle line:84% And we showed that these are actually 00:00:43.720 --> 00:00:47.173 align:middle line:90% limits of subsequences. 00:00:47.173 --> 00:00:48.590 align:middle line:84% Now I'm going to define what looks 00:00:48.590 --> 00:00:54.470 align:middle line:90% like a new class of sequences. 00:00:54.470 --> 00:00:56.210 align:middle line:90% But we'll see it's actually not. 00:00:56.210 --> 00:00:58.040 align:middle line:84% These are called Cauchy sequences. 00:00:58.040 --> 00:01:02.450 align:middle line:90% 00:01:02.450 --> 00:01:09.780 align:middle line:84% So "coe-shee"-- not "couch-ee," not "cawt-shee"-- 00:01:09.780 --> 00:01:12.930 align:middle line:90% Cauchy, so a French guy. 00:01:12.930 --> 00:01:15.960 align:middle line:84% So it's pronounced Cauchy and probably not even pronounced 00:01:15.960 --> 00:01:16.460 align:middle line:90% like that. 00:01:16.460 --> 00:01:24.500 align:middle line:84% It's probably got a different pronunciation 00:01:24.500 --> 00:01:26.480 align:middle line:84% by people who actually speak French. 00:01:26.480 --> 00:01:29.135 align:middle line:84% So what is the definition of a Cauchy sequence? 00:01:29.135 --> 00:01:33.080 align:middle line:84% A Cauchy sequence, intuitively, it's 00:01:33.080 --> 00:01:38.360 align:middle line:84% a sequence so that if you go far enough out in the sequence, 00:01:38.360 --> 00:01:42.710 align:middle line:84% any two entries in that sequence are close together. 00:01:42.710 --> 00:01:45.320 align:middle line:84% So convergent sequences had the property 00:01:45.320 --> 00:01:48.980 align:middle line:84% that if you go far enough out, the entries in the sequence 00:01:48.980 --> 00:01:51.740 align:middle line:84% are getting close to a real number. 00:01:51.740 --> 00:01:54.320 align:middle line:84% Cauchy sequence is that any two entries 00:01:54.320 --> 00:01:55.625 align:middle line:90% are close to each other. 00:01:55.625 --> 00:01:58.720 align:middle line:90% 00:01:58.720 --> 00:02:11.590 align:middle line:84% So a Cauchy sequence, so we say a sequence is Cauchy 00:02:11.590 --> 00:02:18.490 align:middle line:84% if we're all epsilon positive, there 00:02:18.490 --> 00:02:22.030 align:middle line:84% exists an M, natural number, such 00:02:22.030 --> 00:02:29.570 align:middle line:84% that if n is bigger than or equal to M and k 00:02:29.570 --> 00:02:36.465 align:middle line:84% is bigger than or equal to M, then xn minus xk 00:02:36.465 --> 00:02:37.340 align:middle line:90% is less than epsilon. 00:02:37.340 --> 00:02:41.050 align:middle line:90% 00:02:41.050 --> 00:02:43.380 align:middle line:90% So maybe not write if. 00:02:43.380 --> 00:02:45.870 align:middle line:84% I mean, it's the same statement, but since it looks-- 00:02:45.870 --> 00:02:50.120 align:middle line:84% so it'll look a little more like previous statements 00:02:50.120 --> 00:02:51.370 align:middle line:90% when we put a "for all" there. 00:02:51.370 --> 00:02:54.070 align:middle line:90% 00:02:54.070 --> 00:02:57.415 align:middle line:90% So you have a definition here. 00:02:57.415 --> 00:03:00.990 align:middle line:90% 00:03:00.990 --> 00:03:04.420 align:middle line:84% It's the definition of a new thing. 00:03:04.420 --> 00:03:07.300 align:middle line:84% You should now try to look at an example 00:03:07.300 --> 00:03:10.120 align:middle line:84% and then possibly negate the definition to see if you really 00:03:10.120 --> 00:03:11.270 align:middle line:90% understand it. 00:03:11.270 --> 00:03:14.290 align:middle line:84% So an example of a Cauchy sequence 00:03:14.290 --> 00:03:20.660 align:middle line:84% is x of n equals 1 over n, our favorite sequence. 00:03:20.660 --> 00:03:22.930 align:middle line:90% So let's prove this. 00:03:22.930 --> 00:03:24.890 align:middle line:84% So all we have is the definition. 00:03:24.890 --> 00:03:28.300 align:middle line:84% So we have to verify that x of n equals 1 over n 00:03:28.300 --> 00:03:32.080 align:middle line:84% verifies the definition of being Cauchy. 00:03:32.080 --> 00:03:34.780 align:middle line:84% So just like when we try to prove something is convergent, 00:03:34.780 --> 00:03:38.290 align:middle line:84% which is a "for all" epsilon statement, the first thing 00:03:38.290 --> 00:03:41.720 align:middle line:84% you have to do is let epsilon be positive. 00:03:41.720 --> 00:03:44.200 align:middle line:84% And then I have to choose M and show that that capital 00:03:44.200 --> 00:03:47.960 align:middle line:90% M produces this statement here. 00:03:47.960 --> 00:03:57.130 align:middle line:84% So choose M, a natural number, so that 1 over M 00:03:57.130 --> 00:03:59.260 align:middle line:90% is less than epsilon over 2. 00:03:59.260 --> 00:04:02.590 align:middle line:84% So I could phrase that as capital M being bigger than 2 00:04:02.590 --> 00:04:06.640 align:middle line:84% over epsilon, but I'm going to phrase it this way. 00:04:06.640 --> 00:04:08.980 align:middle line:84% Now we have to show that it works-- namely, 00:04:08.980 --> 00:04:11.140 align:middle line:84% if I take n bigger than or equal to capital M 00:04:11.140 --> 00:04:13.270 align:middle line:84% and k bigger than or equal to a capital M, 00:04:13.270 --> 00:04:15.940 align:middle line:84% then this difference is less than epsilon. 00:04:15.940 --> 00:04:21.910 align:middle line:84% Then if n is bigger than or equal to M, 00:04:21.910 --> 00:04:26.620 align:middle line:84% k is bigger than or equal to M, and I look at 1 over n minus 1 00:04:26.620 --> 00:04:31.090 align:middle line:84% over k, this is less than or equal to, by the triangle 00:04:31.090 --> 00:04:34.540 align:middle line:84% inequality, the absolute value of each 00:04:34.540 --> 00:04:39.550 align:middle line:84% of these added together, which is just 1 over n plus 1 over k. 00:04:39.550 --> 00:04:42.610 align:middle line:84% And since these are both bigger than or equal to M, 00:04:42.610 --> 00:04:46.315 align:middle line:84% each 1 is less than or equal to 1 over M 1 00:04:46.315 --> 00:04:50.950 align:middle line:84% over M, so I get 2 over M, which, by our choice of M, 00:04:50.950 --> 00:04:54.360 align:middle line:90% is less than epsilon. 00:04:54.360 --> 00:05:01.690 align:middle line:84% So x of n equals 1 over n is an example of a Cauchy sequence. 00:05:01.690 --> 00:05:10.370 align:middle line:84% So let's negate the definition, and then we'll 00:05:10.370 --> 00:05:12.600 align:middle line:84% look at an example of a sequence which is not Cauchy. 00:05:12.600 --> 00:05:15.225 align:middle line:84% And as you'll probably guess, if this is our favorite sequence, 00:05:15.225 --> 00:05:19.760 align:middle line:84% which converges, our favorite sequence which doesn't converge 00:05:19.760 --> 00:05:23.645 align:middle line:84% will be an example of a sequence which is not Cauchy. 00:05:23.645 --> 00:05:25.520 align:middle line:84% And this should shouldn't come as a surprise. 00:05:25.520 --> 00:05:30.080 align:middle line:84% Because, again, a sequence which is Cauchy, if you 00:05:30.080 --> 00:05:33.410 align:middle line:84% go far enough out, any two entries 00:05:33.410 --> 00:05:35.150 align:middle line:90% are close to each other. 00:05:35.150 --> 00:05:37.370 align:middle line:84% But if we look at, for example, the sequence minus 1 00:05:37.370 --> 00:05:40.010 align:middle line:84% to the n, which is just minus 1 plus 1 minus 1, 00:05:40.010 --> 00:05:44.997 align:middle line:90% any two entries will differ by-- 00:05:44.997 --> 00:05:46.580 align:middle line:84% or you can always choose two entries-- 00:05:46.580 --> 00:05:52.410 align:middle line:90% that differ by 2 in distance. 00:05:52.410 --> 00:06:00.880 align:middle line:84% So let's negate this definition to get what it means 00:06:00.880 --> 00:06:03.280 align:middle line:90% for something not to be Cauchy. 00:06:03.280 --> 00:06:06.700 align:middle line:90% So we'll not write all that out. 00:06:06.700 --> 00:06:13.450 align:middle line:90% 00:06:13.450 --> 00:06:18.100 align:middle line:90% So x of n is not Cauchy if-- 00:06:18.100 --> 00:06:20.290 align:middle line:84% so every time we see a "for all," 00:06:20.290 --> 00:06:24.040 align:middle line:90% it becomes "there exists." 00:06:24.040 --> 00:06:29.140 align:middle line:84% If there exists a bad epsilon 0 positive 00:06:29.140 --> 00:06:34.030 align:middle line:84% such that for all M, a natural number, 00:06:34.030 --> 00:06:38.710 align:middle line:84% you can find two entries further out than M 00:06:38.710 --> 00:06:41.990 align:middle line:84% that are greater than epsilon 0 distance to each other. 00:06:41.990 --> 00:06:45.430 align:middle line:84% So there exists n bigger than or equal to M 00:06:45.430 --> 00:06:53.110 align:middle line:84% and k bigger than or equal to M such that x of n minus x of k 00:06:53.110 --> 00:06:55.000 align:middle line:84% is bigger than or equal to this bad epsilon. 00:06:55.000 --> 00:06:57.720 align:middle line:90% 00:06:57.720 --> 00:07:00.250 align:middle line:90% OK 00:07:00.250 --> 00:07:03.280 align:middle line:84% Again, the definition of Cauchy means 00:07:03.280 --> 00:07:11.610 align:middle line:84% that, as long as I go far enough out in the sequence, 00:07:11.610 --> 00:07:14.330 align:middle line:84% this distance is supposed to be less than epsilon. 00:07:14.330 --> 00:07:17.460 align:middle line:84% So for all epsilon positive, there exists a capital M 00:07:17.460 --> 00:07:19.080 align:middle line:90% so that I have this picture. 00:07:19.080 --> 00:07:23.610 align:middle line:84% If I choose x sub k plus 1, then it should also 00:07:23.610 --> 00:07:28.830 align:middle line:84% be within distance epsilon to x sub n or x sub k. 00:07:28.830 --> 00:07:31.410 align:middle line:84% So they're getting closer and closer together. 00:07:31.410 --> 00:07:34.680 align:middle line:84% The negation means that they're not getting closer and closer 00:07:34.680 --> 00:07:35.830 align:middle line:90% together to each other. 00:07:35.830 --> 00:07:39.100 align:middle line:84% So there exists some small distance 00:07:39.100 --> 00:07:43.230 align:middle line:84% so that you can always go as far out as you want and find 00:07:43.230 --> 00:07:47.700 align:middle line:84% two entries that are greater than epsilon 0 distance 00:07:47.700 --> 00:07:48.480 align:middle line:90% to each other. 00:07:48.480 --> 00:07:51.510 align:middle line:90% 00:07:51.510 --> 00:07:53.570 align:middle line:90% So what's an example of that? 00:07:53.570 --> 00:08:06.750 align:middle line:84% Like I said, the sequence minus 1 to the n is not Cauchy. 00:08:06.750 --> 00:08:17.040 align:middle line:90% 00:08:17.040 --> 00:08:18.360 align:middle line:90% That just doesn't look right. 00:08:18.360 --> 00:08:18.990 align:middle line:90% There we go. 00:08:18.990 --> 00:08:23.980 align:middle line:90% 00:08:23.980 --> 00:08:24.760 align:middle line:90% Now it is. 00:08:24.760 --> 00:08:27.940 align:middle line:90% 00:08:27.940 --> 00:08:29.290 align:middle line:90% So this is not Cauchy. 00:08:29.290 --> 00:08:34.059 align:middle line:84% So that means there should exist some bad epsilon 0. 00:08:34.059 --> 00:08:36.940 align:middle line:84% So I can go as far out as I want and find 00:08:36.940 --> 00:08:40.179 align:middle line:84% two entries in the sequence differing from each other 00:08:40.179 --> 00:08:42.380 align:middle line:90% by epsilon 0 in distance. 00:08:42.380 --> 00:08:48.940 align:middle line:84% So basically, I can always find two entries 00:08:48.940 --> 00:08:52.540 align:middle line:84% in the sequence which differ from each other by 2. 00:08:52.540 --> 00:08:56.050 align:middle line:90% So that'll be my bad epsilon 0. 00:08:56.050 --> 00:09:00.010 align:middle line:90% So if you like, here's a proof. 00:09:00.010 --> 00:09:03.865 align:middle line:90% Choose epsilon 0 equals 2. 00:09:03.865 --> 00:09:06.830 align:middle line:90% 00:09:06.830 --> 00:09:08.250 align:middle line:90% Let M be a natural number. 00:09:08.250 --> 00:09:12.230 align:middle line:84% So now we have to find an element of entries 00:09:12.230 --> 00:09:15.642 align:middle line:84% in the sequence further out than M whose distance to each other 00:09:15.642 --> 00:09:16.850 align:middle line:90% is bigger than or equal to 2. 00:09:16.850 --> 00:09:25.090 align:middle line:84% We can just take M plus 1 and capital M. Choose n 00:09:25.090 --> 00:09:29.680 align:middle line:90% equals M and k equals M plus 1. 00:09:29.680 --> 00:09:33.550 align:middle line:84% So these are both bigger than or equal to M. 00:09:33.550 --> 00:09:39.070 align:middle line:84% Then minus 1 to the n minus 1 to the k, 00:09:39.070 --> 00:09:47.220 align:middle line:84% this is equal to 1 minus minus 1 after I factor out a minus 1 00:09:47.220 --> 00:09:49.410 align:middle line:84% to the capital M, which equals 2. 00:09:49.410 --> 00:09:53.950 align:middle line:90% 00:09:53.950 --> 00:09:58.045 align:middle line:84% So minus 1 to the n is not Cauchy. 00:09:58.045 --> 00:10:05.750 align:middle line:90% 00:10:05.750 --> 00:10:09.440 align:middle line:84% So I, at the beginning, said that this 00:10:09.440 --> 00:10:14.150 align:middle line:84% will look like a definition of a new type of sequence, 00:10:14.150 --> 00:10:15.740 align:middle line:90% but it's not, really. 00:10:15.740 --> 00:10:27.640 align:middle line:84% So as it turns out, the elements of the sequence 00:10:27.640 --> 00:10:30.010 align:middle line:84% are getting closer and closer together 00:10:30.010 --> 00:10:33.020 align:middle line:90% as you go far enough out. 00:10:33.020 --> 00:10:33.520 align:middle line:90% So 00:10:33.520 --> 00:10:36.460 align:middle line:84% They're all kind of clustering near each other, which kind of 00:10:36.460 --> 00:10:38.770 align:middle line:84% makes you think they're all clustering near something 00:10:38.770 --> 00:10:41.410 align:middle line:90% in the real number line. 00:10:41.410 --> 00:10:44.620 align:middle line:84% And therefore, maybe, the sequence is convergent. 00:10:44.620 --> 00:10:50.370 align:middle line:84% Now, this is true-- and we'll prove this-- 00:10:50.370 --> 00:10:54.000 align:middle line:84% that a sequence is Cauchy if and only if it's convergent. 00:10:54.000 --> 00:10:56.430 align:middle line:84% Now, this is true only for the real numbers. 00:10:56.430 --> 00:10:59.377 align:middle line:84% And I'll say a little bit about this in a minute-- 00:10:59.377 --> 00:11:00.960 align:middle line:84% or not only true for the real numbers, 00:11:00.960 --> 00:11:04.690 align:middle line:84% but it's not true for the rational numbers. 00:11:04.690 --> 00:11:06.530 align:middle line:84% And I'll get to that in just a second. 00:11:06.530 --> 00:11:10.090 align:middle line:84% So what we're going to prove is that a sequence is Cauchy if 00:11:10.090 --> 00:11:13.070 align:middle line:90% and only if it is convergent. 00:11:13.070 --> 00:11:18.920 align:middle line:84% So the first thing I want to show 00:11:18.920 --> 00:11:25.695 align:middle line:84% is that Cauchy sequences are bounded. 00:11:25.695 --> 00:11:35.360 align:middle line:90% 00:11:35.360 --> 00:11:38.810 align:middle line:84% So the proof of this statement is essentially 00:11:38.810 --> 00:11:44.160 align:middle line:84% the same as the proof that convergent sequences 00:11:44.160 --> 00:11:44.660 align:middle line:90% are bounded. 00:11:44.660 --> 00:11:48.630 align:middle line:84% So let me draw a picture that goes along with this proof. 00:11:48.630 --> 00:11:51.140 align:middle line:84% So as long as I go far enough out, 00:11:51.140 --> 00:11:57.530 align:middle line:84% there exists an M so that for all n bigger 00:11:57.530 --> 00:12:00.200 align:middle line:84% than or equal to capital M, I can say this. 00:12:00.200 --> 00:12:04.400 align:middle line:84% Let's look at this entry x of M. Then for all n bigger 00:12:04.400 --> 00:12:06.890 align:middle line:84% than or equal to capital M, all of the other entries 00:12:06.890 --> 00:12:10.490 align:middle line:84% have to be within a certain distance to x sub n, 00:12:10.490 --> 00:12:13.160 align:middle line:84% based on the definition of Cauchy. 00:12:13.160 --> 00:12:16.550 align:middle line:84% So let's say I make that distance 1. 00:12:16.550 --> 00:12:20.510 align:middle line:90% 00:12:20.510 --> 00:12:23.900 align:middle line:84% And let's say 0's over here, just for this picture. 00:12:23.900 --> 00:12:28.400 align:middle line:84% So then for all n for all n bigger 00:12:28.400 --> 00:12:31.215 align:middle line:84% than or equal to capital M, x sub n lies in this interval 00:12:31.215 --> 00:12:31.715 align:middle line:90% here. 00:12:31.715 --> 00:12:34.450 align:middle line:90% 00:12:34.450 --> 00:12:43.560 align:middle line:84% And therefore, we'll get that x of n is bounded. 00:12:43.560 --> 00:12:46.700 align:middle line:84% So the way this picture looks, I'm going to write it this way. 00:12:46.700 --> 00:12:47.780 align:middle line:90% It's 1 plus 1. 00:12:47.780 --> 00:12:50.660 align:middle line:90% 00:12:50.660 --> 00:12:54.110 align:middle line:84% Now, that handles all n bigger than or equal to capital M. 00:12:54.110 --> 00:12:56.605 align:middle line:84% So we just need to deal with the first capital 00:12:56.605 --> 00:12:59.960 align:middle line:90% M minus 1 other guy. 00:12:59.960 --> 00:13:04.700 align:middle line:84% So maybe there's capital M minus 1 is over here. 00:13:04.700 --> 00:13:06.290 align:middle line:90% Capital X sub 1 is over there. 00:13:06.290 --> 00:13:08.100 align:middle line:90% Capital X sub 2 is over here. 00:13:08.100 --> 00:13:11.150 align:middle line:84% So then our bound will just be this one, 00:13:11.150 --> 00:13:14.030 align:middle line:84% which handles all of the n bigger than or equal to M 00:13:14.030 --> 00:13:20.135 align:middle line:84% plus the absolute values of these guys that we missed. 00:13:20.135 --> 00:13:24.810 align:middle line:90% 00:13:24.810 --> 00:13:35.140 align:middle line:84% So since xn is Cauchy, there exists 00:13:35.140 --> 00:13:40.050 align:middle line:84% and M, a natural number, such that for all n 00:13:40.050 --> 00:13:44.970 align:middle line:84% bigger than or equal to M, and k bigger 00:13:44.970 --> 00:13:54.130 align:middle line:84% than or equal to M, x sub n minus x sub k is less than 1 00:13:54.130 --> 00:13:56.510 align:middle line:90% in distance. 00:13:56.510 --> 00:13:59.920 align:middle line:84% So this is certainly true for k equals 00:13:59.920 --> 00:14:04.750 align:middle line:84% capital M. So this implies for all n bigger 00:14:04.750 --> 00:14:09.040 align:middle line:84% than or equal to M, x sub n minus x sub capital 00:14:09.040 --> 00:14:11.200 align:middle line:90% M is less than 1. 00:14:11.200 --> 00:14:15.940 align:middle line:84% So now, if I use the triangle inequality, 00:14:15.940 --> 00:14:18.790 align:middle line:84% I can show that the previous implies 00:14:18.790 --> 00:14:21.100 align:middle line:84% that for all n bigger than or equal to M, 00:14:21.100 --> 00:14:23.530 align:middle line:84% if I look at the absolute value of x sub n, 00:14:23.530 --> 00:14:32.020 align:middle line:84% this is equal to x sub n minus x sub capital M plus capital M. 00:14:32.020 --> 00:14:34.330 align:middle line:84% And this is less than or equal to the absolute value 00:14:34.330 --> 00:14:40.300 align:middle line:84% of this guy plus the absolute value of this guy. 00:14:40.300 --> 00:14:42.250 align:middle line:90% And this is bounded by 1. 00:14:42.250 --> 00:14:46.690 align:middle line:90% 00:14:46.690 --> 00:14:52.530 align:middle line:84% So in summary, I've shown that for all n bigger 00:14:52.530 --> 00:14:57.550 align:middle line:84% than or equal to this fixed integer, capital M, x 00:14:57.550 --> 00:15:03.450 align:middle line:84% sub n is less than or equal to x sub capital 00:15:03.450 --> 00:15:07.987 align:middle line:90% M in absolute value plus 1. 00:15:07.987 --> 00:15:09.570 align:middle line:84% So that's for all little n bigger than 00:15:09.570 --> 00:15:11.070 align:middle line:84% or equal to capital M. So now I just 00:15:11.070 --> 00:15:18.280 align:middle line:84% need to pick a big enough number that bounds the first capital M 00:15:18.280 --> 00:15:21.520 align:middle line:84% entries that are not covered by this inequality. 00:15:21.520 --> 00:15:22.525 align:middle line:90% Capital M is fixed. 00:15:22.525 --> 00:15:26.830 align:middle line:90% 00:15:26.830 --> 00:15:32.350 align:middle line:84% So let B be the absolute value of x sub 1 00:15:32.350 --> 00:15:43.770 align:middle line:84% plus absolute value of x sub 2 plus this fixed number now. 00:15:43.770 --> 00:15:47.490 align:middle line:90% 00:15:47.490 --> 00:15:51.890 align:middle line:84% Then for all n bigger than or equal to capital M, 00:15:51.890 --> 00:15:55.170 align:middle line:84% I have, by this inequality up here-- 00:15:55.170 --> 00:15:58.560 align:middle line:90% 00:15:58.560 --> 00:16:00.505 align:middle line:84% this is a sum of non-negative numbers, 00:16:00.505 --> 00:16:02.130 align:middle line:84% so this number is certainly bigger than 00:16:02.130 --> 00:16:03.555 align:middle line:90% or equal to just this part. 00:16:03.555 --> 00:16:09.390 align:middle line:90% 00:16:09.390 --> 00:16:18.610 align:middle line:84% And if I have n bigger than equal to 1 and less than M, 00:16:18.610 --> 00:16:24.120 align:middle line:84% then x of n, the absolute value of this guy 00:16:24.120 --> 00:16:26.850 align:middle line:84% is going to be one of these that appears here, which 00:16:26.850 --> 00:16:30.390 align:middle line:84% is certainly less than or equal to if I add 00:16:30.390 --> 00:16:34.020 align:middle line:84% on this number and the others, which is less than 00:16:34.020 --> 00:16:39.100 align:middle line:84% or equal to B. So now I've found a B which is non-negative which 00:16:39.100 --> 00:16:40.690 align:middle line:90% bounds all the absolute values. 00:16:40.690 --> 00:16:44.860 align:middle line:84% And therefore, this proves that the sequence is bounded. 00:16:44.860 --> 00:16:54.160 align:middle line:90% 00:16:54.160 --> 00:16:58.810 align:middle line:84% So we've shown that a Cauchy sequence is bounded. 00:16:58.810 --> 00:17:01.470 align:middle line:90% 00:17:01.470 --> 00:17:12.349 align:middle line:84% And so what I'm now going to show is the following. 00:17:12.349 --> 00:17:16.099 align:middle line:84% So again, all of the entries are getting close to each other. 00:17:16.099 --> 00:17:18.260 align:middle line:84% They're kind of clustering near each other. 00:17:18.260 --> 00:17:23.609 align:middle line:84% So it kind of feels like they want to converge. 00:17:23.609 --> 00:17:25.950 align:middle line:84% And this next theorem says that, well, 00:17:25.950 --> 00:17:29.880 align:middle line:84% if you've identified a limit along a subsequence, 00:17:29.880 --> 00:17:32.830 align:middle line:84% then, in fact, the entire sequence converges. 00:17:32.830 --> 00:17:36.280 align:middle line:84% So of course, this is not true for an arbitrary sequence. 00:17:36.280 --> 00:17:39.270 align:middle line:90% If a subsequence converges-- 00:17:39.270 --> 00:17:42.540 align:middle line:84% or I should say, for an arbitrary sequence, 00:17:42.540 --> 00:17:45.180 align:middle line:84% it's not true that a subsequence converging implies 00:17:45.180 --> 00:17:47.100 align:middle line:90% a full sequence converging. 00:17:47.100 --> 00:17:51.660 align:middle line:84% We have minus 1 to the n for which a subsequence converges, 00:17:51.660 --> 00:17:53.790 align:middle line:84% but the whole sequence does not converge. 00:17:53.790 --> 00:17:56.190 align:middle line:84% But if we make the additional hypothesis 00:17:56.190 --> 00:17:59.580 align:middle line:84% that the sequence is Cauchy, then the sequence 00:17:59.580 --> 00:18:03.340 align:middle line:84% converges if and only if that subsequence converges. 00:18:03.340 --> 00:18:06.060 align:middle line:84% So the statement of the theorem is following. 00:18:06.060 --> 00:18:12.380 align:middle line:90% 00:18:12.380 --> 00:18:34.620 align:middle line:84% If x sub n is Cauchy and there exists a subsequence which 00:18:34.620 --> 00:18:37.680 align:middle line:90% is converging to some number-- 00:18:37.680 --> 00:18:50.035 align:middle line:84% call it x-- then the whole sequence converges to x. 00:18:50.035 --> 00:18:53.500 align:middle line:90% 00:18:53.500 --> 00:18:56.290 align:middle line:84% So what I was saying right before I stated this theorem is 00:18:56.290 --> 00:18:58.120 align:middle line:84% that if I hide this part and just say, 00:18:58.120 --> 00:19:01.600 align:middle line:84% there exists a subsequence which is converging to x, 00:19:01.600 --> 00:19:04.270 align:middle line:84% this does not imply that the full sequence converges to x. 00:19:04.270 --> 00:19:07.060 align:middle line:84% Because we had this example of minus 1 to the n. 00:19:07.060 --> 00:19:10.900 align:middle line:84% But if I also assume the sequence is Cauchy, 00:19:10.900 --> 00:19:14.110 align:middle line:84% then it does follow that Cauchy plus subsequence 00:19:14.110 --> 00:19:16.720 align:middle line:84% converging implies the full sequence converges. 00:19:16.720 --> 00:19:21.960 align:middle line:90% 00:19:21.960 --> 00:19:26.520 align:middle line:84% So I want to show that xn converges to x. 00:19:26.520 --> 00:19:27.795 align:middle line:90% So want to show-- 00:19:27.795 --> 00:19:36.180 align:middle line:90% 00:19:36.180 --> 00:19:41.310 align:middle line:84% and we're going to do this just by using the definition, 00:19:41.310 --> 00:19:43.200 align:middle line:84% by verifying this through the definition, 00:19:43.200 --> 00:19:47.018 align:middle line:84% not using the squeeze theorem or anything like that. 00:19:47.018 --> 00:19:51.120 align:middle line:90% So let epsilon be positive. 00:19:51.120 --> 00:19:58.260 align:middle line:90% 00:19:58.260 --> 00:20:15.790 align:middle line:84% Since xn is Cauchy, there exist M0, a natural number 00:20:15.790 --> 00:20:20.320 align:middle line:84% such that for all n bigger than or equal to M0 00:20:20.320 --> 00:20:27.145 align:middle line:84% and k bigger than or equal to M0, x 00:20:27.145 --> 00:20:32.200 align:middle line:84% sub n minus x sub k is less than epsilon over 2. 00:20:32.200 --> 00:20:35.200 align:middle line:90% Why this epsilon over 2? 00:20:35.200 --> 00:20:36.730 align:middle line:84% Or why should you not be surprised? 00:20:36.730 --> 00:20:38.720 align:middle line:84% Well, we have two assumptions here. 00:20:38.720 --> 00:20:44.230 align:middle line:84% So like we did when we did convergence of products 00:20:44.230 --> 00:20:49.090 align:middle line:84% of sequences and so on, which had two assumptions, 00:20:49.090 --> 00:20:52.360 align:middle line:84% namely two sequences converged to something, typically, 00:20:52.360 --> 00:20:54.160 align:middle line:84% that means we'll have two integers coming. 00:20:54.160 --> 00:20:58.300 align:middle line:84% We'll choose a bigger integer and then some inequalities 00:20:58.300 --> 00:20:59.240 align:middle line:90% to get an epsilon. 00:20:59.240 --> 00:21:02.320 align:middle line:84% So that's a little bit of a rambling answer 00:21:02.320 --> 00:21:07.240 align:middle line:84% to why we get an epsilon over 2 here, or why we put one there. 00:21:07.240 --> 00:21:12.690 align:middle line:90% 00:21:12.690 --> 00:21:19.920 align:middle line:84% Since the subsequence-- so this subsequence converges to x-- 00:21:19.920 --> 00:21:28.670 align:middle line:90% 00:21:28.670 --> 00:21:33.590 align:middle line:84% there exists another integer, M sub 1 such 00:21:33.590 --> 00:21:41.540 align:middle line:84% that if k is bigger than or equal to M sub 1, 00:21:41.540 --> 00:21:50.840 align:middle line:84% then x sub n sub k minus x is less than epsilon over 2. 00:21:50.840 --> 00:21:53.810 align:middle line:84% So maybe I should have used a different letter here. 00:21:53.810 --> 00:21:55.675 align:middle line:90% Let's use a little m. 00:21:55.675 --> 00:21:57.050 align:middle line:84% Because I don't want you to think 00:21:57.050 --> 00:21:58.430 align:middle line:90% these have to be the same k. 00:21:58.430 --> 00:22:07.950 align:middle line:90% 00:22:07.950 --> 00:22:15.900 align:middle line:84% So now we'll choose an integer bigger than both M1 and M2 00:22:15.900 --> 00:22:17.320 align:middle line:90% and show that it works. 00:22:17.320 --> 00:22:23.190 align:middle line:90% Choose M to be M0 plus M1. 00:22:23.190 --> 00:22:39.470 align:middle line:90% 00:22:39.470 --> 00:22:41.510 align:middle line:90% Now we need to show this works. 00:22:41.510 --> 00:22:53.130 align:middle line:84% And if n is bigger than or equal to M, so let me, 00:22:53.130 --> 00:22:59.150 align:middle line:84% actually, make a first observation 00:22:59.150 --> 00:23:02.720 align:middle line:84% before I go to the n bigger than or equal to capital M. 00:23:02.720 --> 00:23:13.350 align:middle line:84% Then, since n sub k is bigger than or equal to k for all 00:23:13.350 --> 00:23:16.290 align:middle line:84% of k, a natural number-- just because the n sub k is there 00:23:16.290 --> 00:23:19.130 align:middle line:84% in increasing sequence of integers, 00:23:19.130 --> 00:23:20.810 align:middle line:90% which starts at least at 1-- 00:23:20.810 --> 00:23:24.430 align:middle line:90% 00:23:24.430 --> 00:23:28.120 align:middle line:84% and since n sub k is bigger than or equal to k for all k, 00:23:28.120 --> 00:23:33.520 align:middle line:84% this implies that the integer n sub capital M is bigger 00:23:33.520 --> 00:23:35.110 align:middle line:84% than or equal to M, which, remember, 00:23:35.110 --> 00:23:41.080 align:middle line:84% is M0 plus M1, which implies that n sub M is 00:23:41.080 --> 00:23:45.190 align:middle line:84% bigger than or equal to M0 and n sub M is 00:23:45.190 --> 00:23:48.010 align:middle line:90% bigger than or equal to M1. 00:23:48.010 --> 00:23:54.310 align:middle line:84% So I just wanted to make this preliminary observation. 00:23:54.310 --> 00:23:57.145 align:middle line:84% And now we'll go to showing that this capital M works. 00:23:57.145 --> 00:24:00.360 align:middle line:90% 00:24:00.360 --> 00:24:02.700 align:middle line:84% So now, if n is bigger than or equal to capital 00:24:02.700 --> 00:24:08.170 align:middle line:84% M, and I look at x sub n minus x, an absolute value, 00:24:08.170 --> 00:24:20.610 align:middle line:84% and add and subtract x sub n capital M capital M minus x 00:24:20.610 --> 00:24:34.360 align:middle line:84% and use the triangle inequality, then-- 00:24:34.360 --> 00:24:38.020 align:middle line:84% so since n is bigger than or equal to capital M, which 00:24:38.020 --> 00:24:40.360 align:middle line:84% is bigger than or equal to M0, that 00:24:40.360 --> 00:24:43.210 align:middle line:84% means n is bigger than or equal to M0. 00:24:43.210 --> 00:24:46.960 align:middle line:84% And then n sub M we just showed is bigger than or equal to 0. 00:24:46.960 --> 00:24:52.840 align:middle line:84% So by this inequality, I get that the first term 00:24:52.840 --> 00:24:55.780 align:middle line:90% is less than epsilon over 2. 00:24:55.780 --> 00:25:01.840 align:middle line:84% And now, so M is certainly bigger than 00:25:01.840 --> 00:25:03.880 align:middle line:90% or equal to M sub 1. 00:25:03.880 --> 00:25:09.940 align:middle line:84% And therefore, I will get that this part is less than epsilon 00:25:09.940 --> 00:25:14.180 align:middle line:84% over 2 because of this inequality. 00:25:14.180 --> 00:25:22.650 align:middle line:84% So that choice of capital M works. 00:25:22.650 --> 00:25:27.780 align:middle line:90% 00:25:27.780 --> 00:25:35.800 align:middle line:84% And now, we'll prove the following, 00:25:35.800 --> 00:25:39.550 align:middle line:84% that a sequence is convergent if and only if it's Cauchy. 00:25:39.550 --> 00:26:07.140 align:middle line:90% 00:26:07.140 --> 00:26:08.890 align:middle line:90% So this is a two-way street. 00:26:08.890 --> 00:26:12.270 align:middle line:84% So we need to show the left implies the right 00:26:12.270 --> 00:26:13.935 align:middle line:84% and then the right implies the left. 00:26:13.935 --> 00:26:20.800 align:middle line:90% 00:26:20.800 --> 00:26:30.557 align:middle line:84% So this direction is, in fact, easy. 00:26:30.557 --> 00:26:31.640 align:middle line:90% Based on what we've done-- 00:26:31.640 --> 00:26:32.780 align:middle line:90% I shouldn't say it's easy-- 00:26:32.780 --> 00:26:35.750 align:middle line:84% but what we've done so far, it quickly follows. 00:26:35.750 --> 00:26:42.310 align:middle line:90% 00:26:42.310 --> 00:26:45.450 align:middle line:84% So we're assuming x sub n is Cauchy. 00:26:45.450 --> 00:26:46.920 align:middle line:84% I'm trying to show it's convergent. 00:26:46.920 --> 00:26:52.250 align:middle line:90% 00:26:52.250 --> 00:26:55.160 align:middle line:84% So if x sub n is Cauchy, this implies 00:26:55.160 --> 00:27:02.240 align:middle line:84% that x sub n is bounded, the sequence is bounded, 00:27:02.240 --> 00:27:05.720 align:middle line:84% which implies by the Bolzano-Weierstrass theorem 00:27:05.720 --> 00:27:20.830 align:middle line:84% that x sub n has a convergent subsequence. 00:27:20.830 --> 00:27:22.720 align:middle line:84% And by the theorem we just proved, 00:27:22.720 --> 00:27:26.530 align:middle line:84% if a Cauchy sequence has a convergent subsequence, 00:27:26.530 --> 00:27:27.850 align:middle line:90% it must be convergent. 00:27:27.850 --> 00:27:36.050 align:middle line:90% 00:27:36.050 --> 00:27:39.440 align:middle line:84% Now, for the converse direction, that xn 00:27:39.440 --> 00:27:44.570 align:middle line:84% is convergent implies xn is Cauchy, well, so this 00:27:44.570 --> 00:27:45.830 align:middle line:90% should not come as a surprise. 00:27:45.830 --> 00:27:46.747 align:middle line:90% Let me draw a picture. 00:27:46.747 --> 00:27:50.510 align:middle line:90% 00:27:50.510 --> 00:27:55.860 align:middle line:84% Let's suppose x sub n is converging to x, 00:27:55.860 --> 00:27:57.630 align:middle line:90% and epsilon is positive. 00:27:57.630 --> 00:28:00.550 align:middle line:90% 00:28:00.550 --> 00:28:03.550 align:middle line:84% Then, since the xn's are converging 00:28:03.550 --> 00:28:12.850 align:middle line:84% to x, if I draw a little interval around x 00:28:12.850 --> 00:28:15.920 align:middle line:90% of total length epsilon-- 00:28:15.920 --> 00:28:19.300 align:middle line:84% so x minus epsilon over 2 and x plus epsilon over 2-- 00:28:19.300 --> 00:28:24.910 align:middle line:84% then I will find, as long as so then there exists M 00:28:24.910 --> 00:28:29.800 align:middle line:84% so that, for all n bigger than or equal to capital M, 00:28:29.800 --> 00:28:35.470 align:middle line:84% all of the x sub n's lie in this interval. 00:28:35.470 --> 00:28:41.540 align:middle line:90% 00:28:41.540 --> 00:28:43.290 align:middle line:84% They all lie in this interval because they 00:28:43.290 --> 00:28:45.090 align:middle line:84% have to be within distance epsilon over 2 00:28:45.090 --> 00:28:50.550 align:middle line:84% to x if the xn's are converging to x. 00:28:50.550 --> 00:28:52.410 align:middle line:84% And since they lie in this interval, 00:28:52.410 --> 00:28:54.780 align:middle line:84% the distance between any two of them 00:28:54.780 --> 00:28:57.480 align:middle line:84% can only be as big as the length of the interval, which 00:28:57.480 --> 00:29:00.270 align:middle line:90% is epsilon. 00:29:00.270 --> 00:29:04.050 align:middle line:84% So this is essentially the picture 00:29:04.050 --> 00:29:06.720 align:middle line:84% of why a convergence sequence has to be Cauchy. 00:29:06.720 --> 00:29:08.520 align:middle line:84% So now let's turn this picture into math. 00:29:08.520 --> 00:29:13.200 align:middle line:90% 00:29:13.200 --> 00:29:15.540 align:middle line:84% We have to verify xn is Cauchy through the definition. 00:29:15.540 --> 00:29:16.560 align:middle line:90% That's all we have. 00:29:16.560 --> 00:29:20.040 align:middle line:90% So let epsilon be positive. 00:29:20.040 --> 00:29:24.360 align:middle line:84% Since the xn's converge to x, there 00:29:24.360 --> 00:29:28.200 align:middle line:84% exists an integer M sub 0, a natural number, 00:29:28.200 --> 00:29:33.840 align:middle line:84% such that for all n bigger than or equal to M sub 0, x 00:29:33.840 --> 00:29:37.260 align:middle line:84% sub n minus x is less than epsilon over 2. 00:29:37.260 --> 00:29:42.860 align:middle line:90% 00:29:42.860 --> 00:29:47.510 align:middle line:84% And so we'll choose the M for our definition of Cauchy 00:29:47.510 --> 00:29:48.920 align:middle line:90% to be this M sub 0. 00:29:48.920 --> 00:29:52.520 align:middle line:90% 00:29:52.520 --> 00:29:56.180 align:middle line:84% And if n is bigger than or equal to M 00:29:56.180 --> 00:29:59.690 align:middle line:84% and k is bigger than or equal to M 00:29:59.690 --> 00:30:04.310 align:middle line:84% and I look at the absolute value of x sub n minus x sub k 00:30:04.310 --> 00:30:08.450 align:middle line:84% and add and subtract x and use the triangle inequality, 00:30:08.450 --> 00:30:10.100 align:middle line:84% this is less than or equal to x sub 00:30:10.100 --> 00:30:17.030 align:middle line:84% n minus x, an absolute value, plus x minus x sub k. 00:30:17.030 --> 00:30:22.130 align:middle line:84% Each of these is less than epsilon over 2 00:30:22.130 --> 00:30:24.118 align:middle line:84% since n is bigger than or equal to capital M, 00:30:24.118 --> 00:30:25.910 align:middle line:84% and k is bigger than or equal to capital M. 00:30:25.910 --> 00:30:30.110 align:middle line:84% So this is less than epsilon over 2 plus epsilon over 2 00:30:30.110 --> 00:30:31.400 align:middle line:90% equals epsilon. 00:30:31.400 --> 00:30:32.900 align:middle line:90% And therefore, xn is Cauchy. 00:30:32.900 --> 00:30:37.230 align:middle line:90% 00:30:37.230 --> 00:30:41.730 align:middle line:84% Now I want to make a brief remark 00:30:41.730 --> 00:30:44.970 align:middle line:90% about the previous theorem. 00:30:44.970 --> 00:30:47.940 align:middle line:90% 00:30:47.940 --> 00:30:51.120 align:middle line:84% So remember how this whole story started off? 00:30:51.120 --> 00:30:56.070 align:middle line:90% 00:30:56.070 --> 00:31:01.130 align:middle line:84% There was something wrong with the rational numbers, namely, 00:31:01.130 --> 00:31:03.890 align:middle line:84% they didn't contain the square root of 2. 00:31:03.890 --> 00:31:06.230 align:middle line:84% So we couldn't solve the algebraic equation 00:31:06.230 --> 00:31:09.020 align:middle line:90% x squared minus 2 equals 0. 00:31:09.020 --> 00:31:13.760 align:middle line:84% But this also, then, turned into the rationals 00:31:13.760 --> 00:31:17.480 align:middle line:84% not being complete in the sense of order. 00:31:17.480 --> 00:31:20.828 align:middle line:84% Not every non-empty bounded set had a supremum. 00:31:20.828 --> 00:31:22.745 align:middle line:84% It didn't have the least upper bound property. 00:31:22.745 --> 00:31:25.700 align:middle line:90% 00:31:25.700 --> 00:31:30.860 align:middle line:84% But you can also interpret this lack of having square root of 2 00:31:30.860 --> 00:31:37.100 align:middle line:84% as somehow saying that the rationals are 00:31:37.100 --> 00:31:39.030 align:middle line:90% incomplete in this sense. 00:31:39.030 --> 00:31:42.777 align:middle line:84% So hopefully, at the end of this class, 00:31:42.777 --> 00:31:44.360 align:middle line:84% we'll be able to get to metrics basis. 00:31:44.360 --> 00:31:48.390 align:middle line:90% But so what do I mean by that? 00:31:48.390 --> 00:31:50.210 align:middle line:84% Let's say I look at this statement 00:31:50.210 --> 00:31:55.080 align:middle line:84% now within the universe of rational numbers. 00:31:55.080 --> 00:31:59.260 align:middle line:90% 00:31:59.260 --> 00:32:18.290 align:middle line:84% So now, if this sequence is rational numbers-- 00:32:18.290 --> 00:32:22.360 align:middle line:84% meaning sequences are only sequences of rational numbers, 00:32:22.360 --> 00:32:27.370 align:middle line:84% limits are only elements of the rational numbers, 00:32:27.370 --> 00:32:32.470 align:middle line:84% epsilon is only a rational number, and so on-- then 00:32:32.470 --> 00:32:34.480 align:middle line:84% we still have many of the same theorems 00:32:34.480 --> 00:32:36.700 align:middle line:84% that we proved-- not all of them, 00:32:36.700 --> 00:32:39.980 align:middle line:84% and I'll indicate which ones don't hold. 00:32:39.980 --> 00:32:41.530 align:middle line:84% But if we only work in rationals, 00:32:41.530 --> 00:32:52.910 align:middle line:84% then we always do have a convergence implies Cauchy, 00:32:52.910 --> 00:32:57.140 align:middle line:84% meaning convergent sequences are Cauchy. 00:32:57.140 --> 00:33:10.880 align:middle line:84% But Cauchy sequences are not necessarily convergent. 00:33:10.880 --> 00:33:19.320 align:middle line:84% Again, what's the example here, or what's 00:33:19.320 --> 00:33:20.490 align:middle line:90% the intuitive example? 00:33:20.490 --> 00:33:31.540 align:middle line:84% Take x sub n so that x of n is in Q. 00:33:31.540 --> 00:33:38.160 align:middle line:84% And now viewed in the universe of real numbers, 00:33:38.160 --> 00:33:39.975 align:middle line:90% x sub n's converge to root 2. 00:33:39.975 --> 00:33:46.590 align:middle line:90% 00:33:46.590 --> 00:33:50.970 align:middle line:84% Then such a sequence would be a Cauchy sequence. 00:33:50.970 --> 00:33:53.310 align:middle line:90% We just proved that, basically. 00:33:53.310 --> 00:33:55.500 align:middle line:84% So such a sequence would be a Cauchy sequence 00:33:55.500 --> 00:33:57.210 align:middle line:90% of rational numbers. 00:33:57.210 --> 00:34:01.160 align:middle line:84% However, it would not converge in the set of rational numbers. 00:34:01.160 --> 00:34:02.910 align:middle line:84% It would converge to the square root of 2, 00:34:02.910 --> 00:34:04.780 align:middle line:90% which is not a rational number. 00:34:04.780 --> 00:34:09.820 align:middle line:84% So because the square root of 2 is not a rational number, 00:34:09.820 --> 00:34:11.739 align:middle line:84% this shows that the rational numbers do not 00:34:11.739 --> 00:34:16.419 align:middle line:84% have this completeness property that Cauchy sequences converge. 00:34:16.419 --> 00:34:22.340 align:middle line:90% 00:34:22.340 --> 00:34:27.210 align:middle line:84% So there's a whole, still, to this day, kind 00:34:27.210 --> 00:34:31.560 align:middle line:84% of industry of studying spaces for which Cauchy 00:34:31.560 --> 00:34:34.110 align:middle line:90% is equivalent to convergent. 00:34:34.110 --> 00:34:35.909 align:middle line:84% These are called complete metric spaces. 00:34:35.909 --> 00:34:37.742 align:middle line:84% And then if you add a little more structure, 00:34:37.742 --> 00:34:41.889 align:middle line:84% they're called Banach spaces and so on, 00:34:41.889 --> 00:34:43.639 align:middle line:84% which are very important, not just in math 00:34:43.639 --> 00:34:49.909 align:middle line:84% but also for formulating rigorously 00:34:49.909 --> 00:34:56.330 align:middle line:84% a lot of the underlying assumptions 00:34:56.330 --> 00:34:58.190 align:middle line:90% for mathematical physics. 00:34:58.190 --> 00:35:01.960 align:middle line:90% 00:35:01.960 --> 00:35:04.720 align:middle line:84% So if we're just looking inside the rationals, 00:35:04.720 --> 00:35:08.200 align:middle line:84% it does not follow that Cauchy sequences always converge. 00:35:08.200 --> 00:35:10.000 align:middle line:84% And now let's just stop for a minute 00:35:10.000 --> 00:35:13.430 align:middle line:84% and take stock of why this was true for the real numbers. 00:35:13.430 --> 00:35:15.186 align:middle line:90% What did we use going back? 00:35:15.186 --> 00:35:23.560 align:middle line:84% So if you really go back to the proof 00:35:23.560 --> 00:35:27.130 align:middle line:84% of the Bolzano-Weierstrass-- so that's what we used here 00:35:27.130 --> 00:35:29.380 align:middle line:84% to show that Cauchy sequences converge-- 00:35:29.380 --> 00:35:32.110 align:middle line:90% 00:35:32.110 --> 00:35:37.480 align:middle line:84% we use the fact that the lim sup and the lim inf always exists. 00:35:37.480 --> 00:35:41.110 align:middle line:84% And lim sup and lim inf, first off, they're 00:35:41.110 --> 00:35:43.270 align:middle line:84% defined to be sups and infs, which may not always 00:35:43.270 --> 00:35:46.150 align:middle line:84% exist as rational numbers, as we've already shown. 00:35:46.150 --> 00:35:51.430 align:middle line:84% So that's definitely a problem already there. 00:35:51.430 --> 00:35:55.450 align:middle line:84% But even more so, when we prove that every bounded monotone 00:35:55.450 --> 00:35:58.210 align:middle line:84% sequence converges, what we showed 00:35:58.210 --> 00:36:01.480 align:middle line:84% was that this limit is actually a sup of a certain set 00:36:01.480 --> 00:36:04.323 align:middle line:84% or an inf of a certain set, which, again, may or may not 00:36:04.323 --> 00:36:06.490 align:middle line:84% exist if we're just looking in the rational numbers. 00:36:06.490 --> 00:36:09.190 align:middle line:84% Because the rational numbers do not have the least upper bound 00:36:09.190 --> 00:36:10.420 align:middle line:90% property. 00:36:10.420 --> 00:36:14.590 align:middle line:84% So it really is the least upper bound property 00:36:14.590 --> 00:36:18.250 align:middle line:84% that gives us convergence equivalent 00:36:18.250 --> 00:36:20.420 align:middle line:90% to Cauchy for the real numbers. 00:36:20.420 --> 00:36:32.912 align:middle line:84% So for R, the least upper bound property is-- 00:36:32.912 --> 00:36:34.870 align:middle line:84% it has to be because that's the main thing that 00:36:34.870 --> 00:36:39.760 align:middle line:84% separates the two fields, but I'm just reiterating this 00:36:39.760 --> 00:36:41.320 align:middle line:90% here-- 00:36:41.320 --> 00:36:53.420 align:middle line:84% is the reason why convergent is equivalent to Cauchy. 00:36:53.420 --> 00:37:07.750 align:middle line:90% 00:37:07.750 --> 00:37:13.240 align:middle line:84% Now that I've proved that Cauchy is equivalent to convergence, 00:37:13.240 --> 00:37:15.970 align:middle line:84% maybe you'll ask, then why did we introduce it at all? 00:37:15.970 --> 00:37:18.480 align:middle line:90% 00:37:18.480 --> 00:37:23.440 align:middle line:84% If these two notions are the same, 00:37:23.440 --> 00:37:25.630 align:middle line:84% why even introduce them if they're just 00:37:25.630 --> 00:37:27.860 align:middle line:90% convergent sequences already? 00:37:27.860 --> 00:37:34.450 align:middle line:84% And the reason is because to show that a sequence converges, 00:37:34.450 --> 00:37:36.310 align:middle line:84% you have to somehow have your hands 00:37:36.310 --> 00:37:40.780 align:middle line:90% on a candidate for the limit. 00:37:40.780 --> 00:37:45.490 align:middle line:84% If you want to prove that xn converges, 00:37:45.490 --> 00:37:48.880 align:middle line:84% you have to somehow come up with an x that it converges to. 00:37:48.880 --> 00:37:55.114 align:middle line:84% And it's not always clear how to find that x. 00:37:55.114 --> 00:37:58.420 align:middle line:84% But Cauchy, although it's equivalent to a convergent 00:37:58.420 --> 00:38:01.870 align:middle line:84% in the set of real numbers, doesn't 00:38:01.870 --> 00:38:07.150 align:middle line:84% require you to find a candidate for convergence. 00:38:07.150 --> 00:38:09.820 align:middle line:84% All it requires you to do is show 00:38:09.820 --> 00:38:13.430 align:middle line:84% that, as long as you go far enough out, 00:38:13.430 --> 00:38:18.410 align:middle line:84% any two entries in the sequence are close together 00:38:18.410 --> 00:38:21.030 align:middle line:84% without requiring you to come up with a limit. 00:38:21.030 --> 00:38:23.180 align:middle line:84% See, computing limits is quite difficult. 00:38:23.180 --> 00:38:24.890 align:middle line:90% We're about to do series. 00:38:24.890 --> 00:38:28.040 align:middle line:84% And there's maybe, I don't know, five series people 00:38:28.040 --> 00:38:29.977 align:middle line:90% can compute explicitly. 00:38:29.977 --> 00:38:32.060 align:middle line:84% But you do know that there's a ton of other series 00:38:32.060 --> 00:38:34.760 align:middle line:84% that are actually convergent, even though you don't know what 00:38:34.760 --> 00:38:36.080 align:middle line:90% the limit as. 00:38:36.080 --> 00:38:37.730 align:middle line:90% And why do you know that? 00:38:37.730 --> 00:38:39.890 align:middle line:84% This is exactly because and exactly 00:38:39.890 --> 00:38:43.850 align:middle line:84% why people thought of Cauchy sequences 00:38:43.850 --> 00:38:46.440 align:middle line:84% to begin with in much of analysis. 00:38:46.440 --> 00:38:51.770 align:middle line:84% So again, just to summarize this, 00:38:51.770 --> 00:38:53.240 align:middle line:90% convergent sequences are nice. 00:38:53.240 --> 00:38:56.180 align:middle line:84% But in practice, it's difficult to get your hands 00:38:56.180 --> 00:38:58.280 align:middle line:84% on what could be a limit of a sequence, 00:38:58.280 --> 00:39:01.650 align:middle line:84% especially if that sequence is pretty complicated. 00:39:01.650 --> 00:39:05.150 align:middle line:84% So if you're trying to show a certain sequence converges, 00:39:05.150 --> 00:39:09.330 align:middle line:84% it suffices, by what we've done here, to show that it's Cauchy. 00:39:09.330 --> 00:39:12.840 align:middle line:84% And that's a little bit easier to do because that just 00:39:12.840 --> 00:39:15.325 align:middle line:84% requires you to work with the original sequence. 00:39:15.325 --> 00:39:16.950 align:middle line:84% You don't have to come up with a limit. 00:39:16.950 --> 00:39:21.360 align:middle line:84% You can just take your sequence and start playing directly 00:39:21.360 --> 00:39:27.370 align:middle line:84% with the entries rather than try to come up with a limit 00:39:27.370 --> 00:39:28.300 align:middle line:90% explicitly. 00:39:28.300 --> 00:39:32.020 align:middle line:90% 00:39:32.020 --> 00:39:34.480 align:middle line:84% So that's what we're going to move on to 00:39:34.480 --> 00:39:39.940 align:middle line:84% is series now, which, as I said a minute ago, 00:39:39.940 --> 00:39:48.750 align:middle line:84% is original reason why people started developing 00:39:48.750 --> 00:39:50.998 align:middle line:84% the foundations of analysis, what we're talking 00:39:50.998 --> 00:39:52.290 align:middle line:90% about right now, to begin with. 00:39:52.290 --> 00:39:55.640 align:middle line:90% 00:39:55.640 --> 00:40:01.970 align:middle line:84% Because they were just kind of doing very formal things 00:40:01.970 --> 00:40:07.040 align:middle line:84% that ended up not making sense, like they 00:40:07.040 --> 00:40:09.980 align:middle line:84% were adding infinitely many positive numbers 00:40:09.980 --> 00:40:13.028 align:middle line:84% and coming up with a negative number. 00:40:13.028 --> 00:40:14.280 align:middle line:90% Well, that can't be right. 00:40:14.280 --> 00:40:21.590 align:middle line:84% So all of this was created, discovered-- 00:40:21.590 --> 00:40:23.150 align:middle line:90% however you want to phrase it-- 00:40:23.150 --> 00:40:29.293 align:middle line:84% to put on rigorous foundations this next topic, 00:40:29.293 --> 00:40:29.960 align:middle line:90% which is series. 00:40:29.960 --> 00:40:35.200 align:middle line:90% 00:40:35.200 --> 00:40:38.710 align:middle line:84% And you dealt with series in calculus, 00:40:38.710 --> 00:40:40.595 align:middle line:90% so you know what a series is. 00:40:40.595 --> 00:40:42.220 align:middle line:84% Maybe you don't remember all the proofs 00:40:42.220 --> 00:40:43.570 align:middle line:90% of the properties of series. 00:40:43.570 --> 00:40:49.730 align:middle line:84% But suffice it to say, series is a pretty good motivation 00:40:49.730 --> 00:40:54.105 align:middle line:84% since it's one of the most useful things that comes out 00:40:54.105 --> 00:40:54.605 align:middle line:90% of math. 00:40:54.605 --> 00:40:58.790 align:middle line:90% 00:40:58.790 --> 00:41:05.000 align:middle line:84% Series expansions are how you solve ODEs, PDEs, 00:41:05.000 --> 00:41:06.020 align:middle line:90% Taylor expansions. 00:41:06.020 --> 00:41:09.350 align:middle line:84% All these things are, in some sense, a form of series. 00:41:09.350 --> 00:41:14.600 align:middle line:84% So being able to justify them as being real things 00:41:14.600 --> 00:41:17.160 align:middle line:90% is a necessity. 00:41:17.160 --> 00:41:25.820 align:middle line:84% So the definition of a series, for now, 00:41:25.820 --> 00:41:28.520 align:middle line:84% really is just this symbol I'm about to write down. 00:41:28.520 --> 00:41:37.465 align:middle line:84% So given a sequence x sub n, the symbol-- 00:41:37.465 --> 00:41:45.130 align:middle line:90% 00:41:45.130 --> 00:41:48.235 align:middle line:84% or maybe I'll just sometimes write just the sum x sub n-- 00:41:48.235 --> 00:41:50.800 align:middle line:90% 00:41:50.800 --> 00:42:00.110 align:middle line:84% is what's called the series associated to the sequence 00:42:00.110 --> 00:42:00.670 align:middle line:90% x sub n. 00:42:00.670 --> 00:42:03.607 align:middle line:90% 00:42:03.607 --> 00:42:05.065 align:middle line:84% So right now, that's just a symbol. 00:42:05.065 --> 00:42:07.600 align:middle line:90% 00:42:07.600 --> 00:42:10.060 align:middle line:84% We're going to interpret this as a real number 00:42:10.060 --> 00:42:11.590 align:middle line:90% in the following situation. 00:42:11.590 --> 00:42:15.040 align:middle line:90% 00:42:15.040 --> 00:42:28.380 align:middle line:84% We say that the series converges, 00:42:28.380 --> 00:42:48.890 align:middle line:84% if the following sequence given by s sub m equals-- 00:42:48.890 --> 00:42:54.060 align:middle line:90% 00:42:54.060 --> 00:42:57.120 align:middle line:84% so s sub m, this is the element of the sequence. 00:42:57.120 --> 00:42:57.960 align:middle line:90% And what is it? 00:42:57.960 --> 00:42:59.890 align:middle line:90% It is the actual sum. 00:42:59.890 --> 00:43:01.270 align:middle line:90% So this is not a formal thing. 00:43:01.270 --> 00:43:04.830 align:middle line:84% This is just a finite sum from n equals 1 to m. 00:43:04.830 --> 00:43:15.070 align:middle line:84% So this is-- so this sequence now, m equals 1 to infinity. 00:43:15.070 --> 00:43:20.000 align:middle line:90% 00:43:20.000 --> 00:43:34.070 align:middle line:84% And these guys we call partial sums converges. 00:43:34.070 --> 00:43:36.870 align:middle line:90% 00:43:36.870 --> 00:43:41.910 align:middle line:84% So right now, if we just have a sequence x sub n, 00:43:41.910 --> 00:43:46.410 align:middle line:84% the series associated to that sequence is just a symbol. 00:43:46.410 --> 00:43:48.210 align:middle line:84% We say that this series converges 00:43:48.210 --> 00:43:52.930 align:middle line:84% if this sequence of partial sums converge 00:43:52.930 --> 00:44:02.360 align:middle line:90% and if s is this limit. 00:44:02.360 --> 00:44:11.810 align:middle line:90% 00:44:11.810 --> 00:44:25.070 align:middle line:84% And we write s is equal to the series 00:44:25.070 --> 00:44:26.870 align:middle line:84% and treat the series now as a number. 00:44:26.870 --> 00:44:37.030 align:middle line:90% 00:44:37.030 --> 00:44:39.010 align:middle line:84% So in general, if I have a sequence, 00:44:39.010 --> 00:44:42.740 align:middle line:84% I just have this formal symbol, which I'm writing down, 00:44:42.740 --> 00:44:45.700 align:middle line:84% which I call a series associated to it. 00:44:45.700 --> 00:44:49.550 align:middle line:84% In the case that the sequence of partial sums converges, 00:44:49.550 --> 00:44:53.135 align:middle line:84% then I actually identify this series with a real number 00:44:53.135 --> 00:44:54.385 align:middle line:90% and treat it as a real number. 00:44:54.385 --> 00:44:59.120 align:middle line:90% 00:44:59.120 --> 00:45:03.080 align:middle line:84% And so the way I've written this, 00:45:03.080 --> 00:45:04.520 align:middle line:90% the series is starting at 1. 00:45:04.520 --> 00:45:06.530 align:middle line:84% But it doesn't necessarily have to. 00:45:06.530 --> 00:45:11.435 align:middle line:90% So just by shifting the index-- 00:45:11.435 --> 00:45:16.080 align:middle line:90% 00:45:16.080 --> 00:45:25.770 align:middle line:84% so let me just say here that we don't necessarily 00:45:25.770 --> 00:45:35.400 align:middle line:84% have to start a series at n equals 1. 00:45:35.400 --> 00:45:37.650 align:middle line:84% So this could be sum from n equals 0 00:45:37.650 --> 00:45:40.650 align:middle line:84% to infinity, in which case, we have a sequence starting now 00:45:40.650 --> 00:45:42.180 align:middle line:90% at x sub 0. 00:45:42.180 --> 00:45:45.420 align:middle line:84% Or this could be starting at 2, in which place 00:45:45.420 --> 00:45:48.540 align:middle line:84% the sequence of x sub n starts at n equals 2. 00:45:48.540 --> 00:45:50.970 align:middle line:84% And then the sequence of partial sums 00:45:50.970 --> 00:45:54.270 align:middle line:84% would start at not m equals 1 but m equals 0 00:45:54.270 --> 00:45:57.840 align:middle line:84% or m equals 2, depending on where the series starts. 00:45:57.840 --> 00:46:00.860 align:middle line:90% 00:46:00.860 --> 00:46:01.700 align:middle line:90% So some examples. 00:46:01.700 --> 00:46:05.180 align:middle line:90% 00:46:05.180 --> 00:46:14.110 align:middle line:84% The series sum from n equals 1 to infinity 1 over n 00:46:14.110 --> 00:46:21.580 align:middle line:84% plus 1 times n, this is a convergent series. 00:46:21.580 --> 00:46:26.010 align:middle line:90% 00:46:26.010 --> 00:46:27.945 align:middle line:90% So why is this? 00:46:27.945 --> 00:46:29.070 align:middle line:90% So let's look at the proof. 00:46:29.070 --> 00:46:31.770 align:middle line:84% So we look at the m-th partial sum-- 00:46:31.770 --> 00:46:38.700 align:middle line:84% this is the sum from n equals 1 to m of 1 over n plus 1n. 00:46:38.700 --> 00:46:41.430 align:middle line:90% And this is equal to-- 00:46:41.430 --> 00:46:56.540 align:middle line:84% now, if I write 1 over n plus 1 times n as 1 over n minus 1 00:46:56.540 --> 00:47:01.520 align:middle line:84% over n plus 1, this is now the sum of 1 over n 00:47:01.520 --> 00:47:05.390 align:middle line:84% plus-- these are finite sums, so I can always split them up. 00:47:05.390 --> 00:47:07.747 align:middle line:90% 00:47:07.747 --> 00:47:08.705 align:middle line:90% This should be a minus. 00:47:08.705 --> 00:47:11.640 align:middle line:90% 00:47:11.640 --> 00:47:14.550 align:middle line:84% And so now, this is equal to 1 plus 1/2 00:47:14.550 --> 00:47:26.550 align:middle line:84% plus 1 over m minus 1/2 plus 1/3 plus 1 over m 00:47:26.550 --> 00:47:30.960 align:middle line:90% plus 1 over n plus 1. 00:47:30.960 --> 00:47:34.980 align:middle line:90% And you see all of these cancel. 00:47:34.980 --> 00:47:40.800 align:middle line:84% And all that's left is 1 minus 1 over m plus 1. 00:47:40.800 --> 00:47:45.810 align:middle line:84% So the m-th partial sum is equal to 1 minus 1 over m plus 1. 00:47:45.810 --> 00:47:53.290 align:middle line:84% And therefore, the sequence of partial sums 00:47:53.290 --> 00:47:54.910 align:middle line:84% is the limit as m goes to infinity 00:47:54.910 --> 00:47:57.110 align:middle line:90% of this, which is just 1. 00:47:57.110 --> 00:47:59.950 align:middle line:84% And therefore, this series converges. 00:47:59.950 --> 00:48:09.610 align:middle line:90% 00:48:09.610 --> 00:48:14.050 align:middle line:84% Now, our favorite sequence, which does not converge, 00:48:14.050 --> 00:48:16.690 align:middle line:84% will give us a series which does not converge. 00:48:16.690 --> 00:48:22.240 align:middle line:84% So let's look at sum from n equals 1 to infinity minus 1 00:48:22.240 --> 00:48:23.440 align:middle line:90% to the n. 00:48:23.440 --> 00:48:26.500 align:middle line:90% This does not converge. 00:48:26.500 --> 00:48:29.532 align:middle line:90% 00:48:29.532 --> 00:48:30.365 align:middle line:90% So what's the proof? 00:48:30.365 --> 00:48:33.100 align:middle line:90% 00:48:33.100 --> 00:48:39.340 align:middle line:84% The m-th partial sum, this is equal to minus 1 00:48:39.340 --> 00:48:45.480 align:middle line:84% plus 1 plus minus 1 up until I get minus 1 to the m. 00:48:45.480 --> 00:48:48.610 align:middle line:90% 00:48:48.610 --> 00:48:55.060 align:middle line:84% And therefore, this is always equal to one of two things. 00:48:55.060 --> 00:49:02.670 align:middle line:84% If m is odd, then I have an odd number of these guys. 00:49:02.670 --> 00:49:07.050 align:middle line:84% And therefore, the minuses and pluses 00:49:07.050 --> 00:49:10.800 align:middle line:84% cancel, just leaving a minus 1 in the end-- 00:49:10.800 --> 00:49:12.240 align:middle line:90% this last one, the odd one. 00:49:12.240 --> 00:49:15.900 align:middle line:90% 00:49:15.900 --> 00:49:21.760 align:middle line:90% And m is odd and 0 if m is even. 00:49:21.760 --> 00:49:25.960 align:middle line:84% If I add up an even number of these terms, 00:49:25.960 --> 00:49:28.190 align:middle line:84% then all of the minus 1's and 1's cancel out, 00:49:28.190 --> 00:49:29.680 align:middle line:90% so I just get 0. 00:49:29.680 --> 00:49:39.930 align:middle line:84% And therefore, this sequence, which is just minus 1 00:49:39.930 --> 00:49:51.417 align:middle line:84% for m odd, 0 for m even, does not converge. 00:49:51.417 --> 00:49:53.250 align:middle line:84% And therefore, the series does not converge. 00:49:53.250 --> 00:49:56.250 align:middle line:84% So when I write this, this is just a symbol. 00:49:56.250 --> 00:49:58.088 align:middle line:84% This is just chalk on a chalkboard. 00:49:58.088 --> 00:49:59.130 align:middle line:90% It doesn't mean anything. 00:49:59.130 --> 00:50:06.320 align:middle line:90% 00:50:06.320 --> 00:50:12.320 align:middle line:84% So let's go to another series, which does converge. 00:50:12.320 --> 00:50:17.540 align:middle line:84% And this is kind of the one to which we 00:50:17.540 --> 00:50:22.520 align:middle line:84% compare all other series, essentially, as you'll see. 00:50:22.520 --> 00:50:25.700 align:middle line:84% You have all these series tests that you remember, hopefully, 00:50:25.700 --> 00:50:32.700 align:middle line:84% from calculus that tell you when a series converges. 00:50:32.700 --> 00:50:35.280 align:middle line:84% But maybe, if you remember the proof 00:50:35.280 --> 00:50:37.740 align:middle line:84% or don't, how you do that is you converge it 00:50:37.740 --> 00:50:40.320 align:middle line:84% to one series, which you do know how to sum. 00:50:40.320 --> 00:50:45.660 align:middle line:84% So it was just by pure luck we were able to compute the sum, 00:50:45.660 --> 00:50:50.370 align:middle line:84% or compute the explicit series for this guy. 00:50:50.370 --> 00:50:54.010 align:middle line:84% Another one which we can do that for is geometric series. 00:50:54.010 --> 00:50:58.230 align:middle line:84% So the theorem is if I have a real number 00:50:58.230 --> 00:51:06.000 align:middle line:84% with absolute value less than 1, then the series 00:51:06.000 --> 00:51:13.230 align:middle line:84% starting now at 0 R to the n converges. 00:51:13.230 --> 00:51:19.800 align:middle line:84% And I can actually compute the sum of this series. 00:51:19.800 --> 00:51:22.890 align:middle line:90% And this is 1 over 1 minus r. 00:51:22.890 --> 00:51:28.775 align:middle line:90% 00:51:28.775 --> 00:51:31.370 align:middle line:90% So what's the proof? 00:51:31.370 --> 00:51:33.830 align:middle line:90% Let's look at the partial sums. 00:51:33.830 --> 00:51:38.060 align:middle line:84% And we can actually compute these, as well, 00:51:38.060 --> 00:51:40.610 align:middle line:84% just as we were able to do for the first example. 00:51:40.610 --> 00:51:45.630 align:middle line:90% 00:51:45.630 --> 00:51:52.070 align:middle line:84% We compute that the sum from n equals 0 to m of r to the m-- 00:51:52.070 --> 00:51:55.010 align:middle line:84% now, you can prove this by induction. 00:51:55.010 --> 00:51:57.530 align:middle line:84% I cannot exactly remember if I did this-- 00:51:57.530 --> 00:52:00.440 align:middle line:90% 00:52:00.440 --> 00:52:02.240 align:middle line:84% I believe I did-- in the second lecture 00:52:02.240 --> 00:52:04.760 align:middle line:84% on induction, first or second lecture on the induction. 00:52:04.760 --> 00:52:09.170 align:middle line:84% But if I add up some number raised to the n-th-- 00:52:09.170 --> 00:52:12.080 align:middle line:90% so this should be to the n-- 00:52:12.080 --> 00:52:15.440 align:middle line:84% power from 0 to m, this is equal to 1 minus r 00:52:15.440 --> 00:52:20.670 align:middle line:90% to the m plus 1 over 1 minus r. 00:52:20.670 --> 00:52:26.130 align:middle line:84% And I guess two lectures ago, we proved that 00:52:26.130 --> 00:52:28.090 align:middle line:90% if the absolute value-- 00:52:28.090 --> 00:52:30.675 align:middle line:90% so let me state this now. 00:52:30.675 --> 00:52:38.490 align:middle line:90% 00:52:38.490 --> 00:52:46.290 align:middle line:84% Two lectures ago, we proved that if the absolute value of r 00:52:46.290 --> 00:52:55.410 align:middle line:84% is less than 1, then limit as, let's make it m, of r to the m 00:52:55.410 --> 00:53:02.780 align:middle line:90% equals 0, which implies that-- 00:53:02.780 --> 00:53:07.110 align:middle line:84% so this was the m-th partial sum-- 00:53:07.110 --> 00:53:11.930 align:middle line:84% which implies that the limit as m goes to infinity of s sub m 00:53:11.930 --> 00:53:16.220 align:middle line:84% equals the limit as m goes to infinity of this thing, which 00:53:16.220 --> 00:53:17.270 align:middle line:90% is, if you like-- 00:53:17.270 --> 00:53:20.250 align:middle line:90% 00:53:20.250 --> 00:53:24.870 align:middle line:84% so that plus 1 there just multiplies r to the m by r. 00:53:24.870 --> 00:53:29.910 align:middle line:84% And using the algebraic facts we proved 00:53:29.910 --> 00:53:32.610 align:middle line:84% about limits is 1 minus r time 0 over 1 00:53:32.610 --> 00:53:36.540 align:middle line:84% minus r, which equals 1 over 1 minus r. 00:53:36.540 --> 00:53:42.756 align:middle line:90% 00:53:42.756 --> 00:53:46.290 align:middle line:90% So now you ask about the other. 00:53:46.290 --> 00:53:48.720 align:middle line:90% What about r bigger than 1? 00:53:48.720 --> 00:53:52.570 align:middle line:90% 00:53:52.570 --> 00:53:55.300 align:middle line:84% Well, when r equals minus 1, then we 00:53:55.300 --> 00:53:59.440 align:middle line:84% get the second example we looked at. 00:53:59.440 --> 00:54:04.330 align:middle line:84% If we get r equals 1, then that's just summing up 1, 00:54:04.330 --> 00:54:07.570 align:middle line:84% and I'll leave it to you to check that that does not 00:54:07.570 --> 00:54:11.290 align:middle line:84% converge, that the sequence of partial sums, if I just sum up 00:54:11.290 --> 00:54:16.820 align:middle line:84% 1 from n equals 0 to m, is equal 2n plus 1, 00:54:16.820 --> 00:54:17.820 align:middle line:90% which does not converge. 00:54:17.820 --> 00:54:32.670 align:middle line:90% 00:54:32.670 --> 00:54:36.000 align:middle line:84% And let me make just a kind of silly comment. 00:54:36.000 --> 00:54:38.160 align:middle line:84% And maybe I didn't explicitly make 00:54:38.160 --> 00:54:40.170 align:middle line:90% this comment about sequences. 00:54:40.170 --> 00:54:41.980 align:middle line:84% Maybe I forgot to do that, as well. 00:54:41.980 --> 00:54:50.220 align:middle line:84% So for a sequence, you could start your sequence not 00:54:50.220 --> 00:54:52.170 align:middle line:90% necessarily at the first entry. 00:54:52.170 --> 00:54:55.950 align:middle line:84% Maybe you look at a, if you like, new sequence where-- 00:54:55.950 --> 00:55:00.050 align:middle line:90% 00:55:00.050 --> 00:55:04.290 align:middle line:84% well, so we know from sequences that subsequences 00:55:04.290 --> 00:55:05.970 align:middle line:84% of convergent sequences converge. 00:55:05.970 --> 00:55:11.980 align:middle line:84% So if I, instead of looking at the whole sequence, start at, 00:55:11.980 --> 00:55:15.960 align:middle line:84% let's say, x100 and then go x101, x102, 00:55:15.960 --> 00:55:17.700 align:middle line:84% and that's the sequence I look at, 00:55:17.700 --> 00:55:20.490 align:middle line:84% well, that's a subsequence of the original one, which, 00:55:20.490 --> 00:55:23.410 align:middle line:84% if it converges, implies that subsequence converges. 00:55:23.410 --> 00:55:26.760 align:middle line:90% So all that is to say-- 00:55:26.760 --> 00:55:31.500 align:middle line:84% and for this simple way of obtaining this new sequence-- 00:55:31.500 --> 00:55:36.540 align:middle line:84% all of that is to say that to understand 00:55:36.540 --> 00:55:38.400 align:middle line:84% if a sequence converges, I don't have 00:55:38.400 --> 00:55:43.980 align:middle line:84% to consider what happens for the first finitely many terms 00:55:43.980 --> 00:55:49.310 align:middle line:84% in the sequence, meaning a sequence x 00:55:49.310 --> 00:55:54.880 align:middle line:84% sub n converges if and only if a sequence starting now at say, 00:55:54.880 --> 00:56:00.370 align:middle line:84% n equals 100-- and 101, 102, 103 and so on-- converges. 00:56:00.370 --> 00:56:07.100 align:middle line:84% And the same is true for series, that a series converges if 00:56:07.100 --> 00:56:11.780 align:middle line:84% and only if a series converges, now, 00:56:11.780 --> 00:56:14.730 align:middle line:84% starting at a different point along the sequence. 00:56:14.730 --> 00:56:19.250 align:middle line:84% So this is the following theorem. 00:56:19.250 --> 00:56:23.340 align:middle line:90% 00:56:23.340 --> 00:56:38.630 align:middle line:84% Let xn be a sequence, and let capital M be a natural number. 00:56:38.630 --> 00:56:46.640 align:middle line:84% Then n equals 1 to infinity of x sub n. 00:56:46.640 --> 00:56:51.890 align:middle line:84% This converges if and only if the sequence, now starting 00:56:51.890 --> 00:56:58.180 align:middle line:84% at capital M, converges, meaning when 00:56:58.180 --> 00:57:01.210 align:middle line:84% I have to decide whether a series converges or not, 00:57:01.210 --> 00:57:03.670 align:middle line:84% it doesn't matter what's going on 00:57:03.670 --> 00:57:06.680 align:middle line:84% for the first finitely many terms. 00:57:06.680 --> 00:57:08.470 align:middle line:84% What matters is what's going on as I 00:57:08.470 --> 00:57:12.937 align:middle line:84% keep adding terms from further and further out 00:57:12.937 --> 00:57:13.645 align:middle line:90% of this sequence. 00:57:13.645 --> 00:57:16.570 align:middle line:90% 00:57:16.570 --> 00:57:17.500 align:middle line:90% And what's the proof? 00:57:17.500 --> 00:57:20.430 align:middle line:84% The proof is just expressing the partial sums 00:57:20.430 --> 00:57:23.910 align:middle line:84% for this guy in terms of the partial sums for this guy. 00:57:23.910 --> 00:57:42.370 align:middle line:84% So a partial sum satisfied for all m, sum from n 00:57:42.370 --> 00:57:52.040 align:middle line:84% equals 1 to m, and now x sub n as sum from n 00:57:52.040 --> 00:57:58.670 align:middle line:84% equals capital M to m x sub n plus sum from n 00:57:58.670 --> 00:58:05.020 align:middle line:84% equals 1 to capital M minus 1 x sub n. 00:58:05.020 --> 00:58:07.340 align:middle line:84% So this is now just a fixed number. 00:58:07.340 --> 00:58:12.280 align:middle line:84% So this is a sequence of partial sums corresponding 00:58:12.280 --> 00:58:14.200 align:middle line:90% to this series. 00:58:14.200 --> 00:58:16.480 align:middle line:84% This is a sequence of partial sums corresponding 00:58:16.480 --> 00:58:17.470 align:middle line:90% to this series. 00:58:17.470 --> 00:58:19.480 align:middle line:90% And this is just a fixed number. 00:58:19.480 --> 00:58:23.260 align:middle line:84% Therefore, if this converges, then this side 00:58:23.260 --> 00:58:28.120 align:middle line:90% converges to this plus this-- 00:58:28.120 --> 00:58:30.370 align:middle line:84% so maybe I'm going a little quick. 00:58:30.370 --> 00:58:34.660 align:middle line:84% But so if this converges, then this 00:58:34.660 --> 00:58:37.970 align:middle line:84% converges to this minus this number. 00:58:37.970 --> 00:58:43.330 align:middle line:84% And if this converges, then this sequence of partial sums 00:58:43.330 --> 00:58:49.000 align:middle line:84% converges to this limit plus this fixed number. 00:58:49.000 --> 00:58:50.650 align:middle line:84% And that's all I'm going to write. 00:58:50.650 --> 00:58:58.720 align:middle line:90% 00:58:58.720 --> 00:59:14.460 align:middle line:84% Now, coming back to the usefulness of Cauchy sequences, 00:59:14.460 --> 00:59:19.460 align:middle line:84% this is kind of where they really 00:59:19.460 --> 00:59:22.370 align:middle line:84% become useful is in the study of series. 00:59:22.370 --> 00:59:25.580 align:middle line:90% 00:59:25.580 --> 00:59:27.990 align:middle line:84% Because again, it's difficult to sum. 00:59:27.990 --> 00:59:30.800 align:middle line:84% So when I keep talking saying the word sum a series, 00:59:30.800 --> 00:59:35.855 align:middle line:84% I'm talking about find the limit of partial sums. 00:59:35.855 --> 00:59:38.450 align:middle line:90% 00:59:38.450 --> 00:59:41.240 align:middle line:84% But because we have this equivalence between Cauchy 00:59:41.240 --> 00:59:44.870 align:middle line:84% sequences and convergent sequences, 00:59:44.870 --> 00:59:48.830 align:middle line:84% to decide if a series is convergent or not, 00:59:48.830 --> 00:59:53.850 align:middle line:84% we can just decide if, in some sense, it's Cauchy or not. 00:59:53.850 --> 00:59:59.240 align:middle line:90% So let me make this definition. 00:59:59.240 --> 01:00:09.650 align:middle line:84% We say that a series x sub n is Cauchy if the sequence 01:00:09.650 --> 01:00:10.820 align:middle line:90% of partial sums-- 01:00:10.820 --> 01:00:19.480 align:middle line:84% again, I'm just going to put an m up top because this 01:00:19.480 --> 01:00:23.590 align:middle line:84% may start at 0 or n equals 1 or something-- so the sequence 01:00:23.590 --> 01:00:31.320 align:middle line:90% of partial sums is Cauchy. 01:00:31.320 --> 01:00:36.840 align:middle line:90% 01:00:36.840 --> 01:00:42.510 align:middle line:84% And so let me just restate what we proved for sequences 01:00:42.510 --> 01:00:44.130 align:middle line:90% in terms of series. 01:00:44.130 --> 01:00:48.280 align:middle line:84% So we proved that every Cauchy sequence is convergent. 01:00:48.280 --> 01:00:56.010 align:middle line:84% So a series is Cauchy means that the sequence of partial sums 01:00:56.010 --> 01:00:56.790 align:middle line:90% is Cauchy. 01:00:56.790 --> 01:01:00.580 align:middle line:84% But we've proven that Cauchy sequences are convergent. 01:01:00.580 --> 01:01:04.900 align:middle line:84% So if this is Cauchy, then it's convergent and vice versa. 01:01:04.900 --> 01:01:11.890 align:middle line:84% So based on what we've proven already for sequences, 01:01:11.890 --> 01:01:13.875 align:middle line:90% it follows that-- 01:01:13.875 --> 01:01:15.250 align:middle line:84% and this just follows immediately 01:01:15.250 --> 01:01:17.208 align:middle line:84% from what we've proven already, so I'm not even 01:01:17.208 --> 01:01:18.880 align:middle line:90% going to write a proof-- 01:01:18.880 --> 01:01:25.990 align:middle line:84% a series is Cauchy if and only if the series is convergent-- 01:01:25.990 --> 01:01:28.570 align:middle line:90% 01:01:28.570 --> 01:01:30.610 align:middle line:84% again, because both are defined in terms 01:01:30.610 --> 01:01:33.640 align:middle line:84% of the sequence of partial sums associated to the series. 01:01:33.640 --> 01:01:36.490 align:middle line:84% And we've already proven the equivalence between Cauchy 01:01:36.490 --> 01:01:40.030 align:middle line:90% and convergence for sequences. 01:01:40.030 --> 01:01:50.740 align:middle line:90% 01:01:50.740 --> 01:01:53.740 align:middle line:84% Now, let me write what it means to be Cauchy 01:01:53.740 --> 01:01:57.175 align:middle line:90% in a slightly different way. 01:01:57.175 --> 01:02:03.660 align:middle line:90% 01:02:03.660 --> 01:02:04.950 align:middle line:90% And it's the following. 01:02:04.950 --> 01:02:11.860 align:middle line:90% 01:02:11.860 --> 01:02:16.990 align:middle line:84% So before, we had that a sequence is Cauchy, 01:02:16.990 --> 01:02:20.620 align:middle line:84% intuitively, if the elements of the sequence 01:02:20.620 --> 01:02:22.690 align:middle line:90% are getting close to each other. 01:02:22.690 --> 01:02:26.560 align:middle line:84% Now, for a series to be Cauchy, the intuitive way 01:02:26.560 --> 01:02:30.460 align:middle line:84% to think about it is that the tail of the sum 01:02:30.460 --> 01:02:34.180 align:middle line:84% is getting small, is getting arbitrarily 01:02:34.180 --> 01:02:38.110 align:middle line:84% small, the tail of the sum being if I add up 01:02:38.110 --> 01:02:40.990 align:middle line:84% finitely many numbers far enough out. 01:02:40.990 --> 01:02:45.470 align:middle line:84% So a series is Cauchy if and only 01:02:45.470 --> 01:02:53.330 align:middle line:84% if all epsilon positive there exist to M, a natural number, 01:02:53.330 --> 01:03:00.680 align:middle line:84% such that for all integers l bigger than m, which 01:03:00.680 --> 01:03:08.600 align:middle line:84% is bigger than or equal to capital M, the sum from n 01:03:08.600 --> 01:03:13.310 align:middle line:84% equals n plus 1 to l of x sub n-- 01:03:13.310 --> 01:03:17.840 align:middle line:84% so this is a sum involving terms that are pretty far out 01:03:17.840 --> 01:03:22.280 align:middle line:84% there, at least all indexed by something bigger 01:03:22.280 --> 01:03:23.930 align:middle line:90% than or equal to capital M-- 01:03:23.930 --> 01:03:25.790 align:middle line:90% is less than epsilon. 01:03:25.790 --> 01:03:29.180 align:middle line:84% So a series is Cauchy if you're adding up 01:03:29.180 --> 01:03:33.470 align:middle line:84% smaller and smaller pieces, not individual pieces but actual 01:03:33.470 --> 01:03:34.760 align:middle line:90% adding those up. 01:03:34.760 --> 01:03:44.740 align:middle line:90% 01:03:44.740 --> 01:03:49.900 align:middle line:90% So I'll leave it to you to do-- 01:03:49.900 --> 01:03:53.150 align:middle line:84% they're both pretty easy-- but this direction, 01:03:53.150 --> 01:03:55.000 align:middle line:84% I'll leave it to you as an exercise. 01:03:55.000 --> 01:03:59.290 align:middle line:84% And it'll follow immediately from what 01:03:59.290 --> 01:04:02.860 align:middle line:84% I'm going to write for, essentially, this direction. 01:04:02.860 --> 01:04:06.450 align:middle line:90% 01:04:06.450 --> 01:04:14.650 align:middle line:84% So let's suppose-- and let's make things concrete, 01:04:14.650 --> 01:04:17.185 align:middle line:84% starting somewhere-- let's suppose this sum is Cauchy. 01:04:17.185 --> 01:04:21.080 align:middle line:90% 01:04:21.080 --> 01:04:25.310 align:middle line:84% And we want to prove now that it has this property. 01:04:25.310 --> 01:04:30.610 align:middle line:90% So let epsilon be positive. 01:04:30.610 --> 01:04:32.360 align:middle line:84% We now want to produce some capital number 01:04:32.360 --> 01:04:34.370 align:middle line:90% M so that this holds. 01:04:34.370 --> 01:04:43.730 align:middle line:90% 01:04:43.730 --> 01:04:52.220 align:middle line:84% So since the sequence of partial sums s sub m is Cauchy-- 01:04:52.220 --> 01:04:56.210 align:middle line:84% that's what it means for the series to be Cauchy-- 01:04:56.210 --> 01:05:00.350 align:middle line:84% there exists a natural number, capital M, 01:05:00.350 --> 01:05:10.100 align:middle line:84% such that for all M bigger than or equal to M0 01:05:10.100 --> 01:05:23.600 align:middle line:84% and l bigger than or equal to M0, s sub m minus s 01:05:23.600 --> 01:05:25.640 align:middle line:90% sub l is less than epsilon. 01:05:25.640 --> 01:05:31.720 align:middle line:90% 01:05:31.720 --> 01:05:36.190 align:middle line:84% So we're actually going to take capital M to be this M0. 01:05:36.190 --> 01:05:44.840 align:middle line:90% Choose M to be M0. 01:05:44.840 --> 01:05:51.610 align:middle line:84% Then, if l is bigger than m, which 01:05:51.610 --> 01:05:57.730 align:middle line:84% is bigger than or equal to M, which is equal to M0, 01:05:57.730 --> 01:06:05.710 align:middle line:84% if I look at this sum from n equals m plus 1 to l x sub n, 01:06:05.710 --> 01:06:08.278 align:middle line:90% I can write-- 01:06:08.278 --> 01:06:09.550 align:middle line:90% so this is absolute value. 01:06:09.550 --> 01:06:13.730 align:middle line:90% 01:06:13.730 --> 01:06:15.820 align:middle line:84% So in fact, let me remove the absolute value so 01:06:15.820 --> 01:06:17.530 align:middle line:90% that this becomes pretty clear. 01:06:17.530 --> 01:06:20.540 align:middle line:90% 01:06:20.540 --> 01:06:24.830 align:middle line:84% I can write this sum as the l-th partial sum 01:06:24.830 --> 01:06:28.250 align:middle line:90% minus the n-th partial sum. 01:06:28.250 --> 01:06:32.570 align:middle line:84% Because the l-th partial sum sums from n equals 1 up to l. 01:06:32.570 --> 01:06:35.840 align:middle line:84% The m-th partial sums from n equals 1 up to m. 01:06:35.840 --> 01:06:39.890 align:middle line:84% So the sum containing only the terms between n plus 1 and l 01:06:39.890 --> 01:06:41.285 align:middle line:90% is the difference of these guys. 01:06:41.285 --> 01:06:43.910 align:middle line:90% 01:06:43.910 --> 01:06:47.700 align:middle line:84% So now I'll put it on absolute values. 01:06:47.700 --> 01:06:50.160 align:middle line:84% And this thing, because l and m are 01:06:50.160 --> 01:06:53.250 align:middle line:84% bigger than or equal to capital M, which is equal to M0, 01:06:53.250 --> 01:06:55.650 align:middle line:84% and because I have this inequality, 01:06:55.650 --> 01:06:57.480 align:middle line:90% this is less than epsilon. 01:06:57.480 --> 01:06:59.220 align:middle line:84% And the converse direction, again, it 01:06:59.220 --> 01:07:00.960 align:middle line:84% follows immediately, essentially, 01:07:00.960 --> 01:07:04.500 align:middle line:90% from this equality here. 01:07:04.500 --> 01:07:08.570 align:middle line:90% 01:07:08.570 --> 01:07:11.540 align:middle line:84% So to check whether a series converges, 01:07:11.540 --> 01:07:18.230 align:middle line:84% I don't have to somehow come up with a limit for this series, 01:07:18.230 --> 01:07:21.440 align:middle line:90% a sum for this series. 01:07:21.440 --> 01:07:25.160 align:middle line:84% I can just prove that the tail can 01:07:25.160 --> 01:07:27.110 align:middle line:84% be made arbitrarily small, as long 01:07:27.110 --> 01:07:30.215 align:middle line:84% as I go far enough out in the series. 01:07:30.215 --> 01:07:45.970 align:middle line:90% 01:07:45.970 --> 01:07:53.950 align:middle line:84% And from this, we get a pretty simple elementary property, 01:07:53.950 --> 01:07:54.730 align:middle line:90% so a theorem. 01:07:54.730 --> 01:08:00.190 align:middle line:90% 01:08:00.190 --> 01:08:15.170 align:middle line:84% If a series converges, then this implies that the limit as n 01:08:15.170 --> 01:08:21.050 align:middle line:84% goes to infinity of x sub n of the sequence you used 01:08:21.050 --> 01:08:24.410 align:middle line:90% to obtain the series equals 0. 01:08:24.410 --> 01:08:31.310 align:middle line:84% So this should fall in line with a series being convergent 01:08:31.310 --> 01:08:35.180 align:middle line:84% if and only if it satisfies this property here, which is somehow 01:08:35.180 --> 01:08:39.899 align:middle line:84% saying the tail of the sum is getting smaller and smaller, 01:08:39.899 --> 01:08:45.649 align:middle line:84% which means you can't be adding up big things as you go on 01:08:45.649 --> 01:08:47.165 align:middle line:90% out in the series. 01:08:47.165 --> 01:08:53.399 align:middle line:90% 01:08:53.399 --> 01:08:53.940 align:middle line:90% So proof. 01:08:53.940 --> 01:08:59.020 align:middle line:90% 01:08:59.020 --> 01:09:09.569 align:middle line:84% So we'll show this by a simple epsilon M definition. 01:09:09.569 --> 01:09:16.529 align:middle line:90% So suppose xn converges. 01:09:16.529 --> 01:09:19.560 align:middle line:90% 01:09:19.560 --> 01:09:25.505 align:middle line:90% Then xn the series is Cauchy. 01:09:25.505 --> 01:09:28.870 align:middle line:90% 01:09:28.870 --> 01:09:33.460 align:middle line:84% And now I'm going to verify this using the epsilon M definition. 01:09:33.460 --> 01:09:36.760 align:middle line:90% 01:09:36.760 --> 01:09:38.200 align:middle line:90% Let epsilon be positive. 01:09:38.200 --> 01:09:40.760 align:middle line:90% 01:09:40.760 --> 01:09:47.120 align:middle line:84% Since xn is Cauchy, this implies that there 01:09:47.120 --> 01:09:54.500 align:middle line:84% exists a natural number M0 such that for all l bigger 01:09:54.500 --> 01:09:58.610 align:middle line:84% than or equal to m bigger than or equal M0, 01:09:58.610 --> 01:10:08.010 align:middle line:84% I have that condition there, sum from n equals m plus 1 01:10:08.010 --> 01:10:11.670 align:middle line:84% to l x sub n is less than epsilon. 01:10:11.670 --> 01:10:20.740 align:middle line:90% 01:10:20.740 --> 01:10:25.800 align:middle line:90% Choose M to be M0 plus 1. 01:10:25.800 --> 01:10:29.160 align:middle line:90% 01:10:29.160 --> 01:10:31.920 align:middle line:84% So why M0 plus 1 but not exactly M0? 01:10:31.920 --> 01:10:34.680 align:middle line:84% Just because, basically, what I'm going to do 01:10:34.680 --> 01:10:37.530 align:middle line:84% is I'm going to take l to be equal to m plus 1. 01:10:37.530 --> 01:10:40.440 align:middle line:90% And the index gets shifted by 1. 01:10:40.440 --> 01:10:43.320 align:middle line:90% 01:10:43.320 --> 01:10:48.840 align:middle line:84% Then if m is bigger than or equal to M, 01:10:48.840 --> 01:10:52.140 align:middle line:84% I get that the absolute value of x sub m-- 01:10:52.140 --> 01:10:55.380 align:middle line:90% 01:10:55.380 --> 01:10:58.950 align:middle line:84% maybe instead of saying limit as n 01:10:58.950 --> 01:11:01.630 align:middle line:84% goes to infinity, I'll write limit as m goes to infinity, 01:11:01.630 --> 01:11:03.840 align:middle line:84% it's just a change in the dummy variable-- this 01:11:03.840 --> 01:11:15.720 align:middle line:84% is equal to the sum from n equals m to m x sub n. 01:11:15.720 --> 01:11:17.830 align:middle line:90% And so I've shifted this. 01:11:17.830 --> 01:11:23.430 align:middle line:84% So now little m is bigger than or equal to capital M0 plus 1. 01:11:23.430 --> 01:11:27.600 align:middle line:84% So you could write this as, if you like, m minus 1 plus 1. 01:11:27.600 --> 01:11:32.160 align:middle line:84% So little m minus 1 is bigger than or equal to M0. 01:11:32.160 --> 01:11:34.800 align:middle line:84% So by this inequality, this is less than epsilon. 01:11:34.800 --> 01:11:52.960 align:middle line:90% 01:11:52.960 --> 01:11:58.150 align:middle line:84% So we see that if a series converges the terms, 01:11:58.150 --> 01:12:04.180 align:middle line:84% the individual terms, x sub n must converge to 0. 01:12:04.180 --> 01:12:08.680 align:middle line:84% So there's another reason why this series minus 1 to the n 01:12:08.680 --> 01:12:10.660 align:middle line:90% does not converge. 01:12:10.660 --> 01:12:12.190 align:middle line:84% Because those terms do not converge. 01:12:12.190 --> 01:12:15.640 align:middle line:90% 01:12:15.640 --> 01:12:19.720 align:middle line:84% And this also tells us that for this geometric series, when 01:12:19.720 --> 01:12:23.522 align:middle line:84% r is greater than or equal to 1 in absolute value, 01:12:23.522 --> 01:12:24.730 align:middle line:90% the series does not converge. 01:12:24.730 --> 01:12:34.250 align:middle line:90% 01:12:34.250 --> 01:12:36.890 align:middle line:90% It does not converge. 01:12:36.890 --> 01:12:37.670 align:middle line:90% And the proof is-- 01:12:37.670 --> 01:12:41.240 align:middle line:90% 01:12:41.240 --> 01:12:47.950 align:middle line:84% and we proved this, in fact, I think, a few lectures ago, 01:12:47.950 --> 01:12:53.740 align:middle line:84% as well, that if the absolute value of r is bigger than 1, 01:12:53.740 --> 01:12:58.840 align:middle line:84% then the limit as n goes to infinity of r to the n, 01:12:58.840 --> 01:13:02.400 align:middle line:90% this limit does not exist. 01:13:02.400 --> 01:13:05.550 align:middle line:90% 01:13:05.550 --> 01:13:08.940 align:middle line:84% We showed it's, in fact, unbounded. 01:13:08.940 --> 01:13:15.170 align:middle line:84% r to the n is an unbounded sequence. 01:13:15.170 --> 01:13:21.880 align:middle line:90% So it does not convert. 01:13:21.880 --> 01:13:25.090 align:middle line:84% So I'm using that theorem over there 01:13:25.090 --> 01:13:26.440 align:middle line:90% in a little bit roundabout way. 01:13:26.440 --> 01:13:37.890 align:middle line:90% 01:13:37.890 --> 01:13:39.570 align:middle line:84% Let me restate this theorem over here. 01:13:39.570 --> 01:13:46.120 align:middle line:84% This theorem says if the series converges, 01:13:46.120 --> 01:13:50.610 align:middle line:90% then this limit equals 0. 01:13:50.610 --> 01:13:56.760 align:middle line:84% Now, this statement is logically equivalent to the negation 01:13:56.760 --> 01:14:02.520 align:middle line:84% of the converse, or there's an actual word for that, 01:14:02.520 --> 01:14:04.470 align:middle line:90% but I can't remember-- 01:14:04.470 --> 01:14:12.900 align:middle line:84% namely that the negation of this implies the negation of this. 01:14:12.900 --> 01:14:18.960 align:middle line:84% So a logically equivalent way to rewrite that statement over 01:14:18.960 --> 01:14:31.900 align:middle line:84% there is that if this limit does not equal to 0-- 01:14:31.900 --> 01:14:36.580 align:middle line:84% so if it doesn't exist at all, that's also fine-- 01:14:36.580 --> 01:14:43.150 align:middle line:84% this implies that does not converge. 01:14:43.150 --> 01:14:48.070 align:middle line:84% So this restatement of the theorem over there 01:14:48.070 --> 01:14:51.910 align:middle line:90% is really what I used here. 01:14:51.910 --> 01:14:56.350 align:middle line:90% 01:14:56.350 --> 01:14:56.850 align:middle line:90% All right. 01:14:56.850 --> 01:14:59.440 align:middle line:90% And I think I'll stop there. 01:14:59.440 --> 01:15:04.140 align:middle line:84% Next time, we'll see that this theorem here 01:15:04.140 --> 01:15:05.530 align:middle line:90% is a one-way street. 01:15:05.530 --> 01:15:08.460 align:middle line:84% And I think you covered this example in, probably, 01:15:08.460 --> 01:15:12.180 align:middle line:84% calculus, namely that one-way street in the sense 01:15:12.180 --> 01:15:15.810 align:middle line:84% that if x then converges, then this limit is 0. 01:15:15.810 --> 01:15:18.030 align:middle line:90% But the converse does not hold. 01:15:18.030 --> 01:15:21.960 align:middle line:84% Namely, it is not true that if this limit 01:15:21.960 --> 01:15:25.440 align:middle line:90% is 0, then this converges. 01:15:25.440 --> 01:15:31.670 align:middle line:84% And we'll see the famed harmonic series next time. 01:15:31.670 --> 01:15:36.274 align:middle line:90%