WEBVTT 00:00:00.000 --> 00:00:12.630 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:12.630 --> 00:00:15.450 align:middle line:84% CASEY RODRIGUEZ: All right, so last time we 00:00:15.450 --> 00:00:18.800 align:middle line:90% proved the following theorem-- 00:00:18.800 --> 00:00:22.570 align:middle line:90% 00:00:22.570 --> 00:00:30.700 align:middle line:84% that if I have a convergent series, 00:00:30.700 --> 00:00:34.720 align:middle line:84% and this implies that the limit as n 00:00:34.720 --> 00:00:37.765 align:middle line:84% goes to infinity of x sub n equals 0. 00:00:37.765 --> 00:00:41.840 align:middle line:90% 00:00:41.840 --> 00:00:49.650 align:middle line:84% So a natural question is, as a beginning advanced math class, 00:00:49.650 --> 00:00:50.850 align:middle line:90% does the converse hold? 00:00:50.850 --> 00:00:53.530 align:middle line:84% Is this a two-way street or a one-way street? 00:00:53.530 --> 00:01:04.489 align:middle line:84% So if the individual terms in this series converges to 0, 00:01:04.489 --> 00:01:09.740 align:middle line:84% does this imply that the series converges? 00:01:09.740 --> 00:01:16.660 align:middle line:90% 00:01:16.660 --> 00:01:19.900 align:middle line:84% And I'm sure you answered this question in some form 00:01:19.900 --> 00:01:22.520 align:middle line:90% in a previous calculus class. 00:01:22.520 --> 00:01:25.150 align:middle line:84% And the answer to this question is no. 00:01:25.150 --> 00:01:29.830 align:middle line:90% 00:01:29.830 --> 00:01:32.020 align:middle line:90% So what's the counterexample? 00:01:32.020 --> 00:01:35.320 align:middle line:84% It's the so-called harmonic series, 00:01:35.320 --> 00:01:38.590 align:middle line:84% which corresponds to our favorite sequence which 00:01:38.590 --> 00:01:41.760 align:middle line:90% converges to 0. 00:01:41.760 --> 00:01:43.350 align:middle line:84% So we'll state this as a theorem-- 00:01:43.350 --> 00:01:48.360 align:middle line:90% 00:01:48.360 --> 00:01:58.170 align:middle line:84% the series sum from n equals 1 to infinity of 1 over n 00:01:58.170 --> 00:02:01.470 align:middle line:90% does not converge. 00:02:01.470 --> 00:02:06.650 align:middle line:90% 00:02:06.650 --> 00:02:10.940 align:middle line:84% So how are we going to prove this theorem? 00:02:10.940 --> 00:02:13.640 align:middle line:84% We'll prove this theorem by showing 00:02:13.640 --> 00:02:15.350 align:middle line:90% a sequence of partial sums. 00:02:15.350 --> 00:02:19.070 align:middle line:84% Some sequence of partial sums for this guy does not converge. 00:02:19.070 --> 00:02:23.750 align:middle line:84% So if it were to converge, then the sequence of partial sums 00:02:23.750 --> 00:02:27.110 align:middle line:84% converges, and therefore, every subsequence of partial sums 00:02:27.110 --> 00:02:29.900 align:middle line:90% converge. 00:02:29.900 --> 00:02:34.460 align:middle line:90% So what's the strategy? 00:02:34.460 --> 00:02:51.979 align:middle line:84% We're going to show that there exists a subsequence 00:02:51.979 --> 00:03:00.900 align:middle line:90% of partial sums, here s n k-- 00:03:00.900 --> 00:03:04.050 align:middle line:90% let's make this m k-- 00:03:04.050 --> 00:03:06.750 align:middle line:84% so remember, the partial sums are simply 00:03:06.750 --> 00:03:09.420 align:middle line:90% summing up the first-- 00:03:09.420 --> 00:03:12.480 align:middle line:84% so the index here is m sub k, so the first m sub k 00:03:12.480 --> 00:03:21.828 align:middle line:90% terms, 1 over n, diverges. 00:03:21.828 --> 00:03:24.120 align:middle line:84% And this is enough to show that the full series doesn't 00:03:24.120 --> 00:03:25.230 align:middle line:90% converge again. 00:03:25.230 --> 00:03:30.730 align:middle line:84% Because if it did converge, the sequence of partial sums 00:03:30.730 --> 00:03:33.400 align:middle line:84% converges, and therefore, every subsequence of partial sums 00:03:33.400 --> 00:03:34.490 align:middle line:90% converges. 00:03:34.490 --> 00:03:36.910 align:middle line:84% So if we're able to show there exists a subsequence which 00:03:36.910 --> 00:03:39.333 align:middle line:90% diverges, then we're done. 00:03:39.333 --> 00:03:40.750 align:middle line:84% In fact, what we're going to do is 00:03:40.750 --> 00:03:41.990 align:middle line:90% something a little bit stronger. 00:03:41.990 --> 00:03:43.532 align:middle line:84% We're going to show that there exists 00:03:43.532 --> 00:03:48.970 align:middle line:84% a subsequence of partial sums which not only diverges 00:03:48.970 --> 00:03:50.500 align:middle line:90% but is unbounded. 00:03:50.500 --> 00:03:52.720 align:middle line:84% And therefore, the entire sequence of partial sums 00:03:52.720 --> 00:03:53.500 align:middle line:90% is unbounded. 00:03:53.500 --> 00:04:04.010 align:middle line:90% 00:04:04.010 --> 00:04:06.560 align:middle line:84% So if we're able to show there exists a subsequence which 00:04:06.560 --> 00:04:09.470 align:middle line:84% is unbounded, then the entire sequence of partial sums 00:04:09.470 --> 00:04:11.510 align:middle line:84% is unbounded, so it can't converge. 00:04:11.510 --> 00:04:13.790 align:middle line:84% Because remember, convergent sequences 00:04:13.790 --> 00:04:17.660 align:middle line:90% imply bounded sequences. 00:04:17.660 --> 00:04:21.720 align:middle line:90% 00:04:21.720 --> 00:04:25.620 align:middle line:84% So we're going to look at when m sub k is dyadic. 00:04:25.620 --> 00:04:36.690 align:middle line:90% 00:04:36.690 --> 00:04:40.200 align:middle line:84% And for some reason, I switched indices from k 00:04:40.200 --> 00:04:46.350 align:middle line:84% to l in my notes, so instead we're going to go from m sub l. 00:04:46.350 --> 00:04:50.420 align:middle line:90% 00:04:50.420 --> 00:04:55.550 align:middle line:84% So let l be a natural number, and we'll 00:04:55.550 --> 00:05:03.380 align:middle line:84% consider the partial sum corresponding to adding up 00:05:03.380 --> 00:05:05.015 align:middle line:90% the first 2 to the l terms. 00:05:05.015 --> 00:05:11.920 align:middle line:90% 00:05:11.920 --> 00:05:14.130 align:middle line:84% Now, you may ask, why 2 to the l? 00:05:14.130 --> 00:05:16.110 align:middle line:90% Why not 3 to the l? 00:05:16.110 --> 00:05:18.270 align:middle line:84% Well, 4 to the l will be a subsequence of that, 00:05:18.270 --> 00:05:23.070 align:middle line:84% but 2 to the l is-- you could do 3 to the l, 00:05:23.070 --> 00:05:26.250 align:middle line:84% you could do 5 to the l, but 2 to the l 00:05:26.250 --> 00:05:28.140 align:middle line:90% is sufficient for our purposes. 00:05:28.140 --> 00:05:33.160 align:middle line:90% 00:05:33.160 --> 00:05:34.620 align:middle line:84% So what we're going to do is we're 00:05:34.620 --> 00:05:39.630 align:middle line:84% going to take this partial sum and bound it 00:05:39.630 --> 00:05:42.250 align:middle line:84% from below by something which is quite large. 00:05:42.250 --> 00:05:45.150 align:middle line:84% So first off, all of these partial sums 00:05:45.150 --> 00:05:47.190 align:middle line:90% are bounded from below by 0. 00:05:47.190 --> 00:05:49.095 align:middle line:84% They're a sum of non-negative terms. 00:05:49.095 --> 00:05:51.790 align:middle line:90% 00:05:51.790 --> 00:05:57.320 align:middle line:90% So we write s 2 to the l. 00:05:57.320 --> 00:06:02.510 align:middle line:84% This is equal to 1 plus 1/2 plus-- 00:06:02.510 --> 00:06:06.680 align:middle line:84% and I'm going to put parentheses around that-- plus 1/3, 00:06:06.680 --> 00:06:15.710 align:middle line:84% plus 1/4, plus 1/5, plus 1/6, plus 1/7, plus 1/8. 00:06:15.710 --> 00:06:19.640 align:middle line:84% So how I'm grouping these terms is I'm grouping them according 00:06:19.640 --> 00:06:26.330 align:middle line:84% to whether the denominator falls between a power of 2 00:06:26.330 --> 00:06:36.410 align:middle line:84% and the next power of 2, and then plus dot, 00:06:36.410 --> 00:06:46.250 align:middle line:84% dot, dot, 2 to the l minus 1 plus 1 plus 1 over 2 to the l. 00:06:46.250 --> 00:06:51.960 align:middle line:90% 00:06:51.960 --> 00:06:55.108 align:middle line:84% Now I can write this partial sum. 00:06:55.108 --> 00:06:56.400 align:middle line:90% So I've grouped terms this way. 00:06:56.400 --> 00:07:01.220 align:middle line:84% What is this in terms of precise symbols? 00:07:01.220 --> 00:07:09.080 align:middle line:84% This is 1 plus sum from lambda equals 1 to l, sum from n 00:07:09.080 --> 00:07:19.410 align:middle line:84% equals 2 to the lambda minus 1 plus 1 2 to the lambda 1 over n 00:07:19.410 --> 00:07:19.910 align:middle line:90% So. 00:07:19.910 --> 00:07:25.470 align:middle line:84% These lambdas are now parameterizing what power of 2 00:07:25.470 --> 00:07:25.970 align:middle line:90% I'm at. 00:07:25.970 --> 00:07:30.770 align:middle line:84% So when lambda equals 1, I'm at this block. 00:07:30.770 --> 00:07:32.720 align:middle line:84% When lambda equals 2, I'm at this block, 00:07:32.720 --> 00:07:35.960 align:middle line:84% and lambda equals l, I'm at this block. 00:07:35.960 --> 00:07:37.550 align:middle line:84% And then I'm summing up the terms 00:07:37.550 --> 00:07:43.295 align:middle line:84% that have denominator between that power and the next of 2. 00:07:43.295 --> 00:07:49.170 align:middle line:90% 00:07:49.170 --> 00:07:58.530 align:middle line:84% So I have this, and now I bound that sum from below. 00:07:58.530 --> 00:07:59.970 align:middle line:84% Because again, I'm trying to show 00:07:59.970 --> 00:08:06.180 align:middle line:84% that this subsequence of partial sums is unbounded. 00:08:06.180 --> 00:08:07.380 align:middle line:90% So I bound it from below. 00:08:07.380 --> 00:08:10.530 align:middle line:90% 00:08:10.530 --> 00:08:17.790 align:middle line:84% I sum from lambda equals 1 to l, sum from n 00:08:17.790 --> 00:08:26.890 align:middle line:84% equals 2 to the lambda minus 1 plus 1, 2 to the lambda. 00:08:26.890 --> 00:08:31.500 align:middle line:84% And now for each of these, n is between 2 to the-- 00:08:31.500 --> 00:08:33.735 align:middle line:84% so this should be 2 to the lambda. 00:08:33.735 --> 00:08:36.990 align:middle line:90% 00:08:36.990 --> 00:08:42.547 align:middle line:84% Now for n between these two numbers, 1 over n 00:08:42.547 --> 00:08:44.130 align:middle line:84% is always bigger than or equal to when 00:08:44.130 --> 00:08:47.310 align:middle line:84% I plug in the biggest bound here, 2 to the lambda. 00:08:47.310 --> 00:08:56.640 align:middle line:90% 00:08:56.640 --> 00:09:00.340 align:middle line:84% And now I just have a sum, so over n, 00:09:00.340 --> 00:09:01.810 align:middle line:90% but there's no n in this term. 00:09:01.810 --> 00:09:09.510 align:middle line:84% So I just add up all the number of terms in a given block. 00:09:09.510 --> 00:09:10.830 align:middle line:90% And this is equal to-- 00:09:10.830 --> 00:09:14.470 align:middle line:84% so first I have 1 over 2 to the lambda coming from here, 00:09:14.470 --> 00:09:20.820 align:middle line:84% and then sum from n equals 2 to the lambda minus 1 plus 1, 2 00:09:20.820 --> 00:09:22.350 align:middle line:90% to the lambda times 1. 00:09:22.350 --> 00:09:27.430 align:middle line:84% So again, I'm going a little slow here. 00:09:27.430 --> 00:09:34.890 align:middle line:84% And this is just equal to l 1 over 2 00:09:34.890 --> 00:09:38.430 align:middle line:84% to the lambda times the number of terms 00:09:38.430 --> 00:09:42.330 align:middle line:84% I have here, which is 2 to the lambda minus 2 00:09:42.330 --> 00:09:49.470 align:middle line:84% to the lambda minus 1 plus 1, so this minus this plus 1. 00:09:49.470 --> 00:09:57.490 align:middle line:90% 00:09:57.490 --> 00:10:02.770 align:middle line:84% So this is equal to 1 plus lambda equals 1 to l, 00:10:02.770 --> 00:10:04.570 align:middle line:90% 1 over 2 lambda. 00:10:04.570 --> 00:10:07.570 align:middle line:84% And so this 1 cancels with this one. 00:10:07.570 --> 00:10:11.200 align:middle line:84% And then I have 2 the lambda minus 1, which I can bring out. 00:10:11.200 --> 00:10:15.470 align:middle line:90% 00:10:15.470 --> 00:10:16.645 align:middle line:90% And so this is just 1. 00:10:16.645 --> 00:10:19.220 align:middle line:90% 00:10:19.220 --> 00:10:25.490 align:middle line:84% And I get 1 plus sum from lambda equals 1 to l. 00:10:25.490 --> 00:10:28.460 align:middle line:84% And this 2 to the lambda cancels with this 2 to the lambda, 00:10:28.460 --> 00:10:29.730 align:middle line:90% and I'm left with just a 1/2. 00:10:29.730 --> 00:10:35.950 align:middle line:90% 00:10:35.950 --> 00:10:39.340 align:middle line:90% And this is equal to-- 00:10:39.340 --> 00:10:41.920 align:middle line:84% now remember, there's no sum here in lambda, 00:10:41.920 --> 00:10:49.000 align:middle line:90% so this is just 1 plus l over 2. 00:10:49.000 --> 00:10:53.300 align:middle line:90% 00:10:53.300 --> 00:10:55.290 align:middle line:90% So what did we do? 00:10:55.290 --> 00:10:58.370 align:middle line:84% We basically showed that each of these blocks 00:10:58.370 --> 00:11:01.460 align:middle line:90% is bounded from below by a 1/2. 00:11:01.460 --> 00:11:05.420 align:middle line:84% That's this term that we get right here in the end. 00:11:05.420 --> 00:11:08.720 align:middle line:84% And we can see this if we just go through the first three 00:11:08.720 --> 00:11:10.790 align:middle line:90% terms which I have written here. 00:11:10.790 --> 00:11:14.510 align:middle line:84% So 1/2 is clearly bounded below by a 1/2. 00:11:14.510 --> 00:11:18.800 align:middle line:84% 1/3 plus 1/4 is bounded below by 1/4 plus 1/4, 00:11:18.800 --> 00:11:20.570 align:middle line:90% because 1/3 is bigger than that. 00:11:20.570 --> 00:11:23.750 align:middle line:90% So 1/4 plus 1/4 is a 1/2. 00:11:23.750 --> 00:11:28.490 align:middle line:84% If I look at this next block, that's 1 over 5 00:11:28.490 --> 00:11:30.860 align:middle line:84% is bigger than or equal to 1 over 8. 00:11:30.860 --> 00:11:32.030 align:middle line:90% So is 1 over 6. 00:11:32.030 --> 00:11:33.110 align:middle line:90% So is 1 over 7. 00:11:33.110 --> 00:11:37.550 align:middle line:84% So this sum is bigger than or equal to 1/8 plus 1/8 00:11:37.550 --> 00:11:43.423 align:middle line:84% plus 1/8 plus 1/8, which equals 1/2, plus, and then so on. 00:11:43.423 --> 00:11:45.340 align:middle line:84% So maybe I should have said that before I went 00:11:45.340 --> 00:11:46.750 align:middle line:90% into the actual computation. 00:11:46.750 --> 00:11:50.680 align:middle line:84% But in the end, we get that this subsequence of partial sums 00:11:50.680 --> 00:11:51.340 align:middle line:90% s to the 2l. 00:11:51.340 --> 00:11:55.160 align:middle line:90% 00:11:55.160 --> 00:11:58.270 align:middle line:90% So let me just summarize. 00:11:58.270 --> 00:12:04.120 align:middle line:84% This is bigger than or equal to l plus 2 over 2. 00:12:04.120 --> 00:12:06.800 align:middle line:90% 00:12:06.800 --> 00:12:13.290 align:middle line:84% And as l gets very large, this thing gets very large. 00:12:13.290 --> 00:12:19.280 align:middle line:84% So this implies s to the 2l, this subsequence, 00:12:19.280 --> 00:12:24.210 align:middle line:84% equals 1 to infinity is unbounded, 00:12:24.210 --> 00:12:33.240 align:middle line:84% which implies that the full subsequence, 00:12:33.240 --> 00:12:40.200 align:middle line:84% or the full sequence of partial sums, is unbounded. 00:12:40.200 --> 00:12:45.585 align:middle line:84% And therefore, the sequence of partial sums does not converge. 00:12:45.585 --> 00:12:47.460 align:middle line:84% And therefore, that series does not converge. 00:12:47.460 --> 00:12:58.350 align:middle line:90% 00:12:58.350 --> 00:13:09.820 align:middle line:84% So we see that the converse does not hold for that question 00:13:09.820 --> 00:13:12.100 align:middle line:90% or for that theorem. 00:13:12.100 --> 00:13:18.640 align:middle line:84% I will just make a very passing mention to the fact 00:13:18.640 --> 00:13:23.680 align:middle line:84% that there are fields for which that does hold-- 00:13:23.680 --> 00:13:25.737 align:middle line:84% so not ordered fields, because again, 00:13:25.737 --> 00:13:27.820 align:middle line:84% ordered fields with the least upper bound property 00:13:27.820 --> 00:13:30.880 align:middle line:84% have to be R, and therefore, we've 00:13:30.880 --> 00:13:36.900 align:middle line:84% just shown that the converse of that theorem does not hold. 00:13:36.900 --> 00:13:44.160 align:middle line:84% But in fact, if you look at the so-called p-adic numbers, 00:13:44.160 --> 00:13:50.130 align:middle line:84% they do have this property that if the sequence of terms 00:13:50.130 --> 00:13:52.110 align:middle line:84% converges to 0, then the series converges. 00:13:52.110 --> 00:13:53.850 align:middle line:84% But we will never see p-adic numbers. 00:13:53.850 --> 00:13:58.320 align:middle line:84% I just wanted to do a little lip service to that fact-- 00:13:58.320 --> 00:14:01.380 align:middle line:84% that there are at least fields of numbers 00:14:01.380 --> 00:14:03.700 align:middle line:90% that do have this property. 00:14:03.700 --> 00:14:09.560 align:middle line:90% 00:14:09.560 --> 00:14:13.910 align:middle line:84% So we had a theorem about limits of sequences 00:14:13.910 --> 00:14:16.910 align:middle line:84% and how they interact with algebraic operations. 00:14:16.910 --> 00:14:20.810 align:middle line:84% This naturally implies a theorem about series. 00:14:20.810 --> 00:14:36.650 align:middle line:84% So let alpha be in R, and let's suppose 00:14:36.650 --> 00:14:39.935 align:middle line:90% we have two convergent series. 00:14:39.935 --> 00:14:54.070 align:middle line:90% 00:14:54.070 --> 00:14:59.680 align:middle line:84% Then if I look at the series alpha x n plus y n-- 00:14:59.680 --> 00:15:02.620 align:middle line:84% so the terms of my new series are alpha 00:15:02.620 --> 00:15:04.570 align:middle line:90% times x sub n plus y sub n-- 00:15:04.570 --> 00:15:05.980 align:middle line:90% this is a convergent series. 00:15:05.980 --> 00:15:11.210 align:middle line:90% 00:15:11.210 --> 00:15:32.440 align:middle line:84% And the sum of this series is equal to what you expect. 00:15:32.440 --> 00:15:35.650 align:middle line:84% So the sum of the series alpha x n plus y n is equal to alpha 00:15:35.650 --> 00:15:38.695 align:middle line:84% times the sum of x n's plus the sum of y n's. 00:15:38.695 --> 00:15:43.590 align:middle line:90% 00:15:43.590 --> 00:15:47.820 align:middle line:84% So this theorem follows essentially kind of immediately 00:15:47.820 --> 00:15:49.920 align:middle line:90% from what we did for sequences. 00:15:49.920 --> 00:16:01.350 align:middle line:90% 00:16:01.350 --> 00:16:07.560 align:middle line:84% Partial sums satisfy-- if I look at sum stopping at m, 00:16:07.560 --> 00:16:14.490 align:middle line:84% alpha x n plus y n, now just by the linearity of just adding up 00:16:14.490 --> 00:16:16.020 align:middle line:90% finitely many terms-- 00:16:16.020 --> 00:16:19.110 align:middle line:84% I'm not going to put something down below because I really 00:16:19.110 --> 00:16:21.660 align:middle line:90% don't need to-- 00:16:21.660 --> 00:16:27.540 align:middle line:84% this is equal to alpha times the partial sum of x sub n 00:16:27.540 --> 00:16:34.420 align:middle line:84% plus the partial sum corresponding to y sub n. 00:16:34.420 --> 00:16:40.030 align:middle line:84% And so we're assuming this sequence of partial sums 00:16:40.030 --> 00:16:43.400 align:middle line:84% converges and this sequence of partial sums converges. 00:16:43.400 --> 00:16:47.300 align:middle line:84% So therefore, this term on the right-hand side converges, 00:16:47.300 --> 00:16:49.970 align:middle line:84% which implies the left side converges. 00:16:49.970 --> 00:16:58.430 align:middle line:90% 00:16:58.430 --> 00:17:09.079 align:middle line:84% So by the linear properties of limits, namely 00:17:09.079 --> 00:17:16.069 align:middle line:84% that the limit of the sum is the sum of the limits. 00:17:16.069 --> 00:17:22.290 align:middle line:84% And multiplication just by fixed real numbers commutes 00:17:22.290 --> 00:17:26.369 align:middle line:84% with taking limits, so we get that limit as m 00:17:26.369 --> 00:17:30.150 align:middle line:84% goes to infinity, so the partial sum 00:17:30.150 --> 00:17:42.405 align:middle line:84% corresponding to the new series equals alpha plus-- 00:17:42.405 --> 00:17:50.130 align:middle line:90% 00:17:50.130 --> 00:17:57.618 align:middle line:84% and that's just alpha times sum x n plus sum y n. 00:17:57.618 --> 00:17:58.410 align:middle line:90% And that's the end. 00:17:58.410 --> 00:18:13.360 align:middle line:90% 00:18:13.360 --> 00:18:26.620 align:middle line:84% So now, remember we had certain sequences which 00:18:26.620 --> 00:18:28.660 align:middle line:84% we could tell whether they converge, 00:18:28.660 --> 00:18:33.400 align:middle line:84% a little bit easier than just an arbitrary sequence. 00:18:33.400 --> 00:18:34.920 align:middle line:90% A couple of examples of-- 00:18:34.920 --> 00:18:36.430 align:middle line:84% at least one example of a sequence 00:18:36.430 --> 00:18:40.390 align:middle line:84% we could decide if it converges kind of easily 00:18:40.390 --> 00:18:45.700 align:middle line:84% is a monotone increasing sequence. 00:18:45.700 --> 00:18:50.380 align:middle line:84% And we showed that a monotone increasing sequence converges 00:18:50.380 --> 00:18:53.650 align:middle line:90% if and only if it's bounded. 00:18:53.650 --> 00:18:59.290 align:middle line:84% So we're going to use this to be able to say something about 00:18:59.290 --> 00:19:01.870 align:middle line:84% series now-- not sequences, but series-- 00:19:01.870 --> 00:19:07.300 align:middle line:84% that have non-negative terms that I'm adding up. 00:19:07.300 --> 00:19:10.120 align:middle line:84% Because the partial sums corresponding 00:19:10.120 --> 00:19:13.570 align:middle line:84% to a series that has non-negative terms 00:19:13.570 --> 00:19:17.020 align:middle line:84% form a monotone increasing sequence. 00:19:17.020 --> 00:19:19.480 align:middle line:90% And that's not too hard to show. 00:19:19.480 --> 00:19:25.260 align:middle line:90% 00:19:25.260 --> 00:19:28.360 align:middle line:84% So this is the following theorem. 00:19:28.360 --> 00:19:35.260 align:middle line:84% So now, we're going to discuss a little bit about sequences 00:19:35.260 --> 00:19:37.660 align:middle line:84% or look at sequences which have non-negative terms. 00:19:37.660 --> 00:19:40.840 align:middle line:84% So the theorem is the following-- 00:19:40.840 --> 00:19:46.440 align:middle line:84% if for all n a natural number, x then is bigger than or equal 00:19:46.440 --> 00:19:52.060 align:middle line:84% to 0-- so all these terms are non-negative-- 00:19:52.060 --> 00:20:01.280 align:middle line:84% then the series converges if and only 00:20:01.280 --> 00:20:05.810 align:middle line:84% if the sequence of partial sums s sub m-- 00:20:05.810 --> 00:20:18.240 align:middle line:90% 00:20:18.240 --> 00:20:18.795 align:middle line:90% is bounded. 00:20:18.795 --> 00:20:24.370 align:middle line:90% 00:20:24.370 --> 00:20:26.380 align:middle line:84% And again, the way we see this is 00:20:26.380 --> 00:20:30.770 align:middle line:84% just that when these terms are non-negative, 00:20:30.770 --> 00:20:32.630 align:middle line:84% the sequence of partial sums is monotone. 00:20:32.630 --> 00:20:37.060 align:middle line:90% 00:20:37.060 --> 00:20:39.360 align:middle line:90% So here's the proof. 00:20:39.360 --> 00:20:40.750 align:middle line:90% It's quite easy. 00:20:40.750 --> 00:20:51.110 align:middle line:84% So we have for all n a natural number-- 00:20:51.110 --> 00:20:57.010 align:middle line:84% make that m-- if I look at s sub m plus 1, this is equal to-- 00:20:57.010 --> 00:21:00.070 align:middle line:84% so this is the m plus 1 partial sum-- 00:21:00.070 --> 00:21:04.510 align:middle line:84% from m equals 1 to m plus 1, x of n, 00:21:04.510 --> 00:21:13.480 align:middle line:84% this is equal to sum from n equals m x n plus x n plus 1. 00:21:13.480 --> 00:21:16.873 align:middle line:84% And now the x m plus 1 term is not negative. 00:21:16.873 --> 00:21:18.790 align:middle line:84% We're assuming all the terms are non-negative. 00:21:18.790 --> 00:21:21.310 align:middle line:90% So this right-hand side-- 00:21:21.310 --> 00:21:24.460 align:middle line:84% this is certainly bigger than or equal to sum from n 00:21:24.460 --> 00:21:30.520 align:middle line:84% equals 1 to m of x n, and that equals s m. 00:21:30.520 --> 00:21:38.940 align:middle line:84% So just summarizing for all natural numbers m, 00:21:38.940 --> 00:21:42.570 align:middle line:84% the s m plus 1 is bigger than or equal to s m. 00:21:42.570 --> 00:21:44.430 align:middle line:84% If I just keep adding non-negative things, 00:21:44.430 --> 00:21:45.930 align:middle line:84% the partial sums are getting bigger. 00:21:45.930 --> 00:21:50.560 align:middle line:90% 00:21:50.560 --> 00:22:05.530 align:middle line:84% So the partial sums is a monotone. 00:22:05.530 --> 00:22:08.660 align:middle line:90% 00:22:08.660 --> 00:22:12.450 align:middle line:84% So maybe I should've stated this slightly differently just 00:22:12.450 --> 00:22:14.070 align:middle line:84% so that you don't think this is part 00:22:14.070 --> 00:22:15.787 align:middle line:90% of one of the if and only ifs. 00:22:15.787 --> 00:22:17.370 align:middle line:84% I mean, this is the assumption that we 00:22:17.370 --> 00:22:19.360 align:middle line:90% have for this whole statement. 00:22:19.360 --> 00:22:29.160 align:middle line:84% Suppose this, so the conclusion is that this converges if 00:22:29.160 --> 00:22:32.280 align:middle line:84% and only if the sequence of partial sums is bounded. 00:22:32.280 --> 00:22:38.380 align:middle line:84% So based on the assumption that all the terms are non-negative, 00:22:38.380 --> 00:22:40.480 align:middle line:84% we see that the sequence of partial sums 00:22:40.480 --> 00:22:41.860 align:middle line:90% is monotone increasing. 00:22:41.860 --> 00:22:48.970 align:middle line:90% 00:22:48.970 --> 00:22:50.710 align:middle line:84% That's why what we proved for sequences-- 00:22:50.710 --> 00:22:53.748 align:middle line:84% every monotone increasing sequence converges if 00:22:53.748 --> 00:22:54.790 align:middle line:90% and only if it's bounded. 00:22:54.790 --> 00:23:14.557 align:middle line:90% 00:23:14.557 --> 00:23:15.140 align:middle line:90% And that's it. 00:23:15.140 --> 00:23:27.180 align:middle line:90% 00:23:27.180 --> 00:23:32.790 align:middle line:84% Now, not every series we look at does have non-negative terms. 00:23:32.790 --> 00:23:37.530 align:middle line:84% But we can always form a certain series from those terms 00:23:37.530 --> 00:23:42.870 align:middle line:84% to make a new series with non-negative terms, which 00:23:42.870 --> 00:23:46.530 align:middle line:84% gives us information about the original series. 00:23:46.530 --> 00:23:50.680 align:middle line:90% What am I going on about? 00:23:50.680 --> 00:23:54.130 align:middle line:84% And look at the convergence properties of that new series. 00:23:54.130 --> 00:24:01.380 align:middle line:84% So we have the following definition-- 00:24:01.380 --> 00:24:06.690 align:middle line:90% 00:24:06.690 --> 00:24:15.911 align:middle line:84% that a series converges absolutely, 00:24:15.911 --> 00:24:25.250 align:middle line:84% or we say we have absolute convergence, 00:24:25.250 --> 00:24:29.690 align:middle line:84% if the series formed by taking the absolute values 00:24:29.690 --> 00:24:36.880 align:middle line:84% of these terms, if this series converges. 00:24:36.880 --> 00:24:41.560 align:middle line:90% 00:24:41.560 --> 00:24:45.100 align:middle line:84% So what I was trying to get at before I stated 00:24:45.100 --> 00:24:49.420 align:middle line:84% this definition is that absolute convergence 00:24:49.420 --> 00:24:51.940 align:middle line:90% implies usual convergence. 00:24:51.940 --> 00:25:03.230 align:middle line:90% 00:25:03.230 --> 00:25:06.110 align:middle line:84% If I had this series converging absolutely, 00:25:06.110 --> 00:25:13.020 align:middle line:84% then the original series converges. 00:25:13.020 --> 00:25:18.170 align:middle line:90% 00:25:18.170 --> 00:25:20.120 align:middle line:84% Now, before I prove this theorem, 00:25:20.120 --> 00:25:21.890 align:middle line:84% let me prove a little, small theorem. 00:25:21.890 --> 00:25:27.110 align:middle line:84% I can't remember if I gave it for an assignment or not, 00:25:27.110 --> 00:25:30.500 align:middle line:84% but it's essentially a triangle inequality 00:25:30.500 --> 00:25:32.075 align:middle line:90% for however many terms you like. 00:25:32.075 --> 00:25:35.340 align:middle line:90% 00:25:35.340 --> 00:25:38.960 align:middle line:84% So we'll prove this theorem in just a minute. 00:25:38.960 --> 00:25:42.220 align:middle line:84% But first, let me prove the following theorem-- 00:25:42.220 --> 00:25:54.040 align:middle line:84% that if m is bigger than or equal to 2, and x1 up to x m 00:25:54.040 --> 00:25:59.830 align:middle line:84% are in R, then sum from n equals 1 00:25:59.830 --> 00:26:04.450 align:middle line:84% to m x n, take the absolute value of this sum, 00:26:04.450 --> 00:26:09.370 align:middle line:84% this is less than or equal to sum from n equals 1 to m 00:26:09.370 --> 00:26:12.110 align:middle line:90% of the absolute value. 00:26:12.110 --> 00:26:15.950 align:middle line:84% When m equals 2, this is just the usual form 00:26:15.950 --> 00:26:18.330 align:middle line:90% of the triangle inequality. 00:26:18.330 --> 00:26:19.910 align:middle line:90% So m equals 2-- 00:26:19.910 --> 00:26:23.990 align:middle line:84% this is just saying x1 plus x2 is always less 00:26:23.990 --> 00:26:29.930 align:middle line:84% than or equal to x1 plus x2, which is just the triangle 00:26:29.930 --> 00:26:32.450 align:middle line:90% inequality. 00:26:32.450 --> 00:26:39.542 align:middle line:84% But typically how life works, at least in analysis, 00:26:39.542 --> 00:26:41.250 align:middle line:84% if you can do it for two things, then you 00:26:41.250 --> 00:26:47.970 align:middle line:84% can do it for n things or m things, in this case, 00:26:47.970 --> 00:26:48.870 align:middle line:90% by induction. 00:26:48.870 --> 00:26:51.720 align:middle line:84% And so that's how we're going to prove this. 00:26:51.720 --> 00:26:55.350 align:middle line:84% So we'll prove first prove this triangle inequality 00:26:55.350 --> 00:26:56.220 align:middle line:90% by induction. 00:26:56.220 --> 00:27:01.040 align:middle line:90% 00:27:01.040 --> 00:27:06.740 align:middle line:84% Now, in the induction proofs we've done so far, 00:27:06.740 --> 00:27:09.590 align:middle line:84% little n is our thing that we're inducting on. 00:27:09.590 --> 00:27:17.470 align:middle line:84% In this statement, m is the thing, induction on m. 00:27:17.470 --> 00:27:22.470 align:middle line:84% So let's look at the base case, which is m equals 2. 00:27:22.470 --> 00:27:37.090 align:middle line:84% So then this is just the triangle inequality 00:27:37.090 --> 00:27:41.050 align:middle line:84% for two real numbers that we've already proved before-- 00:27:41.050 --> 00:27:44.410 align:middle line:84% x1 plus x2 in absolute value is always 00:27:44.410 --> 00:27:47.080 align:middle line:84% less than or equal to sum of x1 plus the sum of x2-- 00:27:47.080 --> 00:27:49.900 align:middle line:84% I mean, the sum of absolute value of x1 00:27:49.900 --> 00:27:54.100 align:middle line:90% and the absolute value of x2. 00:27:54.100 --> 00:27:59.350 align:middle line:90% So the base case is fine. 00:27:59.350 --> 00:28:01.530 align:middle line:90% So now we do the inductive step. 00:28:01.530 --> 00:28:05.750 align:middle line:90% 00:28:05.750 --> 00:28:09.070 align:middle line:84% So I'll assume the statement that I want to prove. 00:28:09.070 --> 00:28:11.740 align:middle line:84% Usually, I use m, but now I'll go to the next letter 00:28:11.740 --> 00:28:13.855 align:middle line:84% l, going in reverse alphabetical order. 00:28:13.855 --> 00:28:17.220 align:middle line:90% 00:28:17.220 --> 00:28:31.070 align:middle line:90% Suppose if x1, x l in R. 00:28:31.070 --> 00:28:33.845 align:middle line:84% So let's actually, instead of just restating all that, just-- 00:28:33.845 --> 00:28:42.000 align:middle line:90% 00:28:42.000 --> 00:28:43.170 align:middle line:90% I'll just [INAUDIBLE] star. 00:28:43.170 --> 00:28:52.510 align:middle line:84% So suppose star holds for m equals l. 00:28:52.510 --> 00:28:55.450 align:middle line:84% And now we want to prove that star 00:28:55.450 --> 00:28:59.380 align:middle line:90% holds for m equals l plus 1. 00:28:59.380 --> 00:29:09.640 align:middle line:84% Now we want to show star holds for m equals l plus 1. 00:29:09.640 --> 00:29:26.070 align:middle line:90% 00:29:26.070 --> 00:29:34.870 align:middle line:84% Let x1 up to x l plus 1 be in R. Then 00:29:34.870 --> 00:29:43.180 align:middle line:84% if I look at the sum from n equals 1 to l plus 1 x n, 00:29:43.180 --> 00:29:51.430 align:middle line:84% this is equal to sum from n equals 1 to l x n plus x l 00:29:51.430 --> 00:29:53.290 align:middle line:90% plus 1 in absolute value. 00:29:53.290 --> 00:29:55.850 align:middle line:90% 00:29:55.850 --> 00:30:00.120 align:middle line:84% By the usual triangle inequality for two terms, 00:30:00.120 --> 00:30:04.630 align:middle line:84% this is less than or equal to sum from n equals 1 to l 00:30:04.630 --> 00:30:09.950 align:middle line:84% of x n in absolute value plus the absolute value of x l 00:30:09.950 --> 00:30:21.120 align:middle line:84% plus 1 by usual triangle inequality. 00:30:21.120 --> 00:30:24.090 align:middle line:84% And now this term, since I'm assuming m 00:30:24.090 --> 00:30:30.080 align:middle line:84% equals l holds, so the m equals l case says 00:30:30.080 --> 00:30:35.060 align:middle line:84% this is less than or equal to sum from n equals 1 to l 00:30:35.060 --> 00:30:41.570 align:middle line:84% of x n plus x l plus 1 in absolute value. 00:30:41.570 --> 00:30:49.180 align:middle line:84% So this is by inductive hypothesis. 00:30:49.180 --> 00:30:52.040 align:middle line:90% 00:30:52.040 --> 00:30:59.330 align:middle line:84% And this is just equal to sum from n equals 1 to l plus 1 00:30:59.330 --> 00:31:00.980 align:middle line:90% x n. 00:31:00.980 --> 00:31:05.060 align:middle line:84% so we've proven the case for now m equals l plus 1. 00:31:05.060 --> 00:31:18.310 align:middle line:90% 00:31:18.310 --> 00:31:22.690 align:middle line:84% And that concludes the proof of this generalized triangle 00:31:22.690 --> 00:31:26.545 align:middle line:84% inequality with arbitrary number of terms. 00:31:26.545 --> 00:31:31.750 align:middle line:90% 00:31:31.750 --> 00:31:33.730 align:middle line:84% So let's get back to proving this theorem, 00:31:33.730 --> 00:31:38.630 align:middle line:84% that absolute convergence implies convergence. 00:31:38.630 --> 00:31:42.260 align:middle line:84% So we'll do that by proving that absolute convergence implies 00:31:42.260 --> 00:31:43.370 align:middle line:90% that the series is Cauchy. 00:31:43.370 --> 00:31:46.700 align:middle line:90% 00:31:46.700 --> 00:31:57.090 align:middle line:84% So proof-- and this is of the theorem 00:31:57.090 --> 00:31:58.560 align:middle line:84% just before this theorem, we proved 00:31:58.560 --> 00:32:06.480 align:middle line:84% that absolute convergence implies usual convergence. 00:32:06.480 --> 00:32:09.960 align:middle line:90% 00:32:09.960 --> 00:32:24.870 align:middle line:84% So we will prove that in fact, this series is Cauchy, 00:32:24.870 --> 00:32:29.490 align:middle line:90% assuming absolute convergence. 00:32:29.490 --> 00:32:35.610 align:middle line:90% 00:32:35.610 --> 00:32:39.270 align:middle line:84% And from last time, we had approved the statement, 00:32:39.270 --> 00:32:43.680 align:middle line:84% or at least this followed from the statement for sequences, 00:32:43.680 --> 00:32:48.010 align:middle line:90% that a Cauchy series converges-- 00:32:48.010 --> 00:32:51.670 align:middle line:84% that a series is Cauchy if and only if it converges. 00:32:51.670 --> 00:32:55.430 align:middle line:90% 00:32:55.430 --> 00:32:59.870 align:middle line:84% So we have to prove that the series is Cauchy. 00:32:59.870 --> 00:33:07.610 align:middle line:84% Remember, this means for all epsilon positive, 00:33:07.610 --> 00:33:11.270 align:middle line:84% there exists a natural number m such 00:33:11.270 --> 00:33:15.020 align:middle line:84% that for all l bigger than m bigger than 00:33:15.020 --> 00:33:24.710 align:middle line:84% or equal to M, if I look at the sum from n equals m plus 1 to l 00:33:24.710 --> 00:33:27.515 align:middle line:84% of x n, this is less than epsilon. 00:33:27.515 --> 00:33:31.140 align:middle line:90% 00:33:31.140 --> 00:33:33.000 align:middle line:90% So let epsilon be positive. 00:33:33.000 --> 00:33:37.340 align:middle line:90% 00:33:37.340 --> 00:33:42.550 align:middle line:84% So since we're assuming that the series is absolutely 00:33:42.550 --> 00:33:51.920 align:middle line:84% convergent, this implies that this series 00:33:51.920 --> 00:33:54.695 align:middle line:84% with absolute values here is also Cauchy. 00:33:54.695 --> 00:33:58.130 align:middle line:90% 00:33:58.130 --> 00:34:07.770 align:middle line:84% So that means that there exists a natural number m sub 0 00:34:07.770 --> 00:34:11.840 align:middle line:84% such that for all L bigger than m bigger than or equal to M 00:34:11.840 --> 00:34:19.530 align:middle line:84% sub 0, if I look at the sum of absolute values 00:34:19.530 --> 00:34:23.949 align:middle line:84% from m plus 1 to l, this is less than epsilon. 00:34:23.949 --> 00:34:26.699 align:middle line:84% Now, this should have an absolute value on the outside, 00:34:26.699 --> 00:34:28.590 align:middle line:84% but this is a sum of non-negative terms, 00:34:28.590 --> 00:34:31.140 align:middle line:84% so the absolute value can be removed. 00:34:31.140 --> 00:34:39.969 align:middle line:90% 00:34:39.969 --> 00:34:43.730 align:middle line:84% You can essentially see where we're 00:34:43.730 --> 00:34:46.280 align:middle line:84% going based on what's written on the board-- 00:34:46.280 --> 00:34:48.770 align:middle line:84% what we want to prove, and what we know, 00:34:48.770 --> 00:34:52.610 align:middle line:90% and this triangle inequality. 00:34:52.610 --> 00:35:02.790 align:middle line:90% So choose M to be M sub 0. 00:35:02.790 --> 00:35:06.540 align:middle line:90% 00:35:06.540 --> 00:35:15.870 align:middle line:84% Then if l is bigger than m is bigger than or equal to M, then 00:35:15.870 --> 00:35:22.410 align:middle line:84% the absolute value of the sum m plus 1 to l sub n-- 00:35:22.410 --> 00:35:28.410 align:middle line:84% this is less than or equal to the sum from n equals 00:35:28.410 --> 00:35:32.700 align:middle line:84% m plus 1 to l of the absolute values of x sub n 00:35:32.700 --> 00:35:37.990 align:middle line:84% by the theorem we proved just a minute ago. 00:35:37.990 --> 00:35:41.640 align:middle line:84% And this is less than epsilon by our choice of M. M 00:35:41.640 --> 00:35:45.190 align:middle line:84% is equal to M0, and for M0, we have 00:35:45.190 --> 00:35:46.660 align:middle line:90% this inequality right here. 00:35:46.660 --> 00:35:51.170 align:middle line:90% 00:35:51.170 --> 00:35:58.490 align:middle line:84% Thus the series is Cauchy, which implies it converges. 00:35:58.490 --> 00:36:16.310 align:middle line:90% 00:36:16.310 --> 00:36:20.120 align:middle line:84% Basically, the only test you know 00:36:20.120 --> 00:36:23.660 align:middle line:84% for determining when a series is convergent 00:36:23.660 --> 00:36:28.250 align:middle line:90% is in one of two possibilities. 00:36:28.250 --> 00:36:30.960 align:middle line:84% Either A, it has a very simple form, 00:36:30.960 --> 00:36:33.530 align:middle line:84% and so all the terms are non-negative, 00:36:33.530 --> 00:36:36.680 align:middle line:84% but the terms have a very simple form. 00:36:36.680 --> 00:36:38.450 align:middle line:84% It's the alternating series test which 00:36:38.450 --> 00:36:46.460 align:middle line:84% we'll discuss in a little bit, possibly the next lecture. 00:36:46.460 --> 00:36:49.177 align:middle line:84% And then when a series converges absolutely, 00:36:49.177 --> 00:36:50.510 align:middle line:90% we have a lot of tests for that. 00:36:50.510 --> 00:36:54.840 align:middle line:90% 00:36:54.840 --> 00:37:00.770 align:middle line:84% And we'll see that series which converge absolutely 00:37:00.770 --> 00:37:05.540 align:middle line:84% somehow are not fickle, meaning I can rearrange the terms 00:37:05.540 --> 00:37:07.910 align:middle line:84% and the rearranged series will still 00:37:07.910 --> 00:37:10.700 align:middle line:84% converge absolutely, and converge to the same thing 00:37:10.700 --> 00:37:12.995 align:middle line:84% that the original series converged to. 00:37:12.995 --> 00:37:19.890 align:middle line:90% 00:37:19.890 --> 00:37:26.530 align:middle line:84% So let me just make a brief comment after this theorem, 00:37:26.530 --> 00:37:30.720 align:middle line:84% we proved that absolute convergence implies 00:37:30.720 --> 00:37:33.420 align:middle line:84% usual convergence, and tie-in a little bit 00:37:33.420 --> 00:37:34.840 align:middle line:90% to what I just said there. 00:37:34.840 --> 00:37:51.020 align:middle line:84% So we'll show that the series sum from 1 00:37:51.020 --> 00:37:55.790 align:middle line:84% to infinity of minus 1 to the n 1 over n-- 00:37:55.790 --> 00:37:56.840 align:middle line:90% this converges. 00:37:56.840 --> 00:38:01.330 align:middle line:90% 00:38:01.330 --> 00:38:13.590 align:middle line:84% But note that this series does not converge absolutely. 00:38:13.590 --> 00:38:21.510 align:middle line:90% 00:38:21.510 --> 00:38:24.540 align:middle line:84% Because when I take absolute values, I just get sum of 1 00:38:24.540 --> 00:38:28.230 align:middle line:84% over n, which is the harmonic series, which we just showed 00:38:28.230 --> 00:38:30.870 align:middle line:90% a few minutes ago is divergent. 00:38:30.870 --> 00:38:40.770 align:middle line:90% 00:38:40.770 --> 00:38:44.235 align:middle line:84% So now we're going to move on to some convergence tests. 00:38:44.235 --> 00:38:57.674 align:middle line:90% 00:38:57.674 --> 00:39:04.090 align:middle line:84% Now, when it comes to convergence tests, what these 00:39:04.090 --> 00:39:06.100 align:middle line:84% all follow from is basically what 00:39:06.100 --> 00:39:11.050 align:middle line:84% we know about geometric series and the following comparison 00:39:11.050 --> 00:39:13.060 align:middle line:84% test, although when I do the proofs 00:39:13.060 --> 00:39:15.670 align:middle line:84% of the other convergence tests, I won't actually 00:39:15.670 --> 00:39:17.470 align:middle line:84% state that I'm using the comparison test. 00:39:17.470 --> 00:39:20.700 align:middle line:84% But that's kind of what's really getting used. 00:39:20.700 --> 00:39:30.350 align:middle line:84% So the first test we have is the comparison test. 00:39:30.350 --> 00:39:33.450 align:middle line:90% 00:39:33.450 --> 00:39:35.460 align:middle line:84% And the statement is the following-- 00:39:35.460 --> 00:39:38.560 align:middle line:90% 00:39:38.560 --> 00:39:43.530 align:middle line:84% suppose for all n a natural number, 00:39:43.530 --> 00:39:46.470 align:middle line:84% we're looking at non-negative terms 00:39:46.470 --> 00:39:47.925 align:middle line:90% with one smaller than the other. 00:39:47.925 --> 00:39:51.640 align:middle line:90% 00:39:51.640 --> 00:40:04.480 align:middle line:84% Then the conclusion is if the bigger one converges, 00:40:04.480 --> 00:40:06.790 align:middle line:84% this implies that the smaller one converges. 00:40:06.790 --> 00:40:12.100 align:middle line:90% 00:40:12.100 --> 00:40:21.280 align:middle line:84% And if the smaller one diverges, this 00:40:21.280 --> 00:40:25.585 align:middle line:84% implies that the bigger one diverges. 00:40:25.585 --> 00:40:40.350 align:middle line:90% 00:40:40.350 --> 00:40:41.820 align:middle line:84% How we're going to prove this is-- 00:40:41.820 --> 00:40:44.250 align:middle line:84% so we're dealing with terms that are non-negative, 00:40:44.250 --> 00:40:49.920 align:middle line:84% so we'll use this theorem about a series of non-negative terms. 00:40:49.920 --> 00:40:52.748 align:middle line:90% 00:40:52.748 --> 00:40:54.290 align:middle line:84% So we use that theorem, and we proved 00:40:54.290 --> 00:40:57.950 align:middle line:84% that a series of non-negative terms converges if 00:40:57.950 --> 00:41:02.355 align:middle line:84% and only if the sequence of partial sums is bounded. 00:41:02.355 --> 00:41:05.780 align:middle line:90% 00:41:05.780 --> 00:41:16.340 align:middle line:84% So if this series converges, this 00:41:16.340 --> 00:41:28.190 align:middle line:84% implies that the sequence of partial sums 00:41:28.190 --> 00:41:32.030 align:middle line:90% is bounded, which implies-- 00:41:32.030 --> 00:41:38.090 align:middle line:84% that means that there exists a non-negative number such 00:41:38.090 --> 00:41:47.520 align:middle line:84% that for all natural numbers m, sum from n equals 1 to m 00:41:47.520 --> 00:41:57.810 align:middle line:84% of y sub n is less than or equal to B. 00:41:57.810 --> 00:42:01.430 align:middle line:84% But this immediately implies that since all the x n's 00:42:01.430 --> 00:42:03.290 align:middle line:84% are less than or equal to the y n's, we 00:42:03.290 --> 00:42:12.920 align:middle line:84% get that the n-th partial sum corresponding to the x n's, 00:42:12.920 --> 00:42:18.230 align:middle line:84% which is less than or equal to the n-th partial sum 00:42:18.230 --> 00:42:22.280 align:middle line:84% for the y n's, is also less than or equal to B for all m. 00:42:22.280 --> 00:42:30.560 align:middle line:90% 00:42:30.560 --> 00:42:37.815 align:middle line:84% So sequence of partial sums corresponding to the x n's is 00:42:37.815 --> 00:42:38.315 align:middle line:90% bounded. 00:42:38.315 --> 00:42:42.210 align:middle line:90% 00:42:42.210 --> 00:42:44.820 align:middle line:84% And therefore, by the theorem we proved, 00:42:44.820 --> 00:42:55.390 align:middle line:84% which I think I have erased already, the series converges. 00:42:55.390 --> 00:43:01.400 align:middle line:90% 00:43:01.400 --> 00:43:05.760 align:middle line:90% Now, proving 2 is essentially-- 00:43:05.760 --> 00:43:10.393 align:middle line:90% 00:43:10.393 --> 00:43:12.560 align:middle line:84% it's kind of the same thing, except the inequalities 00:43:12.560 --> 00:43:13.580 align:middle line:90% go the other way. 00:43:13.580 --> 00:43:16.370 align:middle line:84% And the x n's are getting bigger, which implies the y n's 00:43:16.370 --> 00:43:17.480 align:middle line:90% are also getting-- 00:43:17.480 --> 00:43:20.420 align:middle line:84% or the partial sums corresponding to the x n's is 00:43:20.420 --> 00:43:23.120 align:middle line:84% getting bigger implying that the partial sums 00:43:23.120 --> 00:43:25.590 align:middle line:84% corresponding to the y n's are also getting bigger. 00:43:25.590 --> 00:43:38.820 align:middle line:84% So now 2, if this series diverges, 00:43:38.820 --> 00:43:49.800 align:middle line:84% then this implies that partial sums is unbounded. 00:43:49.800 --> 00:43:54.100 align:middle line:90% 00:43:54.100 --> 00:43:55.680 align:middle line:84% we'll now prove that this implies 00:43:55.680 --> 00:43:57.930 align:middle line:84% that the partial sums corresponding to the y n's are 00:43:57.930 --> 00:43:58.620 align:middle line:90% unbounded. 00:43:58.620 --> 00:44:14.240 align:middle line:90% 00:44:14.240 --> 00:44:18.110 align:middle line:84% Now, remember what it means for a sequence to be bounded 00:44:18.110 --> 00:44:23.180 align:middle line:84% is that there exists a non-negative number B such 00:44:23.180 --> 00:44:25.040 align:middle line:84% that for all m I have that bound. 00:44:25.040 --> 00:44:28.370 align:middle line:84% So to say it's unbounded means that for all B 00:44:28.370 --> 00:44:31.430 align:middle line:84% there exists a little m bigger than or equal to capital M 00:44:31.430 --> 00:44:33.965 align:middle line:84% such that that inequality is reversed. 00:44:33.965 --> 00:44:36.480 align:middle line:90% 00:44:36.480 --> 00:44:41.100 align:middle line:84% So let me put here in a box what this actually means. 00:44:41.100 --> 00:44:45.450 align:middle line:84% Again, this means that for all B bigger than or equal to 0, 00:44:45.450 --> 00:44:48.230 align:middle line:84% there exists m, a natural number, 00:44:48.230 --> 00:44:57.760 align:middle line:84% so that y n equals 1 to m is bigger than or equal to B. 00:44:57.760 --> 00:44:59.575 align:middle line:90% That's what this means. 00:44:59.575 --> 00:45:02.200 align:middle line:90% 00:45:02.200 --> 00:45:04.240 align:middle line:84% So this is a for all statement, so I 00:45:04.240 --> 00:45:07.720 align:middle line:84% have to be able to prove it for every B, that B be bigger than 00:45:07.720 --> 00:45:09.640 align:middle line:90% or equal to 0. 00:45:09.640 --> 00:45:14.680 align:middle line:84% Now, since we know that the partial sums corresponding 00:45:14.680 --> 00:45:24.340 align:middle line:84% to the x n's is unbounded, this implies 00:45:24.340 --> 00:45:30.310 align:middle line:84% that there exists an m, a natural number, 00:45:30.310 --> 00:45:34.210 align:middle line:84% such that the sum from n equals 1 to m 00:45:34.210 --> 00:45:40.420 align:middle line:84% of x n is bigger than or equal to B. 00:45:40.420 --> 00:45:46.580 align:middle line:84% So let me say again, to show its unbounded I should really 00:45:46.580 --> 00:45:47.180 align:middle line:90% have-- 00:45:47.180 --> 00:45:48.890 align:middle line:84% so for a sequence to be unbounded, 00:45:48.890 --> 00:45:50.510 align:middle line:84% I should have an absolute value here 00:45:50.510 --> 00:45:55.220 align:middle line:84% is bigger than or equal to B. But all of these terms 00:45:55.220 --> 00:45:57.650 align:middle line:84% are non-negative, and therefore, I 00:45:57.650 --> 00:46:02.780 align:middle line:84% can remove the absolute values, and the same thing here. 00:46:02.780 --> 00:46:06.820 align:middle line:84% So there exists a natural number m so that I have this. 00:46:06.820 --> 00:46:09.750 align:middle line:90% 00:46:09.750 --> 00:46:15.200 align:middle line:84% And so we put a little m0 there because we have to somehow show 00:46:15.200 --> 00:46:16.503 align:middle line:90% there exists a little m. 00:46:16.503 --> 00:46:19.750 align:middle line:90% 00:46:19.750 --> 00:46:21.760 align:middle line:90% Choose m to be this m0. 00:46:21.760 --> 00:46:29.800 align:middle line:84% So now, if we look at the partial sums for the y n's, 00:46:29.800 --> 00:46:33.820 align:middle line:84% this is bigger than or equal to-- 00:46:33.820 --> 00:46:37.850 align:middle line:84% because m equals m0, it's this term, 00:46:37.850 --> 00:46:43.010 align:middle line:84% which is bigger than or equal to B. Thus, 00:46:43.010 --> 00:46:52.860 align:middle line:84% this proves the partial sums corresponding to the y n 00:46:52.860 --> 00:46:55.380 align:middle line:90% is unbounded. 00:46:55.380 --> 00:46:58.050 align:middle line:90% 00:46:58.050 --> 00:47:01.035 align:middle line:84% And therefore, this series diverges. 00:47:01.035 --> 00:47:10.580 align:middle line:90% 00:47:10.580 --> 00:47:14.420 align:middle line:84% So let's use the comparison theorem 00:47:14.420 --> 00:47:21.920 align:middle line:84% to consider series like 1 over n p-series, 00:47:21.920 --> 00:47:25.100 align:middle line:90% and prove when they do converge. 00:47:25.100 --> 00:47:29.790 align:middle line:90% 00:47:29.790 --> 00:47:46.240 align:middle line:84% So theorem-- for p, a real number, 00:47:46.240 --> 00:47:51.250 align:middle line:84% sum n equals 1 to infinity 1 over n to the p 00:47:51.250 --> 00:48:00.060 align:middle line:84% converges if and only if p is bigger than 1. 00:48:00.060 --> 00:48:14.930 align:middle line:90% 00:48:14.930 --> 00:48:23.800 align:middle line:84% So for the proof, why does the series converging 00:48:23.800 --> 00:48:27.850 align:middle line:90% imply p has to be bigger than 1? 00:48:27.850 --> 00:48:29.690 align:middle line:90% I'll do this by contradiction. 00:48:29.690 --> 00:48:40.760 align:middle line:84% So suppose 1 over n to the p equals 1 to infinity converges. 00:48:40.760 --> 00:48:45.190 align:middle line:84% So we'll do the proof by contradiction 00:48:45.190 --> 00:48:48.940 align:middle line:90% that p has to be bigger than 1. 00:48:48.940 --> 00:48:51.550 align:middle line:84% Suppose p is less than or equal to 1. 00:48:51.550 --> 00:48:55.360 align:middle line:90% 00:48:55.360 --> 00:48:59.280 align:middle line:90% Then 1 over n to the p-- 00:48:59.280 --> 00:49:01.270 align:middle line:84% where p is less than or equal to 1, 00:49:01.270 --> 00:49:03.880 align:middle line:84% this is bigger than or equal to 1 over n. 00:49:03.880 --> 00:49:09.760 align:middle line:90% 00:49:09.760 --> 00:49:12.970 align:middle line:84% And this implies, since 1 over n-- 00:49:12.970 --> 00:49:15.670 align:middle line:84% since the series corresponding to 1 over n-- 00:49:15.670 --> 00:49:19.610 align:middle line:84% diverges implies that the series corresponding to 1 over n 00:49:19.610 --> 00:49:39.300 align:middle line:84% to the p diverges by the comparison test, which 00:49:39.300 --> 00:49:42.510 align:middle line:84% is a direct contradiction to what we're assuming, 00:49:42.510 --> 00:49:45.520 align:middle line:90% that the series converges. 00:49:45.520 --> 00:49:50.150 align:middle line:90% So this must be false. 00:49:50.150 --> 00:49:51.320 align:middle line:90% p must be bigger than 1. 00:49:51.320 --> 00:49:54.978 align:middle line:84% So we've shown that if this series converges, 00:49:54.978 --> 00:49:56.270 align:middle line:90% then p has to be bigger than 1. 00:49:56.270 --> 00:50:03.380 align:middle line:84% So now let's prove the other direction 00:50:03.380 --> 00:50:08.600 align:middle line:84% and suppose p is bigger than 1, and prove 00:50:08.600 --> 00:50:13.520 align:middle line:84% that the p series, 1 over n to the p, converges. 00:50:13.520 --> 00:50:23.335 align:middle line:90% 00:50:23.335 --> 00:50:24.710 align:middle line:84% So the way we're going to do this 00:50:24.710 --> 00:50:33.270 align:middle line:84% is kind of how we showed that the harmonic series is 00:50:33.270 --> 00:50:34.140 align:middle line:90% divergent. 00:50:34.140 --> 00:50:37.270 align:middle line:90% 00:50:37.270 --> 00:50:39.225 align:middle line:84% So what we're going to do is first, 00:50:39.225 --> 00:50:40.600 align:middle line:84% we're going to show that there is 00:50:40.600 --> 00:50:43.090 align:middle line:84% a subsequence of partial sums corresponding 00:50:43.090 --> 00:50:44.960 align:middle line:90% to this guy that is bounded. 00:50:44.960 --> 00:50:50.800 align:middle line:84% So remember, to prove that this converges, this converges if 00:50:50.800 --> 00:50:55.210 align:middle line:84% and only if the sequence of partial sums is bounded. 00:50:55.210 --> 00:50:57.610 align:middle line:84% And what we're going to first do towards that 00:50:57.610 --> 00:51:01.267 align:middle line:84% is prove that there is a subsequence of partial sums 00:51:01.267 --> 00:51:01.975 align:middle line:90% which is bounded. 00:51:01.975 --> 00:51:11.010 align:middle line:90% 00:51:11.010 --> 00:51:25.480 align:middle line:84% So we make a first claim that the sequence of partial sums-- 00:51:25.480 --> 00:51:35.580 align:middle line:84% so s 2 to the k, this is sum from n goes 1 to 2 to the k 00:51:35.580 --> 00:51:41.910 align:middle line:84% of 1 over n to the p, so k a natural number-- 00:51:41.910 --> 00:51:47.160 align:middle line:84% this partial sum is bounded by a fixed number 00:51:47.160 --> 00:51:56.280 align:middle line:84% depending on p, 1 plus 1 minus 2 to the minus p minus 1. 00:51:56.280 --> 00:51:59.170 align:middle line:90% 00:51:59.170 --> 00:52:04.510 align:middle line:84% In other words, this subsequence of partial sums corresponding 00:52:04.510 --> 00:52:06.565 align:middle line:90% to s 2 to the k is bounded. 00:52:06.565 --> 00:52:10.190 align:middle line:90% 00:52:10.190 --> 00:52:14.740 align:middle line:84% So again, we do this by grouping these terms according 00:52:14.740 --> 00:52:19.480 align:middle line:84% to which power of 2 the denominator is 00:52:19.480 --> 00:52:22.210 align:middle line:84% between, and then estimate from above now, 00:52:22.210 --> 00:52:25.790 align:middle line:84% rather than from below like we did for the harmonic series. 00:52:25.790 --> 00:52:34.940 align:middle line:84% So we have s 2 to the k equals 1. 00:52:34.940 --> 00:52:36.890 align:middle line:84% So again, we're grouping these terms 00:52:36.890 --> 00:52:44.520 align:middle line:90% according to where they fall. 00:52:44.520 --> 00:52:47.210 align:middle line:84% So just write this out one more time-- 00:52:47.210 --> 00:52:57.500 align:middle line:84% this is equal to 1 plus 1/2 to the p plus 1 over 3 to the p 00:52:57.500 --> 00:53:08.280 align:middle line:84% plus 1 over 4 to the p plus 1 over 5 to the p plus 1 over 8 00:53:08.280 --> 00:53:10.200 align:middle line:90% to the p plus-- 00:53:10.200 --> 00:53:12.945 align:middle line:90% and then up until the last term. 00:53:12.945 --> 00:53:22.240 align:middle line:90% 00:53:22.240 --> 00:53:25.870 align:middle line:84% And now I can write this as 1 plus the sum from l 00:53:25.870 --> 00:53:30.280 align:middle line:84% equals 1 to k, so the number of blocks I have here. 00:53:30.280 --> 00:53:39.840 align:middle line:84% and now, the terms that come in each of these 00:53:39.840 --> 00:53:44.700 align:middle line:90% blocks 1 over into the p. 00:53:44.700 --> 00:53:49.300 align:middle line:90% 00:53:49.300 --> 00:53:56.590 align:middle line:84% And so now I estimate 1 over n not from below by this guy, 00:53:56.590 --> 00:54:00.760 align:middle line:84% but from above by putting in the smallest n 00:54:00.760 --> 00:54:03.130 align:middle line:90% that n is in this block. 00:54:03.130 --> 00:54:07.360 align:middle line:84% So this is less than or equal to 1 plus sum 00:54:07.360 --> 00:54:14.950 align:middle line:84% from l equals 1 to k sum from n equals 2 to the l minus 1 00:54:14.950 --> 00:54:20.800 align:middle line:84% plus 1, 2 to the l 1 over 2 to the l minus 1 00:54:20.800 --> 00:54:23.455 align:middle line:90% plus 1 raised to the p-th power. 00:54:23.455 --> 00:54:30.060 align:middle line:90% 00:54:30.060 --> 00:54:35.130 align:middle line:84% Now this plus 1 is just making things bigger on the bottom, 00:54:35.130 --> 00:54:37.350 align:middle line:84% so if I remove it, I've made things bigger 00:54:37.350 --> 00:54:39.310 align:middle line:90% overall for this fraction. 00:54:39.310 --> 00:54:41.490 align:middle line:84% So this is less than or equal to sum from 1 00:54:41.490 --> 00:54:50.610 align:middle line:84% equals l equals 1 to k, sum n equals 2 to the l minus 1, 2 00:54:50.610 --> 00:54:57.960 align:middle line:84% the l times 1 over 2 to the p times l minus 1. 00:54:57.960 --> 00:55:01.118 align:middle line:84% And now this thing here, if we do 00:55:01.118 --> 00:55:02.660 align:middle line:84% the same algebra we did a minute ago, 00:55:02.660 --> 00:55:08.350 align:middle line:84% this is equal to 1 l equals 1 to k. 00:55:08.350 --> 00:55:12.920 align:middle line:90% Now I have this term coming out. 00:55:12.920 --> 00:55:14.800 align:middle line:84% And then the number of terms I have here, 00:55:14.800 --> 00:55:18.430 align:middle line:84% just like I did for the harmonic series, this is equal to 2 00:55:18.430 --> 00:55:24.520 align:middle line:84% to the l minus 2 to the l minus 1 plus 1 plus 1. 00:55:24.520 --> 00:55:30.450 align:middle line:90% 00:55:30.450 --> 00:55:41.900 align:middle line:84% And now this is equal to 1 plus sum from l equals 1 to k. 00:55:41.900 --> 00:55:49.930 align:middle line:84% And so this whole thing here is equal to 2 to the l minus 1. 00:55:49.930 --> 00:55:56.530 align:middle line:84% So I get 2 to the minus p minus 1 l minus 1. 00:55:56.530 --> 00:55:59.550 align:middle line:90% 00:55:59.550 --> 00:56:03.220 align:middle line:90% Now I can shift this index. 00:56:03.220 --> 00:56:10.590 align:middle line:84% Actually, I guess I could have made that sharper 00:56:10.590 --> 00:56:11.530 align:middle line:90% but it doesn't matter. 00:56:11.530 --> 00:56:14.070 align:middle line:90% I could shift this index l by-- 00:56:14.070 --> 00:56:14.670 align:middle line:90% no. 00:56:14.670 --> 00:56:18.120 align:middle line:90% So l starts at 1 and goes to k. 00:56:18.120 --> 00:56:20.340 align:middle line:84% And here, I have the sum l minus 1. 00:56:20.340 --> 00:56:26.520 align:middle line:84% So I can shift this index to go from now l 00:56:26.520 --> 00:56:33.675 align:middle line:84% equals 0 to k minus 1, 2 to the minus p minus 1 l. 00:56:33.675 --> 00:56:36.890 align:middle line:90% 00:56:36.890 --> 00:56:39.335 align:middle line:84% So this is like making a change of variables, 00:56:39.335 --> 00:56:42.050 align:middle line:90% l prime equals l minus 1. 00:56:42.050 --> 00:56:45.380 align:middle line:84% And so let me put l prime instead of l. 00:56:45.380 --> 00:56:51.020 align:middle line:90% 00:56:51.020 --> 00:56:55.100 align:middle line:84% So p is bigger than 1, so this corresponds 00:56:55.100 --> 00:56:58.610 align:middle line:90% to a geometric series now. 00:56:58.610 --> 00:57:07.420 align:middle line:84% So let me actually rewrite this as 1 over 2 to the p minus 1 00:57:07.420 --> 00:57:10.760 align:middle line:90% to the l prime. 00:57:10.760 --> 00:57:13.730 align:middle line:84% When p is bigger than 1, then 1 over 2 to the p minus 1 00:57:13.730 --> 00:57:15.110 align:middle line:90% is less than 1. 00:57:15.110 --> 00:57:20.060 align:middle line:84% So this thing is a k minus 1 partial sum 00:57:20.060 --> 00:57:23.870 align:middle line:84% for the geometric series with this 00:57:23.870 --> 00:57:27.290 align:middle line:84% as R. So this is always bounded above by-- 00:57:27.290 --> 00:57:33.430 align:middle line:90% 00:57:33.430 --> 00:57:39.520 align:middle line:84% if I add up all the terms, which equals that thing that I 00:57:39.520 --> 00:57:45.640 align:middle line:84% have up there, 1 over 1 minus 2 to the minus p minus 1. 00:57:45.640 --> 00:57:56.970 align:middle line:90% 00:57:56.970 --> 00:58:03.120 align:middle line:84% So that proves that along this subsequence, 00:58:03.120 --> 00:58:06.840 align:middle line:84% these partial sums are bounded by this fixed number. 00:58:06.840 --> 00:58:14.920 align:middle line:84% And now I claim that this proves that the whole sequence 00:58:14.920 --> 00:58:19.180 align:middle line:84% of partial sums is bounded, in fact, by the same number. 00:58:19.180 --> 00:58:24.690 align:middle line:90% 00:58:24.690 --> 00:58:30.310 align:middle line:84% For all m, a natural number, s m is less than 00:58:30.310 --> 00:58:32.200 align:middle line:90% or equal to this number again-- 00:58:32.200 --> 00:58:36.100 align:middle line:84% 1 minus 2 to the minus p minus 1. 00:58:36.100 --> 00:58:39.400 align:middle line:90% 00:58:39.400 --> 00:58:44.045 align:middle line:84% So let m be a natural number, so we're 00:58:44.045 --> 00:58:45.170 align:middle line:90% trying to prove this bound. 00:58:45.170 --> 00:58:46.220 align:middle line:90% What do we do? 00:58:46.220 --> 00:58:51.290 align:middle line:84% We find a dyadic number, a number of the form 2 00:58:51.290 --> 00:58:53.630 align:middle line:90% to the k bigger than m. 00:58:53.630 --> 00:58:57.827 align:middle line:84% And since 2 to the m is bigger than m-- 00:58:57.827 --> 00:58:59.660 align:middle line:84% I think that's maybe one of the first things 00:58:59.660 --> 00:59:01.850 align:middle line:90% we did by induction-- 00:59:01.850 --> 00:59:05.850 align:middle line:90% 00:59:05.850 --> 00:59:12.210 align:middle line:84% we get that s sub m-- which is the partial sum 00:59:12.210 --> 00:59:14.940 align:middle line:84% of non-negative terms-- this is going 00:59:14.940 --> 00:59:16.740 align:middle line:84% to be less than or equal to, since this 00:59:16.740 --> 00:59:19.710 align:middle line:84% is a monotone increasing guy, this is going to be less than 00:59:19.710 --> 00:59:23.310 align:middle line:84% or equal s to the 2m, which is less than 00:59:23.310 --> 00:59:29.460 align:middle line:84% or equal to 1 plus 1 minus 2 to the minus p minus 1. 00:59:29.460 --> 00:59:37.230 align:middle line:90% 00:59:37.230 --> 00:59:39.840 align:middle line:84% Thus, the sequence of partial sums 00:59:39.840 --> 00:59:53.450 align:middle line:84% is bounded, which implies this series converges. 00:59:53.450 --> 00:59:57.040 align:middle line:84% And that's the end of the proof, and I think we'll stop there. 00:59:57.040 --> 01:00:03.000 align:middle line:90%