WEBVTT 00:00:00.000 --> 00:00:02.415 align:middle line:90% [SQUEAKING] 00:00:02.415 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:07.728 align:middle line:90% [CLICKING] 00:00:07.728 --> 00:00:10.940 align:middle line:90% 00:00:10.940 --> 00:00:16.040 align:middle line:84% TOBIAS COLDING: So now let's talk 00:00:16.040 --> 00:00:20.330 align:middle line:84% about first the different concepts 00:00:20.330 --> 00:00:23.910 align:middle line:84% that we have talked about in class. 00:00:23.910 --> 00:00:27.740 align:middle line:84% And then in the end, I'll get to just some example 00:00:27.740 --> 00:00:31.850 align:middle line:84% of some problems that you might face on the midterm. 00:00:31.850 --> 00:00:38.690 align:middle line:84% So the first is this-- that if you take a subset of Rn, 00:00:38.690 --> 00:00:49.910 align:middle line:84% to say that A is bounded from above, 00:00:49.910 --> 00:01:06.280 align:middle line:84% it just means that there exists M such that all a in A, a 00:01:06.280 --> 00:01:08.290 align:middle line:84% is less than or equal to M. So this 00:01:08.290 --> 00:01:12.220 align:middle line:84% is what it means to be bounded from above from a subset. 00:01:12.220 --> 00:01:15.720 align:middle line:90% And now the least upper bound. 00:01:15.720 --> 00:01:18.730 align:middle line:90% 00:01:18.730 --> 00:01:22.630 align:middle line:90% And this is a number-- 00:01:22.630 --> 00:01:26.920 align:middle line:90% so M is the least-- 00:01:26.920 --> 00:01:52.420 align:middle line:84% so M is a least upper bound if M is an upper bound 00:01:52.420 --> 00:01:58.667 align:middle line:90% and any other upper bound-- 00:01:58.667 --> 00:02:03.920 align:middle line:90% 00:02:03.920 --> 00:02:05.570 align:middle line:90% let's call it M bar-- 00:02:05.570 --> 00:02:08.330 align:middle line:90% 00:02:08.330 --> 00:02:15.110 align:middle line:84% is bigger or equal to M. So this is 00:02:15.110 --> 00:02:18.752 align:middle line:84% what it means to be a least upper bound. 00:02:18.752 --> 00:02:21.235 align:middle line:90% And so the least upper bound-- 00:02:21.235 --> 00:02:23.700 align:middle line:90% so the least upper bound-- 00:02:23.700 --> 00:02:27.670 align:middle line:84% this is what you denote by sup of A. 00:02:27.670 --> 00:02:44.030 align:middle line:84% This here is the least upper bound, and similarly 00:02:44.030 --> 00:02:44.830 align:middle line:90% For lower bounds. 00:02:44.830 --> 00:02:52.685 align:middle line:90% 00:02:52.685 --> 00:02:55.120 align:middle line:84% And the real numbers have this property 00:02:55.120 --> 00:03:00.550 align:middle line:84% that if a set is bounded from above, 00:03:00.550 --> 00:03:06.700 align:middle line:84% then there is a least upper bound, and likewise for the set 00:03:06.700 --> 00:03:10.435 align:middle line:90% that is bounded below. 00:03:10.435 --> 00:03:11.890 align:middle line:90% So that was sup. 00:03:11.890 --> 00:03:16.345 align:middle line:84% And we'll get to this with the lim sup and lim inf. 00:03:16.345 --> 00:03:20.440 align:middle line:84% I'll just review that a little later also. 00:03:20.440 --> 00:03:23.590 align:middle line:84% But that's another [? performance ?] [? chain. ?] 00:03:23.590 --> 00:03:28.630 align:middle line:84% A sequence-- so another key concept in this class 00:03:28.630 --> 00:03:30.670 align:middle line:90% is sequence. 00:03:30.670 --> 00:03:33.825 align:middle line:84% And let's begin with a sequence of real numbers. 00:03:33.825 --> 00:03:39.100 align:middle line:90% 00:03:39.100 --> 00:03:49.480 align:middle line:84% So this is a map from the positive integers into R. 00:03:49.480 --> 00:03:51.460 align:middle line:90% And again, we denote-- 00:03:51.460 --> 00:03:59.040 align:middle line:84% so the image-- so if you're looking at the image of some n, 00:03:59.040 --> 00:04:04.110 align:middle line:84% we typically denote this or by a n or xn or something 00:04:04.110 --> 00:04:08.910 align:middle line:84% like that where we use the subscript, 00:04:08.910 --> 00:04:13.810 align:middle line:84% but it's just the image under this map So that's a sequence. 00:04:13.810 --> 00:04:19.219 align:middle line:84% So for each positive integer, you get a real number. 00:04:19.219 --> 00:04:25.500 align:middle line:90% 00:04:25.500 --> 00:04:28.470 align:middle line:84% Then for a sequence, one of the first things 00:04:28.470 --> 00:04:31.210 align:middle line:84% you were interested in is convergence. 00:04:31.210 --> 00:04:36.790 align:middle line:90% 00:04:36.790 --> 00:04:45.880 align:middle line:84% So if this here is a sequence, that xn converges to some x 00:04:45.880 --> 00:04:57.050 align:middle line:84% here, this here just means that for all epsilon greater than 0, 00:04:57.050 --> 00:04:58.120 align:middle line:90% there exists-- 00:04:58.120 --> 00:04:59.510 align:middle line:90% I will use this symbol. 00:04:59.510 --> 00:05:03.370 align:middle line:84% I haven't used it until now, but it's just so hard not to use it. 00:05:03.370 --> 00:05:06.100 align:middle line:90% This here means "there exists." 00:05:06.100 --> 00:05:09.550 align:middle line:90% It's not on the midterm. 00:05:09.550 --> 00:05:11.740 align:middle line:84% But it's a totally standard symbol. 00:05:11.740 --> 00:05:20.170 align:middle line:84% So there exists capital N such that if little n is bigger 00:05:20.170 --> 00:05:28.510 align:middle line:84% or equal to capital N, then xn minus x is smaller than epsilon. 00:05:28.510 --> 00:05:33.820 align:middle line:84% So this is what it means for a sequence to converge to x-- 00:05:33.820 --> 00:05:35.770 align:middle line:90% sequence xn converges to x. 00:05:35.770 --> 00:05:39.850 align:middle line:84% And you write it like this, or you can write xn and then 00:05:39.850 --> 00:05:43.000 align:middle line:84% the limit here as n goes to infinity 00:05:43.000 --> 00:05:50.910 align:middle line:84% is equal to-- both of these are standard use. 00:05:50.910 --> 00:06:08.610 align:middle line:84% And then also, if a sequence is not convergent, then 00:06:08.610 --> 00:06:13.757 align:middle line:90% we say it is divergent. 00:06:13.757 --> 00:06:16.500 align:middle line:90% 00:06:16.500 --> 00:06:21.990 align:middle line:84% So a sequence is either convergent, or it's divergent. 00:06:21.990 --> 00:06:29.250 align:middle line:90% And then there's this property-- 00:06:29.250 --> 00:06:34.580 align:middle line:90% useful property, so a lemma. 00:06:34.580 --> 00:06:39.390 align:middle line:90% 00:06:39.390 --> 00:06:50.800 align:middle line:84% If xn is convergent, then the set-- 00:06:50.800 --> 00:06:52.490 align:middle line:90% this here is a set-- 00:06:52.490 --> 00:06:55.190 align:middle line:90% this here is a bounded set. 00:06:55.190 --> 00:07:00.775 align:middle line:90% 00:07:00.775 --> 00:07:03.490 align:middle line:90% So we [? talked ?] about that. 00:07:03.490 --> 00:07:06.850 align:middle line:84% And then another thing about sequences 00:07:06.850 --> 00:07:19.790 align:middle line:84% is the algebraic properties of sequences. 00:07:19.790 --> 00:07:23.560 align:middle line:90% 00:07:23.560 --> 00:07:26.080 align:middle line:84% And what are the algebraic properties? 00:07:26.080 --> 00:07:29.860 align:middle line:84% They are that if you take two sequences a n and bn, 00:07:29.860 --> 00:07:36.430 align:middle line:84% and let's say that a n converges to a and bn converges to b. 00:07:36.430 --> 00:07:40.290 align:middle line:84% Then the algebraic-- and c is just a constant. 00:07:40.290 --> 00:07:43.060 align:middle line:90% 00:07:43.060 --> 00:07:44.780 align:middle line:90% Then you have that-- 00:07:44.780 --> 00:07:49.680 align:middle line:84% so-- and often, one combines-- what I did as 1 and 2 before-- 00:07:49.680 --> 00:07:54.000 align:middle line:84% you combine, and you're looking at a n plus c times bn. 00:07:54.000 --> 00:07:57.810 align:middle line:84% This here converges to a plus cb. 00:07:57.810 --> 00:08:00.940 align:middle line:84% This is the first in this-- and when I state it like this, 00:08:00.940 --> 00:08:03.360 align:middle line:84% it's combining two of the 1 properties 00:08:03.360 --> 00:08:06.390 align:middle line:90% that we talked about earlier. 00:08:06.390 --> 00:08:14.460 align:middle line:84% The second one is that a n times bn will converge to a times b. 00:08:14.460 --> 00:08:22.500 align:middle line:84% Third property is that if these here are not equal to 0 00:08:22.500 --> 00:08:25.650 align:middle line:84% and the limit here is not equal to 0, 00:08:25.650 --> 00:08:28.710 align:middle line:84% then 1 over a n converges to 1/a. 00:08:28.710 --> 00:08:31.500 align:middle line:90% 00:08:31.500 --> 00:08:35.280 align:middle line:84% And so of course, this last one, [INAUDIBLE]-- 00:08:35.280 --> 00:08:38.460 align:middle line:84% this one implies, in particular, of course, that also, 00:08:38.460 --> 00:08:43.010 align:middle line:90% bn over a n converges to b/a. 00:08:43.010 --> 00:08:45.680 align:middle line:84% This is just because this property here 00:08:45.680 --> 00:08:48.080 align:middle line:90% combined with the property 2-- 00:08:48.080 --> 00:08:51.740 align:middle line:90% that will give you this. 00:08:51.740 --> 00:08:58.340 align:middle line:84% So that's the algebraic properties of limits. 00:08:58.340 --> 00:09:06.120 align:middle line:84% Then you have the concept of a subsequence, so a subsequence. 00:09:06.120 --> 00:09:11.810 align:middle line:90% 00:09:11.810 --> 00:09:16.190 align:middle line:84% So we imagine that we have a sequence. 00:09:16.190 --> 00:09:19.770 align:middle line:84% And often we don't go all the way back like this, 00:09:19.770 --> 00:09:24.650 align:middle line:84% but a sequence, again, is a map from the positive integers 00:09:24.650 --> 00:09:32.810 align:middle line:84% into R. And we denote the image of n by, typically, xn. 00:09:32.810 --> 00:09:36.170 align:middle line:84% Then a subsequence of this sequence-- and this 00:09:36.170 --> 00:09:38.930 align:middle line:84% is typically denoted something like this, where 00:09:38.930 --> 00:09:42.630 align:middle line:90% you have a further subscript. 00:09:42.630 --> 00:09:45.480 align:middle line:84% And this is important, again, that this 00:09:45.480 --> 00:09:52.840 align:middle line:84% is equal to f of g of k, where g here-- 00:09:52.840 --> 00:09:55.560 align:middle line:84% and this is the really important property-- 00:09:55.560 --> 00:09:59.850 align:middle line:84% that g is a map from the positive integers 00:09:59.850 --> 00:10:01.840 align:middle line:90% into the positive integers. 00:10:01.840 --> 00:10:05.310 align:middle line:84% And then, really important is that g here 00:10:05.310 --> 00:10:07.480 align:middle line:90% is strictly increasing. 00:10:07.480 --> 00:10:14.190 align:middle line:90% 00:10:14.190 --> 00:10:17.310 align:middle line:84% And so strictly increasing guarantees you 00:10:17.310 --> 00:10:22.950 align:middle line:84% that these numbers here are taken in the same order as 00:10:22.950 --> 00:10:24.750 align:middle line:90% in the original sequence. 00:10:24.750 --> 00:10:27.630 align:middle line:90% But you may skip a bunch. 00:10:27.630 --> 00:10:29.860 align:middle line:84% You take one, and then you may skip a bunch, 00:10:29.860 --> 00:10:32.680 align:middle line:84% and then you take another one, and then you may skip more. 00:10:32.680 --> 00:10:37.260 align:middle line:84% But it's in the same order, and that's what strictly increasing 00:10:37.260 --> 00:10:41.000 align:middle line:90% is guaranteeing. 00:10:41.000 --> 00:10:47.570 align:middle line:84% And then you have this property that-- 00:10:47.570 --> 00:10:58.580 align:middle line:84% so you have the theorem that a sequence xn is convergent if 00:10:58.580 --> 00:11:15.320 align:middle line:84% and only if all subsequences are convergent. 00:11:15.320 --> 00:11:19.640 align:middle line:84% And one of the two directions for this theorem 00:11:19.640 --> 00:11:27.440 align:middle line:84% was trivial, because if you take a sequence, 00:11:27.440 --> 00:11:31.370 align:middle line:84% then it is-- if you're thinking about this map here g 00:11:31.370 --> 00:11:36.090 align:middle line:84% as being the identity, so that would give you a subsequence, 00:11:36.090 --> 00:11:38.710 align:middle line:84% but that really is the original sequence. 00:11:38.710 --> 00:11:42.640 align:middle line:84% So the original sequence is a subsequence of itself. 00:11:42.640 --> 00:11:45.730 align:middle line:84% So if all subsequences are convergent, well, 00:11:45.730 --> 00:11:48.710 align:middle line:84% then, in particular, the sequence itself is convergent. 00:11:48.710 --> 00:11:50.520 align:middle line:84% So one of the two directions was trivial. 00:11:50.520 --> 00:11:53.574 align:middle line:90% 00:11:53.574 --> 00:11:58.060 align:middle line:84% Now, then we had these two theorems, 00:11:58.060 --> 00:12:00.170 align:middle line:90% and they were [INAUDIBLE]. 00:12:00.170 --> 00:12:07.190 align:middle line:84% So the thing in mathematics is that things build on each other. 00:12:07.190 --> 00:12:11.150 align:middle line:84% And hopefully, each step is fairly trivial, 00:12:11.150 --> 00:12:13.840 align:middle line:84% but you build step by step, and then 00:12:13.840 --> 00:12:16.720 align:middle line:84% you end up with something highly non-trivial. 00:12:16.720 --> 00:12:22.870 align:middle line:84% So this here was pretty trivial, but then we used it later on 00:12:22.870 --> 00:12:25.850 align:middle line:90% to prove something non-trivial. 00:12:25.850 --> 00:12:28.490 align:middle line:84% So this was this monotone convergence theorem. 00:12:28.490 --> 00:12:36.920 align:middle line:90% 00:12:36.920 --> 00:12:41.060 align:middle line:84% And there were two versions of this. 00:12:41.060 --> 00:12:50.840 align:middle line:84% You had a sequence xn, and xn was increasing. 00:12:50.840 --> 00:12:52.820 align:middle line:84% So that means that the next element 00:12:52.820 --> 00:12:55.430 align:middle line:84% is bigger or equal to the previous element. 00:12:55.430 --> 00:12:58.580 align:middle line:84% If this here was the case, then xn 00:12:58.580 --> 00:13:13.940 align:middle line:84% is convergent if and only if xn is bounded-- 00:13:13.940 --> 00:13:16.940 align:middle line:90% if the sequence is bounded. 00:13:16.940 --> 00:13:21.470 align:middle line:90% And the sup of the xn-- 00:13:21.470 --> 00:13:29.060 align:middle line:90% 00:13:29.060 --> 00:13:31.020 align:middle line:90% this here is the limit. 00:13:31.020 --> 00:13:41.350 align:middle line:90% 00:13:41.350 --> 00:13:44.980 align:middle line:84% And the second version of this is 00:13:44.980 --> 00:13:48.530 align:middle line:84% if you have a monotone-- so instead of being increasing, 00:13:48.530 --> 00:13:49.760 align:middle line:90% it's now decreasing. 00:13:49.760 --> 00:13:53.000 align:middle line:84% So the next element is smaller or 00:13:53.000 --> 00:13:55.795 align:middle line:90% equal to the previous element. 00:13:55.795 --> 00:14:07.420 align:middle line:84% And then similarly, the sequence xn is convergent if and only 00:14:07.420 --> 00:14:23.290 align:middle line:90% if xn is bounded from below. 00:14:23.290 --> 00:14:25.130 align:middle line:90% And in this case-- 00:14:25.130 --> 00:14:29.260 align:middle line:84% and if it is bounded from below, then the limit here, 00:14:29.260 --> 00:14:33.690 align:middle line:90% [INAUDIBLE] of xn is the inf. 00:14:33.690 --> 00:14:36.620 align:middle line:90% 00:14:36.620 --> 00:14:39.290 align:middle line:84% So that was the monotone convergence theorem. 00:14:39.290 --> 00:14:42.950 align:middle line:84% And it has these two versions, the increasing version 00:14:42.950 --> 00:14:46.160 align:middle line:90% and the decreasing version. 00:14:46.160 --> 00:14:55.340 align:middle line:84% Then another important concept was that of the Cauchy sequence. 00:14:55.340 --> 00:15:00.980 align:middle line:84% And the reason why Cauchy sequence 00:15:00.980 --> 00:15:05.390 align:middle line:84% is useful even for the real numbers-- 00:15:05.390 --> 00:15:08.210 align:middle line:84% later on, we generalize these concepts-- 00:15:08.210 --> 00:15:11.360 align:middle line:84% the most obvious [INAUDIBLE] of the concept of 00:15:11.360 --> 00:15:14.330 align:middle line:84% convergence, but also of that of being a Cauchy 00:15:14.330 --> 00:15:16.490 align:middle line:90% sequence-- to metric spaces. 00:15:16.490 --> 00:15:21.190 align:middle line:84% But even for the real numbers, Cauchy sequence-- 00:15:21.190 --> 00:15:24.590 align:middle line:84% you might be interested in whether a sequence is 00:15:24.590 --> 00:15:28.610 align:middle line:84% convergent, but sometimes it's much easier 00:15:28.610 --> 00:15:31.280 align:middle line:84% to prove that it's a Cauchy sequence 00:15:31.280 --> 00:15:33.490 align:middle line:84% than to prove that it actually is convergent. 00:15:33.490 --> 00:15:37.340 align:middle line:84% And so if you don't really care about what the limit is, 00:15:37.340 --> 00:15:42.660 align:middle line:84% then it's often easier to prove that it's a Cauchy sequence. 00:15:42.660 --> 00:15:47.020 align:middle line:90% 00:15:47.020 --> 00:15:51.260 align:middle line:84% So Cauchy sequence-- so you have a sequence. 00:15:51.260 --> 00:15:55.900 align:middle line:84% Again, we're still in the real numbers. 00:15:55.900 --> 00:15:58.660 align:middle line:90% And this is a Cauchy sequence. 00:15:58.660 --> 00:16:00.820 align:middle line:90% It just means that for all-- 00:16:00.820 --> 00:16:04.180 align:middle line:84% and now I will use also this here. 00:16:04.180 --> 00:16:06.530 align:middle line:84% This symbol just means "for all," 00:16:06.530 --> 00:16:08.270 align:middle line:84% and that's also a standard symbol. 00:16:08.270 --> 00:16:10.900 align:middle line:90% It's an inverted A. 00:16:10.900 --> 00:16:15.580 align:middle line:84% For all epsilon greater than 0, there exists-- 00:16:15.580 --> 00:16:22.750 align:middle line:84% again, this here-- the inverted E means "there exists"-- 00:16:22.750 --> 00:16:23.650 align:middle line:90% delta-- I'm sorry. 00:16:23.650 --> 00:16:34.950 align:middle line:84% There exists capital N such that if little n and little m 00:16:34.950 --> 00:16:39.060 align:middle line:84% are bigger than capital N, then those two elements 00:16:39.060 --> 00:16:45.480 align:middle line:84% are bunched epsilon close together [INAUDIBLE]. 00:16:45.480 --> 00:16:51.690 align:middle line:84% This is what it means for a sequence to be Cauchy sequence-- 00:16:51.690 --> 00:16:55.745 align:middle line:84% that the tail of the sequence lies very, very close together. 00:16:55.745 --> 00:16:58.710 align:middle line:90% 00:16:58.710 --> 00:17:06.780 align:middle line:90% Then let me skip. 00:17:06.780 --> 00:17:09.027 align:middle line:84% In my notes, I have something about contracting, 00:17:09.027 --> 00:17:11.319 align:middle line:84% but actually, there won't be anything about contracting 00:17:11.319 --> 00:17:13.060 align:middle line:90% methods, so I'll skip that. 00:17:13.060 --> 00:17:20.130 align:middle line:90% 00:17:20.130 --> 00:17:24.630 align:middle line:84% Let me now next talk about the Cauchy convergence 00:17:24.630 --> 00:17:29.160 align:middle line:84% theorem and the Bolzano-Weierstrass theorem. 00:17:29.160 --> 00:17:40.000 align:middle line:90% 00:17:40.000 --> 00:17:53.020 align:middle line:84% So Cauchy-- also let me just remind you that, again, 00:17:53.020 --> 00:17:54.080 align:middle line:90% on the midterm-- 00:17:54.080 --> 00:17:56.500 align:middle line:84% so there are these five main problems. 00:17:56.500 --> 00:17:58.870 align:middle line:84% Then there's the six bonus problems. 00:17:58.870 --> 00:18:03.180 align:middle line:90% But make sure that you-- 00:18:03.180 --> 00:18:05.330 align:middle line:84% it doesn't matter which one you do, 00:18:05.330 --> 00:18:09.370 align:middle line:84% so you might as well start with the ones that 00:18:09.370 --> 00:18:14.710 align:middle line:90% seems easiest to you. 00:18:14.710 --> 00:18:18.400 align:middle line:84% And you can see, anyway, it says exactly how many points each 00:18:18.400 --> 00:18:20.690 align:middle line:90% of them get, so you can-- 00:18:20.690 --> 00:18:24.490 align:middle line:84% if you don't have enough time to finish all of it, 00:18:24.490 --> 00:18:26.740 align:middle line:84% you can be sure to try to optimize. 00:18:26.740 --> 00:18:29.350 align:middle line:90% 00:18:29.350 --> 00:18:31.360 align:middle line:84% So what is the Cauchy convergence here? 00:18:31.360 --> 00:18:36.645 align:middle line:84% So again, we are in Rn-- [? R, ?] sorry. 00:18:36.645 --> 00:18:38.800 align:middle line:90% And we have n. 00:18:38.800 --> 00:18:42.180 align:middle line:84% So the Cauchy convergence theorem [INAUDIBLE] theorem. 00:18:42.180 --> 00:18:58.000 align:middle line:84% Let's say that if xn is a Cauchy sequence, then xn is convergent. 00:18:58.000 --> 00:19:02.160 align:middle line:90% 00:19:02.160 --> 00:19:05.160 align:middle line:84% That's the Cauchy convergence theorem. 00:19:05.160 --> 00:19:14.310 align:middle line:84% The Bolzano-Weierstrass theorem-- and this Cauchy 00:19:14.310 --> 00:19:17.220 align:middle line:84% convergence-- we talked about that. 00:19:17.220 --> 00:19:21.920 align:middle line:84% So the concepts-- when we in a minute 00:19:21.920 --> 00:19:26.720 align:middle line:84% review metric spaces, then convergent sequence-- that 00:19:26.720 --> 00:19:28.310 align:middle line:90% makes sense in a metric space. 00:19:28.310 --> 00:19:33.020 align:middle line:84% Cauchy sequence makes sense in a metric space. 00:19:33.020 --> 00:19:36.590 align:middle line:84% But it's not-- and the statement of this theorem 00:19:36.590 --> 00:19:40.730 align:middle line:84% makes sense in a metric space, but it's not always 00:19:40.730 --> 00:19:41.760 align:middle line:90% that it's true. 00:19:41.760 --> 00:19:46.370 align:middle line:84% It holds in R, but it doesn't hold in a general metric space. 00:19:46.370 --> 00:19:53.390 align:middle line:84% The Bolzano-Weierstrass theorem-- statement of that 00:19:53.390 --> 00:19:55.890 align:middle line:84% also makes sense in a general metric space, 00:19:55.890 --> 00:19:59.420 align:middle line:84% but it doesn't hold in a general metric space. 00:19:59.420 --> 00:20:04.780 align:middle line:84% So these are-- again, these are things about the real numbers. 00:20:04.780 --> 00:20:07.580 align:middle line:90% 00:20:07.580 --> 00:20:09.800 align:middle line:84% They extend to more than the real numbers, 00:20:09.800 --> 00:20:12.240 align:middle line:84% but you have to-- for other spaces, 00:20:12.240 --> 00:20:16.660 align:middle line:84% you have to prove it individually. 00:20:16.660 --> 00:20:18.920 align:middle line:84% So Bolzano-Weierstrass theorem says 00:20:18.920 --> 00:20:28.860 align:middle line:84% that if xn is a bounded sequence, 00:20:28.860 --> 00:20:35.710 align:middle line:84% then xn has a convergent subsequence. 00:20:35.710 --> 00:20:43.100 align:middle line:90% 00:20:43.100 --> 00:20:44.850 align:middle line:84% So that's the Bolzano-Weierstrass theorem. 00:20:44.850 --> 00:20:50.400 align:middle line:90% 00:20:50.400 --> 00:20:52.830 align:middle line:84% The next thing I want to review is 00:20:52.830 --> 00:20:55.640 align:middle line:84% what it means for a function to be continuous. 00:20:55.640 --> 00:20:58.930 align:middle line:84% So continuity is another important property. 00:20:58.930 --> 00:21:03.390 align:middle line:84% It's closely tied together with [INAUDIBLE] sequence, 00:21:03.390 --> 00:21:05.075 align:middle line:84% and we'll get to that in just a minute. 00:21:05.075 --> 00:21:24.488 align:middle line:90% 00:21:24.488 --> 00:21:27.320 align:middle line:90% So if you have a function-- 00:21:27.320 --> 00:21:31.400 align:middle line:84% let's say on R, but it could be defined on a subset of R. 00:21:31.400 --> 00:21:33.440 align:middle line:84% So if you have a function like this, 00:21:33.440 --> 00:21:47.020 align:middle line:84% then we say that f here is continuous at x0 if for all-- 00:21:47.020 --> 00:21:50.930 align:middle line:90% 00:21:50.930 --> 00:21:58.400 align:middle line:84% it's continuous at x0 if for all epsilon greater than 0, 00:21:58.400 --> 00:22:11.900 align:middle line:84% there exists a delta greater than 0 such that, if x minus x0 00:22:11.900 --> 00:22:21.800 align:middle line:84% is smaller than delta, then the images are smaller than epsilon. 00:22:21.800 --> 00:22:27.070 align:middle line:90% 00:22:27.070 --> 00:22:30.470 align:middle line:84% So that's what it means for a function to be continuous. 00:22:30.470 --> 00:22:38.830 align:middle line:84% And there was this theorem or lemma that tied things together. 00:22:38.830 --> 00:23:03.250 align:middle line:84% So lemma-- if f is continuous at x0 and xn converges to x0, 00:23:03.250 --> 00:23:09.590 align:middle line:84% then the images of this sequences-- 00:23:09.590 --> 00:23:11.560 align:middle line:90% so that's another sequence-- 00:23:11.560 --> 00:23:15.380 align:middle line:90% converge to f of x0. 00:23:15.380 --> 00:23:18.540 align:middle line:84% So we proved this even in a metric space, 00:23:18.540 --> 00:23:26.090 align:middle line:84% but remember that we're still in R. 00:23:26.090 --> 00:23:31.380 align:middle line:84% And then we had the algebraic properties of functions, 00:23:31.380 --> 00:23:34.340 align:middle line:84% and they were very similar to this. 00:23:34.340 --> 00:23:36.440 align:middle line:90% So let me just review them-- 00:23:36.440 --> 00:24:01.680 align:middle line:84% so algebraic properties of continuous functions. 00:24:01.680 --> 00:24:08.300 align:middle line:90% 00:24:08.300 --> 00:24:13.760 align:middle line:84% So imagine that you have f and g are continuous functions. 00:24:13.760 --> 00:24:18.044 align:middle line:90% 00:24:18.044 --> 00:24:22.300 align:middle line:84% [INAUDIBLE] everything is defined on R. 00:24:22.300 --> 00:24:26.140 align:middle line:90% c here is a constant. 00:24:26.140 --> 00:24:31.060 align:middle line:84% Well, then, property 1 that corresponds to that property 1 00:24:31.060 --> 00:24:35.890 align:middle line:84% is that this function plus c times this function here-- 00:24:35.890 --> 00:24:37.910 align:middle line:90% this here is also continuous. 00:24:37.910 --> 00:24:44.380 align:middle line:90% 00:24:44.380 --> 00:24:50.500 align:middle line:84% And then 2 is that if this function here is never 00:24:50.500 --> 00:24:55.720 align:middle line:84% equal to 0, then 1 over this function 00:24:55.720 --> 00:24:58.430 align:middle line:90% here is also continuous. 00:24:58.430 --> 00:25:01.330 align:middle line:90% 00:25:01.330 --> 00:25:03.700 align:middle line:84% And then-- sorry, that was not the-- 00:25:03.700 --> 00:25:07.660 align:middle line:84% sorry, I should make that the third property-- sorry-- 00:25:07.660 --> 00:25:12.500 align:middle line:84% that f times g is also continuous. 00:25:12.500 --> 00:25:14.720 align:middle line:84% And then the third property would 00:25:14.720 --> 00:25:19.250 align:middle line:84% be that if f is not equal to 0, [INAUDIBLE]. 00:25:19.250 --> 00:25:21.630 align:middle line:84% Then this function here is also continuous. 00:25:21.630 --> 00:25:24.750 align:middle line:90% 00:25:24.750 --> 00:25:26.810 align:middle line:90% And again, I divided it up. 00:25:26.810 --> 00:25:29.240 align:middle line:84% When I did it, I divided it up into four steps 00:25:29.240 --> 00:25:34.010 align:middle line:84% because I separated this out as two steps. 00:25:34.010 --> 00:25:37.820 align:middle line:84% And so from this one, we got that by-- just 00:25:37.820 --> 00:25:41.690 align:middle line:84% because then we proved that the constant function was continuous 00:25:41.690 --> 00:25:45.470 align:middle line:84% and the linear function, the function f of x equal to x, 00:25:45.470 --> 00:25:46.820 align:middle line:90% is continuous. 00:25:46.820 --> 00:25:51.320 align:middle line:84% Once we had that, then we just use 1 and 2 00:25:51.320 --> 00:25:58.100 align:middle line:84% here to prove that all polynomials continuous. 00:25:58.100 --> 00:26:00.740 align:middle line:84% And then if we bring in 3, then we 00:26:00.740 --> 00:26:05.120 align:middle line:84% have this with all rational functions are continuous except, 00:26:05.120 --> 00:26:08.748 align:middle line:84% of course, where the denominator vanishes [INAUDIBLE]. 00:26:08.748 --> 00:26:11.566 align:middle line:90% 00:26:11.566 --> 00:26:15.100 align:middle line:84% So that was how you built continuous functions 00:26:15.100 --> 00:26:19.870 align:middle line:90% from non-continuous functions. 00:26:19.870 --> 00:26:24.550 align:middle line:84% Then we proved the extreme value and the intermediate value 00:26:24.550 --> 00:26:25.100 align:middle line:90% theorem. 00:26:25.100 --> 00:26:26.560 align:middle line:90% We had these two theorems. 00:26:26.560 --> 00:26:27.800 align:middle line:90% And they were proved-- 00:26:27.800 --> 00:26:30.760 align:middle line:84% so one way to prove it that we did last time 00:26:30.760 --> 00:26:37.540 align:middle line:84% was to use this lemma that tied together 00:26:37.540 --> 00:26:40.390 align:middle line:90% continuity and sequences. 00:26:40.390 --> 00:26:45.010 align:middle line:84% And so we had the extreme value theorem. 00:26:45.010 --> 00:26:48.580 align:middle line:84% And this just says that if you have a function-- but now it's 00:26:48.580 --> 00:26:53.560 align:middle line:84% crucial that this here is a bounded interval 00:26:53.560 --> 00:26:56.990 align:middle line:84% and the endpoints are included in this. 00:26:56.990 --> 00:26:59.290 align:middle line:84% So if you have a function like this 00:26:59.290 --> 00:27:05.730 align:middle line:84% and the function is continuous, then 00:27:05.730 --> 00:27:10.050 align:middle line:84% the sup and the inf over the set here-- 00:27:10.050 --> 00:27:19.270 align:middle line:84% this achieves a max or, if you would, max of f and min of f 00:27:19.270 --> 00:27:19.890 align:middle line:90% are achieved. 00:27:19.890 --> 00:27:24.690 align:middle line:90% 00:27:24.690 --> 00:27:28.530 align:middle line:84% In particular, the function is bounded. 00:27:28.530 --> 00:27:31.890 align:middle line:84% So I'll just state it like this, a little bit brief like that. 00:27:31.890 --> 00:27:34.390 align:middle line:84% So that was the extreme value theorem. 00:27:34.390 --> 00:27:37.410 align:middle line:90% The intermediate value theorem-- 00:27:37.410 --> 00:27:40.410 align:middle line:84% it said that if you have a function-- 00:27:40.410 --> 00:27:43.950 align:middle line:84% imagine it starting off here and it ending there. 00:27:43.950 --> 00:27:46.260 align:middle line:90% So this is like a function. 00:27:46.260 --> 00:27:48.300 align:middle line:84% The intermediate value theorem says 00:27:48.300 --> 00:27:51.360 align:middle line:84% that everything in between these two 00:27:51.360 --> 00:27:54.760 align:middle line:90% is also the image of something. 00:27:54.760 --> 00:28:01.260 align:middle line:84% So if-- let's say that f of a is smaller than f of b. 00:28:01.260 --> 00:28:08.800 align:middle line:84% Then f y here is between f of a and f of b, 00:28:08.800 --> 00:28:16.810 align:middle line:84% then there exists x between a and b so that f of x 00:28:16.810 --> 00:28:18.250 align:middle line:90% is equal to y. 00:28:18.250 --> 00:28:22.220 align:middle line:84% Again, the functions we talk about are continuous, 00:28:22.220 --> 00:28:26.230 align:middle line:84% and they are defined on this [? closed ?] interval-- 00:28:26.230 --> 00:28:28.915 align:middle line:90% this interval. 00:28:28.915 --> 00:28:30.540 align:middle line:90% So that was those two theorems. 00:28:30.540 --> 00:28:33.100 align:middle line:90% 00:28:33.100 --> 00:28:40.630 align:middle line:84% And they used some property about that-- 00:28:40.630 --> 00:28:42.430 align:middle line:90% they used two things. 00:28:42.430 --> 00:28:47.680 align:middle line:84% They used the Bolzano-Weierstrass theorem, 00:28:47.680 --> 00:28:50.920 align:middle line:84% and then they used how sequences and continuous functions 00:28:50.920 --> 00:28:52.150 align:middle line:90% were tied together. 00:28:52.150 --> 00:28:56.215 align:middle line:84% The Bolzano-Weierstrass theorem is using something about 00:28:56.215 --> 00:28:58.670 align:middle line:84% that the real numbers are ordered 00:28:58.670 --> 00:29:02.670 align:middle line:84% and like this with sup and inf [? R. ?] 00:29:02.670 --> 00:29:05.655 align:middle line:84% You have the least upper bound property and the greatest lower 00:29:05.655 --> 00:29:06.280 align:middle line:90% bound property. 00:29:06.280 --> 00:29:09.640 align:middle line:90% 00:29:09.640 --> 00:29:13.350 align:middle line:84% Now another thing-- so we talked about-- 00:29:13.350 --> 00:29:17.670 align:middle line:84% so some of the highlights in terms of concepts 00:29:17.670 --> 00:29:22.770 align:middle line:84% are sequences and continuity of functions, 00:29:22.770 --> 00:29:26.410 align:middle line:84% and then various properties that follow from this. 00:29:26.410 --> 00:29:30.720 align:middle line:90% Then another thing is series. 00:29:30.720 --> 00:29:34.940 align:middle line:90% So that's another key concept. 00:29:34.940 --> 00:29:44.940 align:middle line:90% 00:29:44.940 --> 00:29:49.760 align:middle line:84% A series is where you're starting with a sequence. 00:29:49.760 --> 00:29:55.560 align:middle line:90% 00:29:55.560 --> 00:30:01.580 align:middle line:84% So series-- so you're starting with-- so this here 00:30:01.580 --> 00:30:02.815 align:middle line:90% is a sequence. 00:30:02.815 --> 00:30:06.290 align:middle line:90% 00:30:06.290 --> 00:30:12.470 align:middle line:84% And now you form another sequence Sn. 00:30:12.470 --> 00:30:15.290 align:middle line:84% But here you're summing, and it could start 00:30:15.290 --> 00:30:17.360 align:middle line:90% wherever you want to start it. 00:30:17.360 --> 00:30:19.820 align:middle line:90% Let's say it starts from 0. 00:30:19.820 --> 00:30:21.450 align:middle line:90% You're summing the first-- 00:30:21.450 --> 00:30:24.200 align:middle line:84% in this case, it would be-- because you're starting at 0, 00:30:24.200 --> 00:30:27.140 align:middle line:84% you're summing the first n plus 1. 00:30:27.140 --> 00:30:36.230 align:middle line:84% So a i-- this is a0 plus a n-- sorry, n plus 1 in this case. 00:30:36.230 --> 00:30:38.330 align:middle line:84% But you don't have to start at 0. 00:30:38.330 --> 00:30:41.960 align:middle line:84% If you started at 1, it would be the first n numbers. 00:30:41.960 --> 00:30:45.622 align:middle line:84% So you're looking at-- so a series is really 00:30:45.622 --> 00:30:48.080 align:middle line:84% kind of the same thing, but you think about it a little bit 00:30:48.080 --> 00:30:50.270 align:middle line:84% different, but it's really the same thing as a sequence, where 00:30:50.270 --> 00:30:51.937 align:middle line:84% you're starting with a sequence and then 00:30:51.937 --> 00:30:54.110 align:middle line:90% you form a new sequence. 00:30:54.110 --> 00:30:58.360 align:middle line:84% And then the key question here is-- so key question-- 00:30:58.360 --> 00:31:06.180 align:middle line:90% 00:31:06.180 --> 00:31:12.040 align:middle line:90% is a series convergent or not? 00:31:12.040 --> 00:31:18.960 align:middle line:90% 00:31:18.960 --> 00:31:23.160 align:middle line:90% And so that's a key question. 00:31:23.160 --> 00:31:26.340 align:middle line:84% And if it is convergent-- and you often-- 00:31:26.340 --> 00:31:31.110 align:middle line:84% the series is often written like this. 00:31:31.110 --> 00:31:36.450 align:middle line:84% And so this here will symbolize the series. 00:31:36.450 --> 00:31:39.150 align:middle line:84% And if the series is convergent, it 00:31:39.150 --> 00:31:42.695 align:middle line:84% will also symbolize the limit, so it has two meanings. 00:31:42.695 --> 00:31:46.330 align:middle line:90% 00:31:46.330 --> 00:31:49.100 align:middle line:84% But even if it's not convergent, then it [? worked. ?] 00:31:49.100 --> 00:31:51.570 align:middle line:84% Well, if it's not convergent, you would just write down-- 00:31:51.570 --> 00:31:56.030 align:middle line:84% and you would think about this as a series. 00:31:56.030 --> 00:32:05.150 align:middle line:84% Now, there are some series that are of particular importance, 00:32:05.150 --> 00:32:09.260 align:middle line:84% and they can then be used to determine whether or not 00:32:09.260 --> 00:32:13.190 align:middle line:90% other series are convergent. 00:32:13.190 --> 00:32:14.640 align:middle line:90% So let me just review them. 00:32:14.640 --> 00:32:18.430 align:middle line:84% And the most important of all is the geometric series. 00:32:18.430 --> 00:32:25.610 align:middle line:90% 00:32:25.610 --> 00:32:34.940 align:middle line:84% And the geometric series is the series c to the n, n equal to 0 00:32:34.940 --> 00:32:41.670 align:middle line:84% to infinity, and your c is just some fixed number, real number, 00:32:41.670 --> 00:32:44.780 align:middle line:90% and you're looking at this. 00:32:44.780 --> 00:32:48.950 align:middle line:84% And this is the geometric series, 00:32:48.950 --> 00:33:00.580 align:middle line:84% and this is convergent if the absolute value of c 00:33:00.580 --> 00:33:04.296 align:middle line:84% is less than 1 and divergent otherwise. 00:33:04.296 --> 00:33:12.880 align:middle line:90% 00:33:12.880 --> 00:33:19.120 align:middle line:84% And if it is convergent when c in absolute value 00:33:19.120 --> 00:33:30.070 align:middle line:84% is less than 1, then this sum here is equal to 1 minus c. 00:33:30.070 --> 00:33:32.420 align:middle line:90% But remember two things. 00:33:32.420 --> 00:33:35.720 align:middle line:84% One is that when you actually want to evaluate the sum, 00:33:35.720 --> 00:33:37.790 align:middle line:84% it's crucial where you're starting. 00:33:37.790 --> 00:33:41.990 align:middle line:84% If I started at 1, there is not any more equal to this. 00:33:41.990 --> 00:33:46.090 align:middle line:84% But you have to subtract the first element that was missing. 00:33:46.090 --> 00:33:48.770 align:middle line:84% So it's really-- for convergence, 00:33:48.770 --> 00:33:51.880 align:middle line:84% it doesn't matter where you start, but for the limits, 00:33:51.880 --> 00:33:54.200 align:middle line:90% it's key. 00:33:54.200 --> 00:33:55.320 align:middle line:90% That's one thing. 00:33:55.320 --> 00:33:59.840 align:middle line:84% And the other thing is that when you-- again, we 00:33:59.840 --> 00:34:03.740 align:middle line:84% talked about this-- that when you're given a series, even 00:34:03.740 --> 00:34:06.470 align:middle line:84% before there's any question, the first thing-- when you have 00:34:06.470 --> 00:34:09.710 align:middle line:84% a series, the first thing you look at is, does this a n 00:34:09.710 --> 00:34:12.210 align:middle line:90% converge to 0 or not? 00:34:12.210 --> 00:34:15.350 align:middle line:84% If a n does not go to 0, well, the series 00:34:15.350 --> 00:34:16.909 align:middle line:90% is not going to converge. 00:34:16.909 --> 00:34:21.890 align:middle line:84% There's no need to apply any fancy test. 00:34:21.890 --> 00:34:26.100 align:middle line:84% And for a series, the question is always 00:34:26.100 --> 00:34:27.719 align:middle line:90% whether it's convergent or not. 00:34:27.719 --> 00:34:32.677 align:middle line:90% 00:34:32.677 --> 00:34:37.429 align:middle line:90% So that's a geometric series. 00:34:37.429 --> 00:34:41.989 align:middle line:84% And then there are two other series that we talked about. 00:34:41.989 --> 00:34:43.010 align:middle line:90% And let me just-- 00:34:43.010 --> 00:34:45.949 align:middle line:90% 00:34:45.949 --> 00:34:49.690 align:middle line:84% not only talk about, but we also determined whether or not 00:34:49.690 --> 00:34:52.570 align:middle line:84% they were convergent, and then we 00:34:52.570 --> 00:34:56.239 align:middle line:84% used them to determine whether other series were convergent. 00:34:56.239 --> 00:34:58.870 align:middle line:90% 00:34:58.870 --> 00:35:01.810 align:middle line:84% But again, the most important of all of them 00:35:01.810 --> 00:35:03.250 align:middle line:90% is the geometric series. 00:35:03.250 --> 00:35:06.700 align:middle line:84% That's the one that we used to determine 00:35:06.700 --> 00:35:10.730 align:middle line:84% whether power series in general was convergent. 00:35:10.730 --> 00:35:13.570 align:middle line:84% But there are two others that are worth mentioning. 00:35:13.570 --> 00:35:15.120 align:middle line:90% One is the harmonic series. 00:35:15.120 --> 00:35:20.410 align:middle line:90% 00:35:20.410 --> 00:35:26.410 align:middle line:84% The harmonic series is the series that is summing 1/n. 00:35:26.410 --> 00:35:28.250 align:middle line:90% So this here is divergent. 00:35:28.250 --> 00:35:31.210 align:middle line:90% 00:35:31.210 --> 00:35:34.510 align:middle line:90% So this follows second. 00:35:34.510 --> 00:35:39.160 align:middle line:84% And then there is a closely related-- 00:35:39.160 --> 00:35:41.860 align:middle line:90% let's call it 2 and 1/2. 00:35:41.860 --> 00:35:44.492 align:middle line:84% That's called the alternating harmonic series. 00:35:44.492 --> 00:35:53.840 align:middle line:90% 00:35:53.840 --> 00:35:58.100 align:middle line:84% So the alternating was that it was like the harmonic series, 00:35:58.100 --> 00:36:00.450 align:middle line:84% but the sign of it was alternating. 00:36:00.450 --> 00:36:03.800 align:middle line:90% 00:36:03.800 --> 00:36:06.000 align:middle line:84% And this here is actually convergent. 00:36:06.000 --> 00:36:09.080 align:middle line:90% 00:36:09.080 --> 00:36:16.080 align:middle line:84% And then-- so there's 1, 2, 2 and 1/2, and then there's 3, 00:36:16.080 --> 00:36:22.550 align:middle line:90% and that's this series here. 00:36:22.550 --> 00:36:26.638 align:middle line:84% We looked at this series here, and this series was convergent. 00:36:26.638 --> 00:36:30.470 align:middle line:90% 00:36:30.470 --> 00:36:33.815 align:middle line:84% The only one where we determine what the sum was, 00:36:33.815 --> 00:36:35.680 align:middle line:84% was actually the geometric series. 00:36:35.680 --> 00:36:40.355 align:middle line:90% 00:36:40.355 --> 00:36:43.225 align:middle line:90% So that's some key series. 00:36:43.225 --> 00:36:56.800 align:middle line:90% 00:36:56.800 --> 00:36:59.710 align:middle line:84% Then there is-- when you have a series, 00:36:59.710 --> 00:37:03.940 align:middle line:84% there was one concept that proved to be quite useful, 00:37:03.940 --> 00:37:12.112 align:middle line:84% and that was the concept of absolute convergence 00:37:12.112 --> 00:37:15.400 align:middle line:90% [INAUDIBLE]. 00:37:15.400 --> 00:37:19.060 align:middle line:84% And so you were looking at a series. 00:37:19.060 --> 00:37:27.050 align:middle line:84% And absolute convergence was that this sum here 00:37:27.050 --> 00:37:28.360 align:middle line:90% was convergent. 00:37:28.360 --> 00:37:39.430 align:middle line:84% So this here is absolute convergent if this series, 00:37:39.430 --> 00:37:42.850 align:middle line:84% where you just-- the a n is replaced by the absolute value-- 00:37:42.850 --> 00:37:44.200 align:middle line:90% is convergent. 00:37:44.200 --> 00:37:48.570 align:middle line:84% So of course, for a series of non-negative numbers, 00:37:48.570 --> 00:37:51.150 align:middle line:84% it's automatically-- if it's convergent, 00:37:51.150 --> 00:37:54.150 align:middle line:84% it's automatically absolute convergent. 00:37:54.150 --> 00:37:59.226 align:middle line:84% And we talked about it a little bit 00:37:59.226 --> 00:38:02.160 align:middle line:84% that one of the things about an absolute convergent 00:38:02.160 --> 00:38:06.430 align:middle line:84% is that you could reorder the way you're summing things, 00:38:06.430 --> 00:38:09.090 align:middle line:84% and it wouldn't matter for the limit. 00:38:09.090 --> 00:38:12.658 align:middle line:84% But if it's not absolute convergence that really matters. 00:38:12.658 --> 00:38:13.450 align:middle line:90% We talked about it. 00:38:13.450 --> 00:38:15.130 align:middle line:84% We didn't really prove anything there, 00:38:15.130 --> 00:38:18.720 align:middle line:84% but I think it was clear that you could do this. 00:38:18.720 --> 00:38:21.540 align:middle line:84% One thing where we used it was when we looked 00:38:21.540 --> 00:38:23.640 align:middle line:90% at the product of series. 00:38:23.640 --> 00:38:27.870 align:middle line:84% Then the product of series was convergent if each of them 00:38:27.870 --> 00:38:29.350 align:middle line:90% was absolute convergence. 00:38:29.350 --> 00:38:33.240 align:middle line:84% And there you saw that we estimated some remainder term. 00:38:33.240 --> 00:38:36.420 align:middle line:84% And there, it was really key that we 00:38:36.420 --> 00:38:39.226 align:middle line:90% had absolute convergence. 00:38:39.226 --> 00:38:42.100 align:middle line:84% So anyway, so absolute convergence is an important-- 00:38:42.100 --> 00:38:45.010 align:middle line:90% 00:38:45.010 --> 00:38:48.710 align:middle line:84% for series, it's a stronger property than convergence, 00:38:48.710 --> 00:38:52.160 align:middle line:84% so absolute convergence implies convergence. 00:38:52.160 --> 00:38:54.280 align:middle line:84% Maybe I should write this as a theorem. 00:38:54.280 --> 00:38:57.193 align:middle line:84% So we had the theorem-- so absolute convergence. 00:38:57.193 --> 00:39:03.820 align:middle line:90% 00:39:03.820 --> 00:39:07.242 align:middle line:84% So this implies convergent, but it's stronger. 00:39:07.242 --> 00:39:08.325 align:middle line:90% In general, it's stronger. 00:39:08.325 --> 00:39:11.620 align:middle line:90% 00:39:11.620 --> 00:39:17.590 align:middle line:84% And now-- then we had some tests. 00:39:17.590 --> 00:39:18.860 align:middle line:90% Then we had some tests. 00:39:18.860 --> 00:39:23.560 align:middle line:84% And so we talked in particular about-- 00:39:23.560 --> 00:39:27.010 align:middle line:84% there were a few other [? tests ?] that we talked 00:39:27.010 --> 00:39:30.620 align:middle line:90% about, but I didn't-- 00:39:30.620 --> 00:39:34.210 align:middle line:84% they're not actually going to be on the midterm. 00:39:34.210 --> 00:39:37.030 align:middle line:84% And some of them, we'll get back to. 00:39:37.030 --> 00:39:40.530 align:middle line:84% This one here won't be on the midterm [INAUDIBLE] alternating. 00:39:40.530 --> 00:39:43.920 align:middle line:84% But we talked about how to improve the convergence. 00:39:43.920 --> 00:39:47.400 align:middle line:84% And the important-- and when we talked about it, 00:39:47.400 --> 00:39:50.700 align:middle line:84% then we discovered that this thing here-- 00:39:50.700 --> 00:39:53.520 align:middle line:84% that there was a stronger statement that if you have 00:39:53.520 --> 00:39:58.800 align:middle line:84% alternating and the a n are all non-negative and these guys, 00:39:58.800 --> 00:40:00.480 align:middle line:90% it's decreasing-- 00:40:00.480 --> 00:40:03.390 align:middle line:90% goes decreasing down to 0-- 00:40:03.390 --> 00:40:09.141 align:middle line:84% then actually, any alternating sequence like that 00:40:09.141 --> 00:40:11.550 align:middle line:90% would be convergent. 00:40:11.550 --> 00:40:15.510 align:middle line:84% Anyway-- but for the test we really talked about 00:40:15.510 --> 00:40:17.190 align:middle line:90% was two tests. 00:40:17.190 --> 00:40:18.695 align:middle line:90% One of them had two parts. 00:40:18.695 --> 00:40:23.010 align:middle line:90% 00:40:23.010 --> 00:40:27.630 align:middle line:84% And then, in a minute, we'll talk about how we apply them 00:40:27.630 --> 00:40:31.840 align:middle line:90% to power series, [INAUDIBLE]. 00:40:31.840 --> 00:40:40.700 align:middle line:84% So we had some tests to determine 00:40:40.700 --> 00:40:44.305 align:middle line:90% whether or not a series-- 00:40:44.305 --> 00:40:52.470 align:middle line:90% or not-- a series is convergent. 00:40:52.470 --> 00:40:57.180 align:middle line:90% 00:40:57.180 --> 00:40:58.290 align:middle line:90% So what are the tests? 00:40:58.290 --> 00:41:00.475 align:middle line:84% So there are the comparison tests. 00:41:00.475 --> 00:41:03.920 align:middle line:90% 00:41:03.920 --> 00:41:06.740 align:middle line:84% Let me just first write down the test. 00:41:06.740 --> 00:41:09.650 align:middle line:84% So the comparison test-- and there was two versions, version 00:41:09.650 --> 00:41:19.640 align:middle line:84% A and version B. Then there was the ratio test, 00:41:19.640 --> 00:41:21.400 align:middle line:84% and then there was the root test. 00:41:21.400 --> 00:41:28.820 align:middle line:90% 00:41:28.820 --> 00:41:31.050 align:middle line:84% The ratio test and the root test-- 00:41:31.050 --> 00:41:36.390 align:middle line:84% they are both offsprings of the geometric series. 00:41:36.390 --> 00:41:39.780 align:middle line:84% In both of these, you got-- let's just talk 00:41:39.780 --> 00:41:41.850 align:middle line:90% about these first. 00:41:41.850 --> 00:41:45.780 align:middle line:84% The ratio test was when we were looking at this-- 00:41:45.780 --> 00:41:54.270 align:middle line:90% the ratio of a n plus 1 over an. 00:41:54.270 --> 00:41:59.070 align:middle line:84% So you are implicitly assuming that the a n does not vanish. 00:41:59.070 --> 00:42:00.270 align:middle line:90% I'm not sure. 00:42:00.270 --> 00:42:03.120 align:middle line:90% So you can form this ratio. 00:42:03.120 --> 00:42:07.560 align:middle line:84% And then if this limit here as n goes to infinity-- 00:42:07.560 --> 00:42:13.530 align:middle line:84% if this limit here is less than 1, 00:42:13.530 --> 00:42:20.730 align:middle line:84% then the ratio test says that the series was convergent. 00:42:20.730 --> 00:42:26.070 align:middle line:84% If the ratio was bigger than 1, then this ordinary element a n 00:42:26.070 --> 00:42:28.690 align:middle line:90% would not even go to 0. 00:42:28.690 --> 00:42:30.960 align:middle line:90% So this here is-- 00:42:30.960 --> 00:42:32.400 align:middle line:90% so in this case here-- 00:42:32.400 --> 00:42:33.930 align:middle line:90% maybe I'll write it over here. 00:42:33.930 --> 00:42:42.200 align:middle line:90% 00:42:42.200 --> 00:42:43.735 align:middle line:90% So for the ratio test-- 00:42:43.735 --> 00:42:48.429 align:middle line:90% 00:42:48.429 --> 00:42:52.130 align:middle line:84% so the ratio test-- you assume that the a n-- 00:42:52.130 --> 00:42:54.890 align:middle line:90% none of the a n equals 0. 00:42:54.890 --> 00:43:00.680 align:middle line:84% And you're looking at a n plus 1 over a n. 00:43:00.680 --> 00:43:04.670 align:middle line:90% And if this limit here-- 00:43:04.670 --> 00:43:06.230 align:middle line:90% let's call this limit-- 00:43:06.230 --> 00:43:08.130 align:middle line:84% let's assume that the limit exists. 00:43:08.130 --> 00:43:10.340 align:middle line:90% Of course, it may not exist 00:43:10.340 --> 00:43:13.370 align:middle line:90% But if the limit existed-- 00:43:13.370 --> 00:43:19.260 align:middle line:90% so if c here is less than 1-- 00:43:19.260 --> 00:43:22.250 align:middle line:84% it's always non-negative because these are non-negative numbers-- 00:43:22.250 --> 00:43:26.287 align:middle line:84% if c is less than 1, then the series is convergent. 00:43:26.287 --> 00:43:30.860 align:middle line:90% 00:43:30.860 --> 00:43:34.560 align:middle line:84% If this c here is strictly bigger than 1, 00:43:34.560 --> 00:43:41.700 align:middle line:84% this implies that the series is divergent. 00:43:41.700 --> 00:43:45.185 align:middle line:84% And if the c was equal to 1, then it's unclear-- 00:43:45.185 --> 00:43:48.870 align:middle line:90% 00:43:48.870 --> 00:43:52.685 align:middle line:84% and so unclear, which means that further tests are needed. 00:43:52.685 --> 00:43:55.320 align:middle line:90% 00:43:55.320 --> 00:44:02.320 align:middle line:84% And so now-- so that's the ratio test. 00:44:02.320 --> 00:44:04.220 align:middle line:90% And the root test was similar. 00:44:04.220 --> 00:44:06.870 align:middle line:90% 00:44:06.870 --> 00:44:08.260 align:middle line:90% The root test was-- 00:44:08.260 --> 00:44:10.920 align:middle line:90% so you're looking at the a n. 00:44:10.920 --> 00:44:13.710 align:middle line:84% You take the-- so this is the root test. 00:44:13.710 --> 00:44:17.760 align:middle line:84% And again, this is tested by comparing it 00:44:17.760 --> 00:44:19.840 align:middle line:90% to the geometric series-- 00:44:19.840 --> 00:44:22.410 align:middle line:90% so the root test. 00:44:22.410 --> 00:44:29.420 align:middle line:84% So you take the a n, take the n-th root of this, 00:44:29.420 --> 00:44:31.760 align:middle line:84% and you're looking at the limit here. 00:44:31.760 --> 00:44:36.320 align:middle line:84% And we actually looked at the lim sup. 00:44:36.320 --> 00:44:38.510 align:middle line:90% I'm writing the lim sup. 00:44:38.510 --> 00:44:45.110 align:middle line:84% The reason why I want the lim sup is that I want to make sure 00:44:45.110 --> 00:44:46.250 align:middle line:90% that it's-- 00:44:46.250 --> 00:44:49.020 align:middle line:84% later on, I want to apply more generally. 00:44:49.020 --> 00:44:52.170 align:middle line:90% 00:44:52.170 --> 00:44:58.070 align:middle line:84% So if the lim sup here was less than 1-- 00:44:58.070 --> 00:45:02.000 align:middle line:84% let's call the lim sup-- let's call it c. 00:45:02.000 --> 00:45:07.290 align:middle line:84% If c here is less than 1, then the series is convergent. 00:45:07.290 --> 00:45:12.230 align:middle line:90% 00:45:12.230 --> 00:45:16.940 align:middle line:84% 2-- if the c is bigger than 1, series is divergent. 00:45:16.940 --> 00:45:19.190 align:middle line:90% This fails the most basic test-- 00:45:19.190 --> 00:45:21.680 align:middle line:90% divergent. 00:45:21.680 --> 00:45:26.580 align:middle line:84% And again, 3-- if c is equal to 1, then it's unclear. 00:45:26.580 --> 00:45:27.778 align:middle line:90% Further tests are needed. 00:45:27.778 --> 00:45:32.590 align:middle line:90% 00:45:32.590 --> 00:45:35.020 align:middle line:90% That was those two tests. 00:45:35.020 --> 00:45:38.420 align:middle line:84% The comparison tests-- let's get back to that. 00:45:38.420 --> 00:45:43.690 align:middle line:84% The comparison test has two parts that you had. 00:45:43.690 --> 00:45:46.990 align:middle line:90% Suppose you had a n and bn. 00:45:46.990 --> 00:45:49.420 align:middle line:84% The first part is this statement. 00:45:49.420 --> 00:45:52.390 align:middle line:84% If you have two series, one based 00:45:52.390 --> 00:45:55.550 align:middle line:84% on the a n, another series based on the bn-- 00:45:55.550 --> 00:45:57.370 align:middle line:90% if you have this-- 00:45:57.370 --> 00:46:01.366 align:middle line:90% if bn-- so this here-- 00:46:01.366 --> 00:46:10.120 align:middle line:84% if this series here is convergent, then [INAUDIBLE] 00:46:10.120 --> 00:46:11.351 align:middle line:90% is convergent. 00:46:11.351 --> 00:46:14.650 align:middle line:90% 00:46:14.650 --> 00:46:19.910 align:middle line:84% And this is sometimes used both exactly like this, 00:46:19.910 --> 00:46:22.840 align:middle line:84% and it's also sometimes used where you know that this one 00:46:22.840 --> 00:46:26.194 align:middle line:84% here is divergent, and so you compute that this one is 00:46:26.194 --> 00:46:31.220 align:middle line:84% [? divergent. ?] The other part of the comparison test was 00:46:31.220 --> 00:46:37.490 align:middle line:84% if you had the limit here, if this limit here exists and it 00:46:37.490 --> 00:46:43.800 align:middle line:84% was something L, then actually, it [? says ?] that-- 00:46:43.800 --> 00:46:45.720 align:middle line:84% let's say L here is not equal to 0. 00:46:45.720 --> 00:46:50.951 align:middle line:90% 00:46:50.951 --> 00:46:56.730 align:middle line:84% Then if one of them is convergent, 00:46:56.730 --> 00:46:58.200 align:middle line:84% the other one is also convergent. 00:46:58.200 --> 00:47:01.290 align:middle line:90% So if a n-- 00:47:01.290 --> 00:47:06.490 align:middle line:84% or bn, let's say-- bn is convergent, 00:47:06.490 --> 00:47:10.016 align:middle line:84% then it implies that a n is convergent. 00:47:10.016 --> 00:47:12.890 align:middle line:90% 00:47:12.890 --> 00:47:14.900 align:middle line:90% And vice versa-- this actually-- 00:47:14.900 --> 00:47:18.372 align:middle line:84% [INAUDIBLE] this is convergent [INAUDIBLE]. 00:47:18.372 --> 00:47:21.110 align:middle line:90% 00:47:21.110 --> 00:47:24.800 align:middle line:84% So those were the three tests that we really went through. 00:47:24.800 --> 00:47:26.930 align:middle line:84% There were a few other tests, like, again, 00:47:26.930 --> 00:47:30.110 align:middle line:84% this alternating series test that we saw a proof, 00:47:30.110 --> 00:47:31.970 align:middle line:84% but it's not going to be on midterms, 00:47:31.970 --> 00:47:33.880 align:middle line:84% and we will return to this later. 00:47:33.880 --> 00:47:34.810 align:middle line:90% Yeah? 00:47:34.810 --> 00:47:37.120 align:middle line:84% AUDIENCE: Are there patterns of when 00:47:37.120 --> 00:47:39.020 align:middle line:90% we should apply which tests? 00:47:39.020 --> 00:47:40.360 align:middle line:90% TOBIAS COLDING: Yeah-- no. 00:47:40.360 --> 00:47:46.335 align:middle line:84% And I want to get to this in just a second. 00:47:46.335 --> 00:47:50.575 align:middle line:90% 00:47:50.575 --> 00:47:52.250 align:middle line:84% Let me just say a couple of words first, 00:47:52.250 --> 00:47:56.210 align:middle line:84% and then let me jump to some of the examples of problems, 00:47:56.210 --> 00:48:00.151 align:middle line:84% and then I'll come back, and then we'll do a little bit more. 00:48:00.151 --> 00:48:05.950 align:middle line:84% But let's just talk about power series, and maybe-- 00:48:05.950 --> 00:48:10.090 align:middle line:84% let me think [INAUDIBLE] and whether-- 00:48:10.090 --> 00:48:14.740 align:middle line:84% just now when you were asking the question-- 00:48:14.740 --> 00:48:16.910 align:middle line:84% no, let's just review the power series. 00:48:16.910 --> 00:48:21.060 align:middle line:90% 00:48:21.060 --> 00:48:24.540 align:middle line:84% So our power series was a series, 00:48:24.540 --> 00:48:28.791 align:middle line:84% but it was something that also defined a function. 00:48:28.791 --> 00:48:30.930 align:middle line:90% And so you take-- 00:48:30.930 --> 00:48:40.805 align:middle line:90% so you have-- so a power change. 00:48:40.805 --> 00:48:46.260 align:middle line:90% 00:48:46.260 --> 00:48:49.194 align:middle line:90% So you have a sequence a n. 00:48:49.194 --> 00:48:51.540 align:middle line:90% And then you think for each x-- 00:48:51.540 --> 00:48:54.210 align:middle line:84% for x, think about x as being fixed-- 00:48:54.210 --> 00:49:00.130 align:middle line:84% you form a series, and this is the series that you form. 00:49:00.130 --> 00:49:03.270 align:middle line:90% 00:49:03.270 --> 00:49:06.210 align:middle line:84% And so for each x you can ask, is 00:49:06.210 --> 00:49:08.910 align:middle line:90% this series convergent or not? 00:49:08.910 --> 00:49:13.230 align:middle line:84% This here is called a power series. 00:49:13.230 --> 00:49:19.000 align:middle line:84% And the key point for power series is, again, of course, 00:49:19.000 --> 00:49:21.640 align:middle line:84% whether or not this is convergent. 00:49:21.640 --> 00:49:25.310 align:middle line:84% And this gave rise to the radius of convergence. 00:49:25.310 --> 00:49:31.600 align:middle line:90% 00:49:31.600 --> 00:49:36.340 align:middle line:84% And so the radius of convergence was-- 00:49:36.340 --> 00:49:48.220 align:middle line:84% that we defined-- we looked at lim sup of the n-th root of a n, 00:49:48.220 --> 00:49:50.630 align:middle line:90% like that. 00:49:50.630 --> 00:49:54.040 align:middle line:84% And we looked at when this thing here-- 00:49:54.040 --> 00:49:57.910 align:middle line:90% if this thing here is-- 00:49:57.910 --> 00:50:03.760 align:middle line:84% actually, we defined this thing here, and-- 00:50:03.760 --> 00:50:05.270 align:middle line:90% so we got this thing here. 00:50:05.270 --> 00:50:07.670 align:middle line:84% And then we defined the radius of convergence. 00:50:07.670 --> 00:50:15.070 align:middle line:90% So the radius of convergence-- 00:50:15.070 --> 00:50:20.796 align:middle line:84% this is R, and the R is 1 over this lim sup. 00:50:20.796 --> 00:50:23.490 align:middle line:90% 00:50:23.490 --> 00:50:27.600 align:middle line:84% But there was an understanding in this. 00:50:27.600 --> 00:50:32.310 align:middle line:84% The understanding is that if this number here-- 00:50:32.310 --> 00:50:36.930 align:middle line:84% this number potentially could be infinite. 00:50:36.930 --> 00:50:42.360 align:middle line:84% If it was infinite, it meant that this thing here was 0. 00:50:42.360 --> 00:50:46.590 align:middle line:90% If this one here is infinity-- 00:50:46.590 --> 00:50:54.018 align:middle line:84% if this here is 0, then this here would be infinity. 00:50:54.018 --> 00:50:57.690 align:middle line:84% So that's like a convention in this particular case. 00:50:57.690 --> 00:51:00.760 align:middle line:84% Usually, we don't want to divide by 0 [INAUDIBLE]. 00:51:00.760 --> 00:51:05.370 align:middle line:84% But this is a convention in this particular case. 00:51:05.370 --> 00:51:08.280 align:middle line:84% And so using this radius of convergence, 00:51:08.280 --> 00:51:11.140 align:middle line:84% we saw that if you take a power series, 00:51:11.140 --> 00:51:15.200 align:middle line:84% if you have any power series like this-- 00:51:15.200 --> 00:51:16.850 align:middle line:90% here you have 0-- 00:51:16.850 --> 00:51:20.940 align:middle line:84% any power series like this, it's going to be-- 00:51:20.940 --> 00:51:23.510 align:middle line:90% if this here is R-- 00:51:23.510 --> 00:51:27.860 align:middle line:84% and again, R could be 0 or it could be infinity. 00:51:27.860 --> 00:51:30.350 align:middle line:90% But if this is R-- 00:51:30.350 --> 00:51:33.170 align:middle line:84% sorry, I should write this with a minus sign. 00:51:33.170 --> 00:51:37.190 align:middle line:90% Then for x in between-- 00:51:37.190 --> 00:51:41.690 align:middle line:84% for x not including the endpoints, but in between-- 00:51:41.690 --> 00:51:43.880 align:middle line:90% the series is convergent. 00:51:43.880 --> 00:51:45.680 align:middle line:90% The power series is convergent. 00:51:45.680 --> 00:51:49.740 align:middle line:84% And outside here, it's divergent. 00:51:49.740 --> 00:51:52.820 align:middle line:90% 00:51:52.820 --> 00:51:54.360 align:middle line:90% Outside is divergent. 00:51:54.360 --> 00:51:55.742 align:middle line:90% Here it's convergent. 00:51:55.742 --> 00:51:58.640 align:middle line:90% 00:51:58.640 --> 00:52:04.216 align:middle line:84% And at those two points here, unclear. 00:52:04.216 --> 00:52:06.750 align:middle line:90% [INAUDIBLE] 00:52:06.750 --> 00:52:08.840 align:middle line:90% So this is always this picture. 00:52:08.840 --> 00:52:13.480 align:middle line:84% So this comes back to your question about which tests-- 00:52:13.480 --> 00:52:15.550 align:middle line:84% of course, you can use any test you want, 00:52:15.550 --> 00:52:18.850 align:middle line:84% but when I-- like if you're looking at a series, 00:52:18.850 --> 00:52:20.830 align:middle line:84% you can use whatever test you want. 00:52:20.830 --> 00:52:23.850 align:middle line:84% But if it was a power series, you already 00:52:23.850 --> 00:52:31.240 align:middle line:84% know that there's some interval around 0 where it's convergent, 00:52:31.240 --> 00:52:33.730 align:middle line:84% and then there's a question about the endpoints. 00:52:33.730 --> 00:52:41.555 align:middle line:84% And so now let's look at some examples of this. 00:52:41.555 --> 00:52:44.100 align:middle line:90% 00:52:44.100 --> 00:52:47.500 align:middle line:90% And suppose you look at-- 00:52:47.500 --> 00:52:59.280 align:middle line:84% and so It's usually not too hard to use either-- 00:52:59.280 --> 00:53:02.670 align:middle line:90% 00:53:02.670 --> 00:53:05.160 align:middle line:84% this here was based on the root test, 00:53:05.160 --> 00:53:08.290 align:middle line:84% so you use the root test or the ratio test for power 00:53:08.290 --> 00:53:12.260 align:middle line:84% [? convergence. ?] But you can take advantage of it sometimes 00:53:12.260 --> 00:53:14.130 align:middle line:90% it's just slightly easier. 00:53:14.130 --> 00:53:16.310 align:middle line:84% So let's look at two examples here. 00:53:16.310 --> 00:53:32.420 align:middle line:84% So suppose that you're asked to find the radius of convergence 00:53:32.420 --> 00:53:33.540 align:middle line:90% for the power series. 00:53:33.540 --> 00:53:40.250 align:middle line:90% 00:53:40.250 --> 00:53:45.920 align:middle line:84% And then let's say that this is n to the fifth power, xn. 00:53:45.920 --> 00:53:50.840 align:middle line:84% So suppose that you are-- this is a power series you're given, 00:53:50.840 --> 00:53:55.710 align:middle line:84% and you want to find the radius of convergence. 00:53:55.710 --> 00:54:03.950 align:middle line:84% You can either just plug it into this test, 00:54:03.950 --> 00:54:06.750 align:middle line:84% or you can just say, OK, we'll just apply it. 00:54:06.750 --> 00:54:09.240 align:middle line:84% We know that it's some interval, and I just 00:54:09.240 --> 00:54:12.570 align:middle line:84% want to see for which x does this here converge. 00:54:12.570 --> 00:54:14.790 align:middle line:84% So if I looked-- in this case, you 00:54:14.790 --> 00:54:20.160 align:middle line:84% might find it slightly easier to do the root test. 00:54:20.160 --> 00:54:23.780 align:middle line:84% So if you're looking at the n plus 1 element in this series-- 00:54:23.780 --> 00:54:26.730 align:middle line:90% 00:54:26.730 --> 00:54:29.900 align:middle line:84% n elements would be this to the fifth power. 00:54:29.900 --> 00:54:34.470 align:middle line:84% Think about x as being fixed, and you're asking for which x. 00:54:34.470 --> 00:54:36.600 align:middle line:84% And then you want to-- for a given x, 00:54:36.600 --> 00:54:39.190 align:middle line:84% you want to determine whether the series-- 00:54:39.190 --> 00:54:44.310 align:middle line:84% this series here-- is convergent. 00:54:44.310 --> 00:54:46.980 align:middle line:84% So if you were using the root test, 00:54:46.980 --> 00:54:48.720 align:middle line:84% you would look at this thing here. 00:54:48.720 --> 00:54:51.120 align:middle line:84% And you might find in this case slightly easier 00:54:51.120 --> 00:54:53.610 align:middle line:90% to use the root test. 00:54:53.610 --> 00:54:56.340 align:middle line:84% So if you're looking at this, this thing 00:54:56.340 --> 00:55:02.820 align:middle line:84% here is now n plus 1 to the fifth power times x to the n 00:55:02.820 --> 00:55:10.100 align:middle line:84% plus 1 divided by n to the fifth power times x to the n. 00:55:10.100 --> 00:55:12.230 align:middle line:90% That's that ratio. 00:55:12.230 --> 00:55:15.380 align:middle line:84% And so of course, you can write this here. 00:55:15.380 --> 00:55:22.520 align:middle line:84% This is like-- sorry, n plus 1 multiplied by itself five times, 00:55:22.520 --> 00:55:27.350 align:middle line:84% divided by n multiplied by itself five times. 00:55:27.350 --> 00:55:31.070 align:middle line:84% And then this thing here-- you can write it as x to the n, 00:55:31.070 --> 00:55:36.980 align:middle line:90% times x, over x to the n. 00:55:36.980 --> 00:55:41.180 align:middle line:84% So you see, this thing here, you can then think about as n 00:55:41.180 --> 00:55:54.030 align:middle line:84% plus 1 over n to the fifth power, and times x. 00:55:54.030 --> 00:55:58.160 align:middle line:84% And I can write this thing here in the usual way. 00:55:58.160 --> 00:56:02.760 align:middle line:84% If you want, you can write it as 1 plus 1/n. 00:56:02.760 --> 00:56:04.940 align:middle line:84% I'm dividing by n in both denominator. 00:56:04.940 --> 00:56:07.330 align:middle line:84% And denominator, you can write it like this-- 00:56:07.330 --> 00:56:12.040 align:middle line:90% to the fifth power times x. 00:56:12.040 --> 00:56:15.325 align:middle line:90% And so now-- the root test-- 00:56:15.325 --> 00:56:17.870 align:middle line:90% 00:56:17.870 --> 00:56:19.540 align:middle line:90% it's asking whether this-- 00:56:19.540 --> 00:56:22.450 align:middle line:84% the absolute value of this root test 00:56:22.450 --> 00:56:29.080 align:middle line:84% would be then asking if the limit here of x n plus 1 over-- 00:56:29.080 --> 00:56:32.020 align:middle line:90% I'm sorry, a n plus 1 over a n-- 00:56:32.020 --> 00:56:34.480 align:middle line:90% if this here has a limit. 00:56:34.480 --> 00:56:39.200 align:middle line:84% And then it's asking you, if that limit is less than 1, 00:56:39.200 --> 00:56:41.360 align:middle line:84% if it has a limit and the limit is less than 1, 00:56:41.360 --> 00:56:43.640 align:middle line:84% then the series is convergent, and it's divergent 00:56:43.640 --> 00:56:44.890 align:middle line:90% if it's bigger than 1. 00:56:44.890 --> 00:56:50.950 align:middle line:84% But in this case, this thing here, this ratio here, 00:56:50.950 --> 00:56:57.970 align:middle line:84% is 1 plus 1/n to the fifth power, times x. 00:56:57.970 --> 00:57:00.550 align:middle line:84% But this is all non-negative, so the absolute value is just 00:57:00.550 --> 00:57:01.620 align:middle line:90% going over here. 00:57:01.620 --> 00:57:05.090 align:middle line:90% 00:57:05.090 --> 00:57:06.960 align:middle line:84% The algebraic property of the limit 00:57:06.960 --> 00:57:10.790 align:middle line:84% says that this one here goes to 0, 00:57:10.790 --> 00:57:13.920 align:middle line:84% and so raised to the fifth power still goes to 0. 00:57:13.920 --> 00:57:18.120 align:middle line:84% It's multiplying a bunch of sequences five times together. 00:57:18.120 --> 00:57:18.890 align:middle line:90% That gives you 0. 00:57:18.890 --> 00:57:22.700 align:middle line:84% To 1, [? too-- ?] so this here this goes to 1, 00:57:22.700 --> 00:57:25.230 align:middle line:84% and so even raised to the fifth power, still goes to 1. 00:57:25.230 --> 00:57:29.720 align:middle line:84% So this here goes to the norm of x. 00:57:29.720 --> 00:57:33.920 align:middle line:84% And so the ratio test would be that if x here-- 00:57:33.920 --> 00:57:35.540 align:middle line:90% so the ratio test gives you-- 00:57:35.540 --> 00:57:39.244 align:middle line:90% 00:57:39.244 --> 00:57:42.530 align:middle line:84% is that if this norm here is less than 1, then 00:57:42.530 --> 00:57:45.380 align:middle line:90% this series is convergent. 00:57:45.380 --> 00:57:51.620 align:middle line:84% If the norm here is bigger than 1, the ratio is divergent. 00:57:51.620 --> 00:57:55.230 align:middle line:84% And it's unclear if the norm is equal to 1. 00:57:55.230 --> 00:57:58.520 align:middle line:84% So in other words, you see that the ratio test here 00:57:58.520 --> 00:58:02.520 align:middle line:84% gives you directly that the radius of convergence is 1. 00:58:02.520 --> 00:58:05.026 align:middle line:90% 00:58:05.026 --> 00:58:10.312 align:middle line:84% And of course, you could apply the root test. 00:58:10.312 --> 00:58:13.920 align:middle line:84% The root test is great on this here. 00:58:13.920 --> 00:58:15.550 align:middle line:90% It's a little bit more-- 00:58:15.550 --> 00:58:17.618 align:middle line:84% no, you have to do a little bit more argument. 00:58:17.618 --> 00:58:18.410 align:middle line:90% It's not very hard. 00:58:18.410 --> 00:58:21.520 align:middle line:84% But it actually-- in this case, the ratio test 00:58:21.520 --> 00:58:24.070 align:middle line:84% would be easier to-- it would be less 00:58:24.070 --> 00:58:27.580 align:middle line:90% to argue than in the root test. 00:58:27.580 --> 00:58:31.360 align:middle line:84% Let's do one more example of this. 00:58:31.360 --> 00:58:34.360 align:middle line:84% So maybe I'll just switch a little bit [? to-- ?] 00:58:34.360 --> 00:58:39.130 align:middle line:84% and then I'll come back and review some more stuff. 00:58:39.130 --> 00:58:41.410 align:middle line:84% I'll just talk about a few of the problems 00:58:41.410 --> 00:58:45.140 align:middle line:84% that I added to the end of my review notes. 00:58:45.140 --> 00:58:49.375 align:middle line:84% Again, I'll post this with these problems. 00:58:49.375 --> 00:58:52.040 align:middle line:90% And obviously, they are-- 00:58:52.040 --> 00:58:54.730 align:middle line:84% it's not exactly problems on the midterm, 00:58:54.730 --> 00:58:57.190 align:middle line:84% but [INAUDIBLE] it's certainly relevant to the midterm. 00:58:57.190 --> 00:59:06.274 align:middle line:90% 00:59:06.274 --> 00:59:07.890 align:middle line:84% And also, another thing I want to say 00:59:07.890 --> 00:59:15.930 align:middle line:84% is that the homework is graded up to and including 4. 00:59:15.930 --> 00:59:18.580 align:middle line:84% So I wasn't actually aware of, over the weekend, 00:59:18.580 --> 00:59:20.640 align:middle line:84% they were supposed to be handed back within, 00:59:20.640 --> 00:59:23.770 align:middle line:84% latest, a week of when you handed it in, 00:59:23.770 --> 00:59:28.320 align:middle line:84% and I just discovered it over the weekend that it was-- 00:59:28.320 --> 00:59:33.840 align:middle line:84% some of them were late, so-- but it's fixed. 00:59:33.840 --> 00:59:35.400 align:middle line:90% I'm not the grader. 00:59:35.400 --> 00:59:40.650 align:middle line:84% There's two [INAUDIBLE] that are grading and a grader. 00:59:40.650 --> 00:59:42.130 align:middle line:90% But it's all fixed now. 00:59:42.130 --> 00:59:45.400 align:middle line:90% So the first four are-- 00:59:45.400 --> 00:59:51.150 align:middle line:84% you can review those also for the midterm. 00:59:51.150 --> 00:59:58.450 align:middle line:84% So the other power series I want to look at is-- 00:59:58.450 --> 01:00:01.839 align:middle line:84% so again, what is the radius of convergence? 01:00:01.839 --> 01:00:09.530 align:middle line:90% 01:00:09.530 --> 01:00:11.200 align:middle line:84% The nice thing about power series 01:00:11.200 --> 01:00:16.210 align:middle line:84% is that power series also test about series. 01:00:16.210 --> 01:00:16.960 align:middle line:90% [INAUDIBLE] 01:00:16.960 --> 01:00:21.730 align:middle line:90% 01:00:21.730 --> 01:00:28.900 align:middle line:84% So what is the radius of convergence for-- 01:00:28.900 --> 01:00:32.650 align:middle line:84% and now the series is a slightly different series. 01:00:32.650 --> 01:00:33.890 align:middle line:90% It's this series here-- 01:00:33.890 --> 01:00:38.950 align:middle line:90% 01:00:38.950 --> 01:00:40.010 align:middle line:90% so this series. 01:00:40.010 --> 01:00:42.895 align:middle line:90% 01:00:42.895 --> 01:00:45.010 align:middle line:90% And so now let me-- 01:00:45.010 --> 01:00:48.050 align:middle line:84% again, you could apply the root test. 01:00:48.050 --> 01:00:52.970 align:middle line:84% The root test is the one that just gives you an answer. 01:00:52.970 --> 01:00:56.070 align:middle line:84% It always gives you an answer, but you 01:00:56.070 --> 01:01:01.610 align:middle line:84% may have to work harder to find the n-th root, 01:01:01.610 --> 01:01:04.690 align:middle line:84% and sometimes it's just easier to use the ratio test. 01:01:04.690 --> 01:01:07.210 align:middle line:84% So let's use the ratio test again. 01:01:07.210 --> 01:01:12.060 align:middle line:84% So a n here is n to the fifth power, x to the n, 01:01:12.060 --> 01:01:13.800 align:middle line:90% over n factorial. 01:01:13.800 --> 01:01:19.887 align:middle line:84% So if you're looking at a n plus 1 over a n-- maybe I should-- 01:01:19.887 --> 01:01:21.220 align:middle line:90% maybe let me write it over here. 01:01:21.220 --> 01:01:27.180 align:middle line:90% 01:01:27.180 --> 01:01:33.040 align:middle line:84% So if I'm looking at a n plus 1 over a n-- 01:01:33.040 --> 01:01:36.840 align:middle line:90% 01:01:36.840 --> 01:01:44.220 align:middle line:84% so then I have the n plus 1 to the fifth power, times 01:01:44.220 --> 01:01:55.740 align:middle line:84% x to the power n plus 1, over n plus 1 factorial. 01:01:55.740 --> 01:01:57.590 align:middle line:90% That is a n plus 1. 01:01:57.590 --> 01:02:00.060 align:middle line:84% And then I have to divide by this, 01:02:00.060 --> 01:02:02.300 align:middle line:84% but when I divide by a fraction, it's 01:02:02.300 --> 01:02:06.990 align:middle line:84% like multiplying by the inverse of this. 01:02:06.990 --> 01:02:09.470 align:middle line:90% So like this thing here-- 01:02:09.470 --> 01:02:15.260 align:middle line:84% [INAUDIBLE] the n power, x to the n power-- 01:02:15.260 --> 01:02:17.540 align:middle line:90% like that. 01:02:17.540 --> 01:02:27.110 align:middle line:84% And so again, [INAUDIBLE] I can write this thing here as-- 01:02:27.110 --> 01:02:30.300 align:middle line:84% so if I just do like I did before, 01:02:30.300 --> 01:02:36.270 align:middle line:84% I would have that n plus 1 over n, to the fifth power. 01:02:36.270 --> 01:02:40.730 align:middle line:84% That's when I take this one here and that one together. 01:02:40.730 --> 01:02:43.710 align:middle line:84% Then I have-- when I take those two together, 01:02:43.710 --> 01:02:49.070 align:middle line:84% I can get rid of the n here, and I just get an x. 01:02:49.070 --> 01:02:53.700 align:middle line:84% And then when I look at the ratio of these two, 01:02:53.700 --> 01:02:55.470 align:middle line:90% this one here-- 01:02:55.470 --> 01:02:58.260 align:middle line:90% n plus 1 factorial-- 01:02:58.260 --> 01:03:01.200 align:middle line:84% you can think about that as n plus 1 01:03:01.200 --> 01:03:03.360 align:middle line:90% and then times n factorial. 01:03:03.360 --> 01:03:08.080 align:middle line:84% And so this n factorial cancel out with that, 01:03:08.080 --> 01:03:10.680 align:middle line:84% and you're left with an n plus 1. 01:03:10.680 --> 01:03:14.430 align:middle line:90% So this is what you get. 01:03:14.430 --> 01:03:16.120 align:middle line:90% And this one here-- 01:03:16.120 --> 01:03:18.580 align:middle line:84% just writing it as we did before, 01:03:18.580 --> 01:03:26.550 align:middle line:84% dividing this fraction by 1/n in both numerator and denominator, 01:03:26.550 --> 01:03:33.720 align:middle line:84% getting this thing here to the power of 5 times x n plus 1. 01:03:33.720 --> 01:03:36.180 align:middle line:90% And now you see that-- 01:03:36.180 --> 01:03:38.430 align:middle line:84% and this here, I guess it should have 01:03:38.430 --> 01:03:40.350 align:middle line:84% been absolute value [? applied, ?] 01:03:40.350 --> 01:03:47.670 align:middle line:84% so it's absolute value on this, like that. 01:03:47.670 --> 01:03:51.110 align:middle line:90% This one here converged to 1. 01:03:51.110 --> 01:03:54.680 align:middle line:84% This one is just x, so this converged to x. 01:03:54.680 --> 01:03:57.270 align:middle line:84% And then this here goes to infinity, 01:03:57.270 --> 01:04:00.180 align:middle line:90% so the ratio here goes to 0. 01:04:00.180 --> 01:04:03.380 align:middle line:84% I'm using the algebraic property of limits. 01:04:03.380 --> 01:04:06.590 align:middle line:90% So this is converging to 0. 01:04:06.590 --> 01:04:11.810 align:middle line:84% And so you see that it does have a limit, so the ratio test 01:04:11.810 --> 01:04:16.400 align:middle line:84% [? applied ?] because that limit is 0 for all x, 01:04:16.400 --> 01:04:18.480 align:middle line:84% it's going to be convergent for all x. 01:04:18.480 --> 01:04:21.130 align:middle line:84% So the radius of convergence in this case-- 01:04:21.130 --> 01:04:29.960 align:middle line:90% 01:04:29.960 --> 01:04:33.620 align:middle line:84% and of course, on the midterm, don't just 01:04:33.620 --> 01:04:36.560 align:middle line:84% say that the radius of convergence is infinite. 01:04:36.560 --> 01:04:39.660 align:middle line:84% Explain this thing here, and the more detail, 01:04:39.660 --> 01:04:43.940 align:middle line:84% the better, because even if there's some typos, 01:04:43.940 --> 01:04:46.650 align:middle line:84% you definitely get should definitely get-- 01:04:46.650 --> 01:04:48.130 align:middle line:84% I'm not grading the midterm either, 01:04:48.130 --> 01:04:50.400 align:middle line:84% but you definitely should get partial credit. 01:04:50.400 --> 01:04:51.390 align:middle line:90% Yeah? 01:04:51.390 --> 01:04:53.370 align:middle line:84% AUDIENCE: So to get R, is it always 01:04:53.370 --> 01:04:55.150 align:middle line:90% like you divide by the lim sup? 01:04:55.150 --> 01:04:57.360 align:middle line:84% Because-- is the thing on the bottom one 01:04:57.360 --> 01:04:58.810 align:middle line:90% the lim sup or just the lim? 01:04:58.810 --> 01:04:59.685 align:middle line:90% TOBIAS COLDING: Yeah. 01:04:59.685 --> 01:05:02.580 align:middle line:90% 01:05:02.580 --> 01:05:11.361 align:middle line:84% So when you take just a complete general series, 01:05:11.361 --> 01:05:15.300 align:middle line:84% then the radius of convergence is by definition this. 01:05:15.300 --> 01:05:16.785 align:middle line:90% This is related to-- 01:05:16.785 --> 01:05:19.830 align:middle line:84% you asked this question a couple of times ago that-- 01:05:19.830 --> 01:05:22.530 align:middle line:84% because we started with the ratio test. 01:05:22.530 --> 01:05:24.480 align:middle line:84% We did that before we did the root test. 01:05:24.480 --> 01:05:27.390 align:middle line:84% And you were noting that in the ratio test, 01:05:27.390 --> 01:05:32.610 align:middle line:84% you're always comparing an element to the previous one. 01:05:32.610 --> 01:05:35.700 align:middle line:84% And in general, the series can be convergent 01:05:35.700 --> 01:05:38.160 align:middle line:84% even though the next element is slightly 01:05:38.160 --> 01:05:40.020 align:middle line:90% larger than the previous. 01:05:40.020 --> 01:05:43.090 align:middle line:84% If that keeps happening, it's not convergent, 01:05:43.090 --> 01:05:47.140 align:middle line:84% but it could happen a few times, and then it goes really down, 01:05:47.140 --> 01:05:48.680 align:middle line:90% and that's fine. 01:05:48.680 --> 01:05:54.370 align:middle line:84% And so that's why I like the root test-- 01:05:54.370 --> 01:05:59.020 align:middle line:84% gives you this general result about that it's always-- 01:05:59.020 --> 01:05:59.890 align:middle line:90% it always works. 01:05:59.890 --> 01:06:05.220 align:middle line:84% It doesn't assume that the a n's are not 0, et cetera. 01:06:05.220 --> 01:06:07.950 align:middle line:84% For any particular-- and it really 01:06:07.950 --> 01:06:09.760 align:middle line:90% should be the lim sup here. 01:06:09.760 --> 01:06:11.835 align:middle line:90% And then it's defined. 01:06:11.835 --> 01:06:13.565 align:middle line:84% It's a number that's always defined. 01:06:13.565 --> 01:06:16.760 align:middle line:90% 01:06:16.760 --> 01:06:20.640 align:middle line:84% And again, the way it's stated here, the way the proof goes, 01:06:20.640 --> 01:06:26.290 align:middle line:84% is it's not comparing two consecutive a n. 01:06:26.290 --> 01:06:28.860 align:middle line:90% 01:06:28.860 --> 01:06:32.310 align:middle line:84% In any particular example, the root test 01:06:32.310 --> 01:06:35.850 align:middle line:84% may not be the easiest way to argue. 01:06:35.850 --> 01:06:40.500 align:middle line:84% Often, if you can, for a power series-- 01:06:40.500 --> 01:06:45.940 align:middle line:84% almost always, I would say, in a power series, 01:06:45.940 --> 01:06:51.050 align:middle line:84% it's almost always easier to use the ratio test, 01:06:51.050 --> 01:06:56.770 align:middle line:84% but it requires that the power series has some nice property 01:06:56.770 --> 01:06:58.180 align:middle line:90% for this to apply. 01:06:58.180 --> 01:06:59.540 align:middle line:90% [INAUDIBLE] Yeah? 01:06:59.540 --> 01:07:01.915 align:middle line:84% AUDIENCE: So [INAUDIBLE] R is infinity for that example, 01:07:01.915 --> 01:07:04.540 align:middle line:84% it was because it was convergent for all x [INAUDIBLE]. 01:07:04.540 --> 01:07:07.200 align:middle line:84% TOBIAS COLDING: So in this example here-- 01:07:07.200 --> 01:07:10.130 align:middle line:84% then yeah, this here is infinity, 01:07:10.130 --> 01:07:13.750 align:middle line:84% which means that this thing here would be 0. 01:07:13.750 --> 01:07:15.310 align:middle line:90% Yeah? 01:07:15.310 --> 01:07:16.800 align:middle line:84% AUDIENCE: But you know [INAUDIBLE]. 01:07:16.800 --> 01:07:18.800 align:middle line:84% TOBIAS COLDING: But I didn't actually calculate. 01:07:18.800 --> 01:07:21.820 align:middle line:84% You see, I didn't calculate the [INAUDIBLE] in this example. 01:07:21.820 --> 01:07:24.920 align:middle line:84% But if I had calculated, I would actually-- 01:07:24.920 --> 01:07:26.420 align:middle line:84% and it's not very hard to calculate, 01:07:26.420 --> 01:07:29.140 align:middle line:84% but it needs some small tricks, and so-- 01:07:29.140 --> 01:07:31.720 align:middle line:84% and we haven't-- yeah, we haven't talked about. 01:07:31.720 --> 01:07:34.240 align:middle line:90% 01:07:34.240 --> 01:07:37.210 align:middle line:90% So that's why I [INAUDIBLE]. 01:07:37.210 --> 01:07:41.760 align:middle line:84% Let me talk about a few other things, and then-- 01:07:41.760 --> 01:07:48.310 align:middle line:84% so again, I don't think I can [INAUDIBLE] review all 01:07:48.310 --> 01:07:51.740 align:middle line:90% what I had in my review notes. 01:07:51.740 --> 01:07:54.900 align:middle line:84% But let me just talk about a few of the problems that I had. 01:07:54.900 --> 01:07:58.440 align:middle line:84% But again-- so the things I want-- 01:07:58.440 --> 01:08:01.730 align:middle line:84% let me just quickly mention the things I may or may not 01:08:01.730 --> 01:08:05.600 align:middle line:84% get to that I have in my review notes. 01:08:05.600 --> 01:08:10.940 align:middle line:84% So I had also [INAUDIBLE] property-- 01:08:10.940 --> 01:08:12.350 align:middle line:90% what is lim sup? 01:08:12.350 --> 01:08:13.590 align:middle line:90% What is lim inf? 01:08:13.590 --> 01:08:17.750 align:middle line:84% And so definitely review that also. 01:08:17.750 --> 01:08:21.350 align:middle line:84% Talked about the radius of convergence. 01:08:21.350 --> 01:08:23.660 align:middle line:84% Well, maybe this is all, and then talk 01:08:23.660 --> 01:08:25.910 align:middle line:84% a little bit about the power series 01:08:25.910 --> 01:08:28.109 align:middle line:90% for the exponential functions. 01:08:28.109 --> 01:08:32.840 align:middle line:84% But again, all the stuff we did on the power series 01:08:32.840 --> 01:08:36.319 align:middle line:90% for the exponential functions-- 01:08:36.319 --> 01:08:39.279 align:middle line:84% at this point, we have established this. 01:08:39.279 --> 01:08:44.029 align:middle line:90% This is like-- it's more like-- 01:08:44.029 --> 01:08:46.729 align:middle line:84% there won't be anything particular about them. 01:08:46.729 --> 01:08:50.359 align:middle line:84% The exponential function itself could be on the midterm, 01:08:50.359 --> 01:08:54.279 align:middle line:84% but it's not like you would have to prove the properties 01:08:54.279 --> 01:08:56.029 align:middle line:90% of the exponential function. 01:08:56.029 --> 01:08:58.569 align:middle line:84% This property-- that it has the property 01:08:58.569 --> 01:09:01.779 align:middle line:84% that the exponential function of x plus y 01:09:01.779 --> 01:09:05.420 align:middle line:84% is the product of that kind of stuff-- it's not on the midterm. 01:09:05.420 --> 01:09:09.069 align:middle line:84% So I didn't-- so it's not part of the review. 01:09:09.069 --> 01:09:12.430 align:middle line:84% The other thing that I probably won't get to, 01:09:12.430 --> 01:09:15.050 align:middle line:84% because in a minute I'll just turn to these other problems, 01:09:15.050 --> 01:09:16.840 align:middle line:90% is that metric spaces-- 01:09:16.840 --> 01:09:19.180 align:middle line:90% definitely review metric spaces. 01:09:19.180 --> 01:09:20.800 align:middle line:90% It was in the last lecture. 01:09:20.800 --> 01:09:23.214 align:middle line:90% So there's a bunch of-- 01:09:23.214 --> 01:09:25.910 align:middle line:84% there's three properties for a metric space. 01:09:25.910 --> 01:09:27.649 align:middle line:84% The distance should be non-negative, 01:09:27.649 --> 01:09:30.880 align:middle line:84% it should be symmetric, and the distance between two points 01:09:30.880 --> 01:09:33.470 align:middle line:84% should be 0 if and only if those points are the same. 01:09:33.470 --> 01:09:37.750 align:middle line:84% And then the crucial property in the metric space-- 01:09:37.750 --> 01:09:40.229 align:middle line:84% then that's like the powerful property-- 01:09:40.229 --> 01:09:43.630 align:middle line:90% is the triangle inequality. 01:09:43.630 --> 01:09:46.020 align:middle line:90% And so there was a few examples. 01:09:46.020 --> 01:09:48.330 align:middle line:90% And review those examples. 01:09:48.330 --> 01:09:50.399 align:middle line:90% They are in my review notes. 01:09:50.399 --> 01:09:53.790 align:middle line:84% The most important example, I would say, is-- 01:09:53.790 --> 01:09:59.050 align:middle line:84% the French rail metric is kind of a crazy metric. 01:09:59.050 --> 01:10:00.940 align:middle line:84% That's not one of the important ones. 01:10:00.940 --> 01:10:05.010 align:middle line:84% So I just did review notes as just the most important ones. 01:10:05.010 --> 01:10:06.720 align:middle line:84% And then there's this with, what does 01:10:06.720 --> 01:10:09.220 align:middle line:84% it mean to be convergent in a metric space, 01:10:09.220 --> 01:10:12.930 align:middle line:84% and what does it mean to be a Cauchy sequence? 01:10:12.930 --> 01:10:18.120 align:middle line:84% And so now let's get to a couple of the other problems 01:10:18.120 --> 01:10:21.690 align:middle line:84% that I suggested that you look at. 01:10:21.690 --> 01:10:28.080 align:middle line:84% And so here are two other problems. 01:10:28.080 --> 01:10:30.720 align:middle line:84% And again, I won't get to-- there's also 01:10:30.720 --> 01:10:33.060 align:middle line:84% problems about metric spaces, and you should definitely 01:10:33.060 --> 01:10:35.310 align:middle line:84% look at it, but I don't have the time 01:10:35.310 --> 01:10:39.430 align:middle line:84% now, even though it's relevant for the midterm. 01:10:39.430 --> 01:10:40.620 align:middle line:90% I just have to pick some. 01:10:40.620 --> 01:10:44.440 align:middle line:90% 01:10:44.440 --> 01:10:46.690 align:middle line:84% Again, all of the problems that I-- 01:10:46.690 --> 01:10:48.590 align:middle line:90% I have four problems here. 01:10:48.590 --> 01:10:51.340 align:middle line:84% All of these are relevant for the midterm. 01:10:51.340 --> 01:10:56.840 align:middle line:84% So now-- suppose you take a sequence. 01:10:56.840 --> 01:10:59.440 align:middle line:90% So I have a sequence here, xn. 01:10:59.440 --> 01:11:03.790 align:middle line:84% And so often, a sequence is defined recursively. 01:11:03.790 --> 01:11:18.520 align:middle line:84% So recursive-- so recursively defined [INAUDIBLE] sequences. 01:11:18.520 --> 01:11:20.090 align:middle line:90% So what does it mean? 01:11:20.090 --> 01:11:22.270 align:middle line:84% It means that typically, you have-- 01:11:22.270 --> 01:11:25.330 align:middle line:84% the first one is defined to be whatever, 01:11:25.330 --> 01:11:29.680 align:middle line:84% and then the next one is defined from the previous one. 01:11:29.680 --> 01:11:34.230 align:middle line:84% And so the one I was suggesting looking at here 01:11:34.230 --> 01:11:44.535 align:middle line:84% is that-- you see this here, the first one [INAUDIBLE]. 01:11:44.535 --> 01:11:45.910 align:middle line:84% And then-- actually, [INAUDIBLE]. 01:11:45.910 --> 01:11:48.255 align:middle line:84% And then the next one-- let me just do this one here. 01:11:48.255 --> 01:11:50.380 align:middle line:84% It's actually simpler than what I have in my notes, 01:11:50.380 --> 01:11:53.400 align:middle line:84% but maybe that one, but I just want to-- 01:11:53.400 --> 01:12:00.180 align:middle line:84% so the next one is defined to be xn over n. 01:12:00.180 --> 01:12:08.310 align:middle line:84% And so then-- so now, the question 01:12:08.310 --> 01:12:27.452 align:middle line:84% would be, for what a does this have a limit-- the sequence have 01:12:27.452 --> 01:12:30.450 align:middle line:90% a limit. 01:12:30.450 --> 01:12:32.630 align:middle line:84% And so you see that in this case-- 01:12:32.630 --> 01:12:35.060 align:middle line:84% we only have four minutes left, so I want to [INAUDIBLE]. 01:12:35.060 --> 01:12:37.190 align:middle line:90% But this is also-- 01:12:37.190 --> 01:12:40.840 align:middle line:84% I just wanted you to review what it means for a sequence 01:12:40.840 --> 01:12:44.360 align:middle line:90% to be defined recursively. 01:12:44.360 --> 01:12:47.950 align:middle line:84% And so you see that in this case, 01:12:47.950 --> 01:12:50.860 align:middle line:90% you have that this thing here-- 01:12:50.860 --> 01:12:52.340 align:middle line:90% it is going down. 01:12:52.340 --> 01:12:54.250 align:middle line:90% It's actually going down. 01:12:54.250 --> 01:12:57.080 align:middle line:84% It even could go down by a fraction. 01:12:57.080 --> 01:13:00.610 align:middle line:84% But now, it's like-- if you're looking at this in here, 01:13:00.610 --> 01:13:05.200 align:middle line:84% you're looking at xn, this here is definitely going to converge, 01:13:05.200 --> 01:13:07.750 align:middle line:90% whatever you start with. 01:13:07.750 --> 01:13:16.630 align:middle line:84% So I claim that you will always get that xn here converges to 0. 01:13:16.630 --> 01:13:20.650 align:middle line:84% And it's just-- you see that if you-- when you have something 01:13:20.650 --> 01:13:25.810 align:middle line:84% recursively, it often can pay to just write out 01:13:25.810 --> 01:13:27.410 align:middle line:90% the first few terms. 01:13:27.410 --> 01:13:29.250 align:middle line:90% So you have [? x1, ?] a. 01:13:29.250 --> 01:13:32.535 align:middle line:90% x2 is equal to a. 01:13:32.535 --> 01:13:36.300 align:middle line:90% 01:13:36.300 --> 01:13:37.620 align:middle line:90% It's also a. 01:13:37.620 --> 01:13:45.224 align:middle line:84% But the previous one, [? x3 ?] is now a/2. 01:13:45.224 --> 01:13:58.270 align:middle line:84% x4 is then a/2, divided by 3, so this is a/6, and so on. 01:13:58.270 --> 01:14:00.100 align:middle line:84% And so if you continue like this, 01:14:00.100 --> 01:14:02.415 align:middle line:84% you will see that they get-- that it's 01:14:02.415 --> 01:14:05.970 align:middle line:84% like a divided by some larger and larger number. 01:14:05.970 --> 01:14:10.070 align:middle line:84% But you have to, of course, write it nicely out as a proof. 01:14:10.070 --> 01:14:13.380 align:middle line:90% But this is proving that. 01:14:13.380 --> 01:14:16.450 align:middle line:84% But again, this here-- if you just did what I did, 01:14:16.450 --> 01:14:18.660 align:middle line:84% that would not be enough to get full credit, 01:14:18.660 --> 01:14:20.590 align:middle line:90% so definitely write out. 01:14:20.590 --> 01:14:23.470 align:middle line:84% The other thing, in the last minutes here-- 01:14:23.470 --> 01:14:32.340 align:middle line:84% the other thing-- the other example here [INAUDIBLE], 01:14:32.340 --> 01:14:35.250 align:middle line:84% suppose you take a series like this, 01:14:35.250 --> 01:14:40.580 align:middle line:84% and suppose that a n here is less than 1 over n squared. 01:14:40.580 --> 01:14:45.530 align:middle line:84% And then the question could be, is 01:14:45.530 --> 01:14:48.920 align:middle line:84% the series-- suppose you knew this, but that's all you know. 01:14:48.920 --> 01:14:58.140 align:middle line:84% Is the series convergent or divergent? 01:14:58.140 --> 01:15:03.380 align:middle line:90% 01:15:03.380 --> 01:15:07.140 align:middle line:84% And then I say in the notes here-- 01:15:07.140 --> 01:15:09.770 align:middle line:90% I mean, in the review here-- 01:15:09.770 --> 01:15:14.780 align:middle line:84% that either you should prove that for all such series, 01:15:14.780 --> 01:15:18.060 align:middle line:84% it's convergent, or if that's not the case, 01:15:18.060 --> 01:15:20.630 align:middle line:84% you should give an example of a series 01:15:20.630 --> 01:15:23.150 align:middle line:84% with this property that is divergent. 01:15:23.150 --> 01:15:26.560 align:middle line:84% Because obviously, if all the a n were 0, 01:15:26.560 --> 01:15:28.620 align:middle line:90% it would be convergent. 01:15:28.620 --> 01:15:31.620 align:middle line:84% But it still could be another series 01:15:31.620 --> 01:15:36.210 align:middle line:84% with a n's that have that property that was divergent. 01:15:36.210 --> 01:15:40.180 align:middle line:84% But in this case, it is going to be convergent. 01:15:40.180 --> 01:15:41.860 align:middle line:90% And why is it convergent? 01:15:41.860 --> 01:15:45.900 align:middle line:84% Because this series here is convergent. 01:15:45.900 --> 01:15:47.160 align:middle line:90% We have already seen that. 01:15:47.160 --> 01:15:50.250 align:middle line:84% So this here would be like the simple comparison test, 01:15:50.250 --> 01:15:52.810 align:middle line:90% like the comparison test. 01:15:52.810 --> 01:15:58.830 align:middle line:90% 01:15:58.830 --> 01:16:04.050 align:middle line:84% So these are just examples of things you should be reviewing. 01:16:04.050 --> 01:16:09.555 align:middle line:84% They are-- like these things here, the two things to the end, 01:16:09.555 --> 01:16:11.400 align:middle line:90% like this one here-- 01:16:11.400 --> 01:16:14.580 align:middle line:84% this is super fast, so don't expect it to be 01:16:14.580 --> 01:16:16.970 align:middle line:90% quite as fast on the midterm. 01:16:16.970 --> 01:16:22.310 align:middle line:84% But it just gives you an idea of what you should be doing. 01:16:22.310 --> 01:16:25.000 align:middle line:90%