WEBVTT 00:00:00.000 --> 00:00:02.430 align:middle line:90% [SQUEAKING] 00:00:02.430 --> 00:00:03.402 align:middle line:90% [RUSTLING] 00:00:03.402 --> 00:00:04.374 align:middle line:90% 00:00:04.374 --> 00:00:05.346 align:middle line:90% [CLICKING] 00:00:05.346 --> 00:00:11.180 align:middle line:90% 00:00:11.180 --> 00:00:12.410 align:middle line:90% BRYNMOR CHAPMAN: Hello. 00:00:12.410 --> 00:00:13.920 align:middle line:90% Can people hear me? 00:00:13.920 --> 00:00:15.750 align:middle line:90% [SIDE CONVERSATION] 00:00:15.750 --> 00:00:16.250 align:middle line:90% Hello? 00:00:16.250 --> 00:00:19.160 align:middle line:90% [SIDE CONVERSATION] 00:00:19.160 --> 00:00:20.880 align:middle line:90% Getting some thumbs up. 00:00:20.880 --> 00:00:23.754 align:middle line:90% [SIDE CONVERSATION] 00:00:23.754 --> 00:00:29.990 align:middle line:90% 00:00:29.990 --> 00:00:30.660 align:middle line:90% OK. 00:00:30.660 --> 00:00:32.090 align:middle line:90% Why don't we get started? 00:00:32.090 --> 00:00:33.060 align:middle line:90% Hello, everybody. 00:00:33.060 --> 00:00:34.310 align:middle line:90% I'm Brynmor. 00:00:34.310 --> 00:00:35.730 align:middle line:84% If you've been coming to lecture, 00:00:35.730 --> 00:00:38.280 align:middle line:84% you've already met Erik and Zach. 00:00:38.280 --> 00:00:41.360 align:middle line:84% I'm the third instructor for this course. 00:00:41.360 --> 00:00:43.740 align:middle line:84% Today, we are going to switch gears a little bit. 00:00:43.740 --> 00:00:46.470 align:middle line:84% So we're going to move on from state machines. 00:00:46.470 --> 00:00:51.210 align:middle line:84% And we're going to start a new unit on algorithmic analysis. 00:00:51.210 --> 00:00:56.510 align:middle line:84% So we'll be looking at that until the end of next week. 00:00:56.510 --> 00:01:00.170 align:middle line:84% This lecture is going to be about sums. 00:01:00.170 --> 00:01:00.810 align:middle line:90% OK. 00:01:00.810 --> 00:01:03.920 align:middle line:90% So sums are quite useful. 00:01:03.920 --> 00:01:05.990 align:middle line:84% As I hinted at a moment ago, they're 00:01:05.990 --> 00:01:07.470 align:middle line:90% good for algorithmic analysis. 00:01:07.470 --> 00:01:12.230 align:middle line:84% You'll use everything in this unit quite a bit if you take 00:01:12.230 --> 00:01:15.650 align:middle line:84% 6.1210 or any other algorithms course. 00:01:15.650 --> 00:01:20.332 align:middle line:84% They're also useful for things like probability, counting, 00:01:20.332 --> 00:01:22.790 align:middle line:84% machine learning-- everything's useful for machine learning 00:01:22.790 --> 00:01:24.020 align:middle line:90% nowadays-- 00:01:24.020 --> 00:01:27.140 align:middle line:84% recurrences and a bunch of other things. 00:01:27.140 --> 00:01:31.530 align:middle line:90% So let's start with an example. 00:01:31.530 --> 00:01:32.042 align:middle line:90% OK. 00:01:32.042 --> 00:01:33.625 align:middle line:84% Has anybody here ever won the lottery? 00:01:33.625 --> 00:01:36.320 align:middle line:90% 00:01:36.320 --> 00:01:36.920 align:middle line:90% Hands. 00:01:36.920 --> 00:01:37.670 align:middle line:90% Nobody? 00:01:37.670 --> 00:01:38.520 align:middle line:90% Really? 00:01:38.520 --> 00:01:39.020 align:middle line:90% Oh. 00:01:39.020 --> 00:01:39.810 align:middle line:90% Come on, guys. 00:01:39.810 --> 00:01:41.210 align:middle line:90% OK. 00:01:41.210 --> 00:01:43.670 align:middle line:84% Well, does anybody know how it works? 00:01:43.670 --> 00:01:47.360 align:middle line:84% So if you win $1 million in the lottery, 00:01:47.360 --> 00:01:50.510 align:middle line:84% it turns out you don't actually get a check for $1 million 00:01:50.510 --> 00:01:51.090 align:middle line:90% in the MAIL. 00:01:51.090 --> 00:01:54.230 align:middle line:90% Or you do, but it won't cash. 00:01:54.230 --> 00:01:56.850 align:middle line:84% It actually comes in installments. 00:01:56.850 --> 00:01:57.440 align:middle line:90% OK. 00:01:57.440 --> 00:02:04.040 align:middle line:84% So imagine you get $50,000 now and $50,000 next year 00:02:04.040 --> 00:02:06.020 align:middle line:90% and so on for the next 20 years. 00:02:06.020 --> 00:02:11.300 align:middle line:84% So in total, it ends up being $1 million in some sense. 00:02:11.300 --> 00:02:13.480 align:middle line:90% Right? 00:02:13.480 --> 00:02:16.270 align:middle line:90% But does it have the same value? 00:02:16.270 --> 00:02:20.050 align:middle line:84% Who would prefer to have $1 million right now? 00:02:20.050 --> 00:02:20.870 align:middle line:90% Lots of hands. 00:02:20.870 --> 00:02:23.740 align:middle line:84% Who would prefer to have the installments, 00:02:23.740 --> 00:02:26.590 align:middle line:90% $50,000 per year for 20 years? 00:02:26.590 --> 00:02:27.850 align:middle line:90% Well, a few hands. 00:02:27.850 --> 00:02:30.197 align:middle line:90% Can I get some reasons? 00:02:30.197 --> 00:02:32.030 align:middle line:84% AUDIENCE: So you don't waste it all at once. 00:02:32.030 --> 00:02:34.030 align:middle line:84% BRYNMOR CHAPMAN: Oh, that's an excellent reason. 00:02:34.030 --> 00:02:36.130 align:middle line:90% 00:02:36.130 --> 00:02:36.920 align:middle line:90% Supposed to be. 00:02:36.920 --> 00:02:38.680 align:middle line:90% Anybody else? 00:02:38.680 --> 00:02:39.620 align:middle line:90% A reason either way. 00:02:39.620 --> 00:02:40.120 align:middle line:90% Yeah? 00:02:40.120 --> 00:02:41.920 align:middle line:84% AUDIENCE: Inflation would make [INAUDIBLE]. 00:02:41.920 --> 00:02:43.188 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:02:43.188 --> 00:02:45.730 align:middle line:84% AUDIENCE: Inflation will make the installments less valuable. 00:02:45.730 --> 00:02:46.510 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:02:46.510 --> 00:02:47.010 align:middle line:90% OK. 00:02:47.010 --> 00:02:49.510 align:middle line:84% So that's an argument for getting the million dollars 00:02:49.510 --> 00:02:50.120 align:middle line:90% right now. 00:02:50.120 --> 00:02:51.371 align:middle line:90% Right? 00:02:51.371 --> 00:02:59.110 align:middle line:84% The $50,000 you get in 20 years, if a gallon of milk costs $4 00:02:59.110 --> 00:03:03.460 align:middle line:84% today and $10 in 20 years, that $50,000 isn't going to go 00:03:03.460 --> 00:03:05.822 align:middle line:90% so far. 00:03:05.822 --> 00:03:10.150 align:middle line:84% So at least in terms of the actual buying 00:03:10.150 --> 00:03:13.930 align:middle line:84% power of the money, you'd really prefer to have that $1 million 00:03:13.930 --> 00:03:18.310 align:middle line:84% right now because money now is more valuable than money 00:03:18.310 --> 00:03:20.730 align:middle line:84% in the future, inflation or, relatedly, you 00:03:20.730 --> 00:03:22.570 align:middle line:84% can invest this money and it'll end up 00:03:22.570 --> 00:03:25.690 align:middle line:84% being more worth more in 20 years. 00:03:25.690 --> 00:03:31.105 align:middle line:84% So this is exactly how a loan works, right? 00:03:31.105 --> 00:03:33.250 align:middle line:84% You get a lump sump of money now. 00:03:33.250 --> 00:03:35.920 align:middle line:84% And you pay it back in installments. 00:03:35.920 --> 00:03:38.560 align:middle line:90% And you account for interest. 00:03:38.560 --> 00:03:40.530 align:middle line:84% So this is what's known as an annuity. 00:03:40.530 --> 00:03:50.242 align:middle line:90% 00:03:50.242 --> 00:03:53.410 align:middle line:84% So for an annuity, we're going to make some fairly simplified 00:03:53.410 --> 00:03:54.080 align:middle line:90% assumptions. 00:03:54.080 --> 00:03:56.040 align:middle line:84% We're going to assume a fixed interest rate. 00:03:56.040 --> 00:04:04.570 align:middle line:90% 00:04:04.570 --> 00:04:06.730 align:middle line:90% Let's call it p. 00:04:06.730 --> 00:04:10.030 align:middle line:84% So as an example, the Federal Reserve 00:04:10.030 --> 00:04:14.560 align:middle line:84% says that interest rate right now is about 5.33% 00:04:14.560 --> 00:04:20.560 align:middle line:84% So let's just take that as a working example, 00:04:20.560 --> 00:04:23.060 align:middle line:84% or at least that's what it was at the beginning of the week. 00:04:23.060 --> 00:04:26.350 align:middle line:90% I haven't checked it today. 00:04:26.350 --> 00:04:38.670 align:middle line:84% And we're going to get m dollars every year for n years. 00:04:38.670 --> 00:04:41.260 align:middle line:90% 00:04:41.260 --> 00:04:46.660 align:middle line:84% So in the example that we were just looking at, m is 50k, 00:04:46.660 --> 00:04:52.270 align:middle line:84% n is 20 years, and we've got this interest rate p, 5.33%. 00:04:52.270 --> 00:04:53.030 align:middle line:90% OK. 00:04:53.030 --> 00:04:59.350 align:middle line:84% So what does this mean about the value of money over time? 00:04:59.350 --> 00:05:06.100 align:middle line:84% Well $1 now, if we get this $1 now, it's going to be worth 00:05:06.100 --> 00:05:09.610 align:middle line:84% or it's equivalent to getting 1 plus p 00:05:09.610 --> 00:05:13.065 align:middle line:90% dollars one year from now. 00:05:13.065 --> 00:05:19.000 align:middle line:90% 00:05:19.000 --> 00:05:21.160 align:middle line:84% Does that make sense to everybody? 00:05:21.160 --> 00:05:24.260 align:middle line:84% If I take my $1 now and I invest it, 00:05:24.260 --> 00:05:27.920 align:middle line:84% I'll end up with $1.05 in a year. 00:05:27.920 --> 00:05:28.750 align:middle line:90% OK. 00:05:28.750 --> 00:05:33.850 align:middle line:84% Or in two years, it'll be 1 plus p squared or, in k years, 00:05:33.850 --> 00:05:35.980 align:middle line:90% it'll be 1 plus p to the k. 00:05:35.980 --> 00:05:41.650 align:middle line:84% Conversely, what if I get that dollar in 10 years? 00:05:41.650 --> 00:05:45.370 align:middle line:84% How much is that worth in today's money? 00:05:45.370 --> 00:05:47.310 align:middle line:90% Anybody? 00:05:47.310 --> 00:05:47.810 align:middle line:90% Hand. 00:05:47.810 --> 00:05:51.510 align:middle line:90% 00:05:51.510 --> 00:05:54.150 align:middle line:90% No? 00:05:54.150 --> 00:05:55.300 align:middle line:90% Are we still asleep? 00:05:55.300 --> 00:05:57.150 align:middle line:90% Too early in the morning? 00:05:57.150 --> 00:05:57.660 align:middle line:90% OK. 00:05:57.660 --> 00:06:01.990 align:middle line:84% Well, conversely, if I get my dollar in 10 years, 00:06:01.990 --> 00:06:05.470 align:middle line:84% it's going to be worth 1 divided by 1 plus p to the 10th. 00:06:05.470 --> 00:06:08.250 align:middle line:84% Does that make sense to everybody? 00:06:08.250 --> 00:06:11.340 align:middle line:90% Yes, no, kind of, thumbs up? 00:06:11.340 --> 00:06:12.310 align:middle line:90% A few thumbs up. 00:06:12.310 --> 00:06:12.870 align:middle line:90% OK. 00:06:12.870 --> 00:06:13.920 align:middle line:90% Good. 00:06:13.920 --> 00:06:16.020 align:middle line:90% I'll take it. 00:06:16.020 --> 00:06:17.940 align:middle line:90% OK. 00:06:17.940 --> 00:06:24.720 align:middle line:84% So basically, what's going on is we're getting m dollars now, 00:06:24.720 --> 00:06:31.950 align:middle line:84% m dollars in one year, m dollars in two years, OK, 00:06:31.950 --> 00:06:37.260 align:middle line:90% dot, dot, dot, up to n years. 00:06:37.260 --> 00:06:43.290 align:middle line:84% And we'd like to figure out what the equivalent value is today. 00:06:43.290 --> 00:06:49.325 align:middle line:84% So this m dollars now, now this is worth m dollars today. 00:06:49.325 --> 00:06:55.230 align:middle line:90% 00:06:55.230 --> 00:06:56.700 align:middle line:90% OK. 00:06:56.700 --> 00:07:02.490 align:middle line:84% This m dollars in one year is going to be worth-- 00:07:02.490 --> 00:07:08.400 align:middle line:84% whoops-- m times 1 plus p to the minus 1. 00:07:08.400 --> 00:07:09.150 align:middle line:90% OK. 00:07:09.150 --> 00:07:14.820 align:middle line:84% This m dollars in two years is going to be m times 1 plus p 00:07:14.820 --> 00:07:19.080 align:middle line:90% to the minus 2, et cetera. 00:07:19.080 --> 00:07:24.480 align:middle line:84% And the m dollars that we get in n years is going to be m times 1 00:07:24.480 --> 00:07:29.010 align:middle line:90% plus p to the minus n. 00:07:29.010 --> 00:07:31.930 align:middle line:84% I guess this is n minus 1, right? 00:07:31.930 --> 00:07:33.210 align:middle line:90% Yeah. 00:07:33.210 --> 00:07:34.760 align:middle line:90% We're starting at 0 years. 00:07:34.760 --> 00:07:40.788 align:middle line:90% 00:07:40.788 --> 00:07:44.580 align:middle line:84% Does that make sense to everybody? 00:07:44.580 --> 00:07:46.420 align:middle line:90% So what is the total value? 00:07:46.420 --> 00:07:50.670 align:middle line:84% The total equivalent value in today's money? 00:07:50.670 --> 00:07:52.410 align:middle line:90% OK. 00:07:52.410 --> 00:07:54.760 align:middle line:90% We'd like this in a closed form. 00:07:54.760 --> 00:07:57.990 align:middle line:90% So right now we've kind of got-- 00:07:57.990 --> 00:08:00.150 align:middle line:84% we're trying to sum all of this up. 00:08:00.150 --> 00:08:01.050 align:middle line:90% Right. 00:08:01.050 --> 00:08:02.650 align:middle line:84% But it's not a very nice expression. 00:08:02.650 --> 00:08:03.150 align:middle line:90% OK. 00:08:03.150 --> 00:08:04.740 align:middle line:84% We'd like a closed form, something 00:08:04.740 --> 00:08:07.320 align:middle line:84% that you can put into one of those old-fashioned calculators 00:08:07.320 --> 00:08:08.820 align:middle line:84% if you've ever seen it, those things 00:08:08.820 --> 00:08:11.250 align:middle line:84% that you hold in your hand and push the buttons on it? 00:08:11.250 --> 00:08:12.480 align:middle line:90% No. 00:08:12.480 --> 00:08:13.210 align:middle line:90% Oh, yeah. 00:08:13.210 --> 00:08:15.550 align:middle line:84% There are some old souls in the audience. 00:08:15.550 --> 00:08:17.026 align:middle line:90% OK. 00:08:17.026 --> 00:08:17.770 align:middle line:90% But yeah. 00:08:17.770 --> 00:08:21.750 align:middle line:84% We don't want any ellipses like this or sigma notation 00:08:21.750 --> 00:08:22.810 align:middle line:90% or anything like that. 00:08:22.810 --> 00:08:24.450 align:middle line:84% We just want a nice, clean expression 00:08:24.450 --> 00:08:26.020 align:middle line:84% that you can put into a calculator. 00:08:26.020 --> 00:08:28.890 align:middle line:84% So how can we go about computing it? 00:08:28.890 --> 00:08:29.570 align:middle line:90% Any ideas? 00:08:29.570 --> 00:08:40.179 align:middle line:90% 00:08:40.179 --> 00:08:43.510 align:middle line:84% What if we were to replace this 1 plus p with an x? 00:08:43.510 --> 00:08:46.010 align:middle line:90% Would that make it any easier? 00:08:46.010 --> 00:08:58.700 align:middle line:84% So we've got m plus mx plus mx squared plus mx 00:08:58.700 --> 00:09:01.520 align:middle line:90% to the n minus 1. 00:09:01.520 --> 00:09:03.450 align:middle line:84% Are people happier with this expression? 00:09:03.450 --> 00:09:06.140 align:middle line:90% 00:09:06.140 --> 00:09:07.260 align:middle line:90% Getting some nods. 00:09:07.260 --> 00:09:08.715 align:middle line:90% OK. 00:09:08.715 --> 00:09:11.510 align:middle line:90% What is it? 00:09:11.510 --> 00:09:12.565 align:middle line:90% Anybody remember? 00:09:12.565 --> 00:09:16.250 align:middle line:90% 00:09:16.250 --> 00:09:17.090 align:middle line:90% Yeah? 00:09:17.090 --> 00:09:18.506 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:09:18.506 --> 00:09:19.920 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:09:19.920 --> 00:09:21.450 align:middle line:90% It's a geometric series. 00:09:21.450 --> 00:09:23.242 align:middle line:90% OK. 00:09:23.242 --> 00:09:31.610 align:middle line:84% So this is going to be m times 1 minus x. 00:09:31.610 --> 00:09:33.830 align:middle line:90% Oops. 00:09:33.830 --> 00:09:36.325 align:middle line:90% n over 1 minus x. 00:09:36.325 --> 00:09:41.272 align:middle line:90% 00:09:41.272 --> 00:09:45.060 align:middle line:84% So we knew this expression already. 00:09:45.060 --> 00:09:47.480 align:middle line:90% How could we check it? 00:09:47.480 --> 00:09:48.810 align:middle line:90% Or how did you check it? 00:09:48.810 --> 00:09:51.500 align:middle line:90% I believe you already have. 00:09:51.500 --> 00:09:52.910 align:middle line:90% Yeah? 00:09:52.910 --> 00:09:55.190 align:middle line:84% AUDIENCE: Are you sure that's the [iNAUDIBLE]? 00:09:55.190 --> 00:09:59.157 align:middle line:90% 00:09:59.157 --> 00:09:59.990 align:middle line:90% BRYNMOR CHAPMAN: Oh. 00:09:59.990 --> 00:10:03.350 align:middle line:90% 00:10:03.350 --> 00:10:04.030 align:middle line:90% Thank you. 00:10:04.030 --> 00:10:14.310 align:middle line:90% 00:10:14.310 --> 00:10:14.810 align:middle line:90% Yeah. 00:10:14.810 --> 00:10:20.230 align:middle line:90% 00:10:20.230 --> 00:10:20.730 align:middle line:90% Yep. 00:10:20.730 --> 00:10:21.272 align:middle line:90% You're right. 00:10:21.272 --> 00:10:22.340 align:middle line:90% Thank you. 00:10:22.340 --> 00:10:24.777 align:middle line:90% So how could you prove this? 00:10:24.777 --> 00:10:26.360 align:middle line:84% Or how would I have failed to prove it 00:10:26.360 --> 00:10:28.440 align:middle line:90% if I tried to do it just now? 00:10:28.440 --> 00:10:28.940 align:middle line:90% Yeah? 00:10:28.940 --> 00:10:30.084 align:middle line:90% AUDIENCE: Subtract it? 00:10:30.084 --> 00:10:31.001 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:10:31.001 --> 00:10:40.630 align:middle line:90% 00:10:40.630 --> 00:10:42.490 align:middle line:90% OK? 00:10:42.490 --> 00:10:46.280 align:middle line:84% So this is what we call the guess and check method. 00:10:46.280 --> 00:10:52.510 align:middle line:84% So if we've got a sum like this, and we know 00:10:52.510 --> 00:10:56.680 align:middle line:84% or we can reasonably guess a nice closed form for it, 00:10:56.680 --> 00:10:59.620 align:middle line:84% then we can prove it by induction. 00:10:59.620 --> 00:11:02.090 align:middle line:90% That's fairly straightforward. 00:11:02.090 --> 00:11:06.100 align:middle line:84% The hard part is figuring out what that closed form should be. 00:11:06.100 --> 00:11:09.310 align:middle line:90% 00:11:09.310 --> 00:11:12.100 align:middle line:84% And in this case, the inductive step 00:11:12.100 --> 00:11:15.820 align:middle line:84% reduces to some fairly simple algebra as you already saw, 00:11:15.820 --> 00:11:16.820 align:middle line:90% I think in recitation. 00:11:16.820 --> 00:11:18.100 align:middle line:90% Is that right? 00:11:18.100 --> 00:11:20.830 align:middle line:90% Rec 02 maybe? 00:11:20.830 --> 00:11:21.430 align:middle line:90% Yeah. 00:11:21.430 --> 00:11:22.490 align:middle line:90% Yes. 00:11:22.490 --> 00:11:23.540 align:middle line:90% Thumbs up from Zach. 00:11:23.540 --> 00:11:26.110 align:middle line:84% Even if you don't remember, Zach does. 00:11:26.110 --> 00:11:27.490 align:middle line:90% [LAUGHS] 00:11:27.490 --> 00:11:31.480 align:middle line:90% 00:11:31.480 --> 00:11:35.200 align:middle line:84% So how might we discover this closed form if we didn't already 00:11:35.200 --> 00:11:36.910 align:middle line:90% know it, though? 00:11:36.910 --> 00:11:37.590 align:middle line:90% Any ideas? 00:11:37.590 --> 00:11:43.450 align:middle line:90% 00:11:43.450 --> 00:11:46.300 align:middle line:90% Yeah? 00:11:46.300 --> 00:11:49.570 align:middle line:84% AUDIENCE: Multiply the series by [INAUDIBLE]. 00:11:49.570 --> 00:11:52.600 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:11:52.600 --> 00:11:55.875 align:middle line:84% So this is what we call the perturbation method. 00:11:55.875 --> 00:12:09.070 align:middle line:90% 00:12:09.070 --> 00:12:13.690 align:middle line:84% So the answer was if we multiply this entire thing by x 00:12:13.690 --> 00:12:17.650 align:middle line:84% and subtract it, then things will cancel. 00:12:17.650 --> 00:12:23.440 align:middle line:84% So this is one example of what we call the perturbation method. 00:12:23.440 --> 00:12:25.090 align:middle line:90% This is something that-- 00:12:25.090 --> 00:12:27.790 align:middle line:84% does anybody know the story of Gauss 00:12:27.790 --> 00:12:29.420 align:middle line:84% in the kindergarten or first grade 00:12:29.420 --> 00:12:34.853 align:middle line:84% when he was given a massive sum to evaluate by his teacher and 3 00:12:34.853 --> 00:12:37.520 align:middle line:84% seconds, he gave the teacher his answer and the teacher is like, 00:12:37.520 --> 00:12:38.103 align:middle line:90% what the hell? 00:12:38.103 --> 00:12:39.580 align:middle line:90% No, go back to your seat. 00:12:39.580 --> 00:12:40.290 align:middle line:90% OK. 00:12:40.290 --> 00:12:40.790 align:middle line:90% Yeah. 00:12:40.790 --> 00:12:42.770 align:middle line:84% So perturbation method in action. 00:12:42.770 --> 00:12:47.400 align:middle line:84% So Gauss was asked to evaluate 1 plus 2 plus-- 00:12:47.400 --> 00:12:50.620 align:middle line:90% 00:12:50.620 --> 00:12:52.100 align:middle line:90% I think the story was 100. 00:12:52.100 --> 00:12:53.410 align:middle line:90% Not actually sure. 00:12:53.410 --> 00:12:59.140 align:middle line:84% And Gauss observed that if you take this series and perturb it 00:12:59.140 --> 00:13:09.160 align:middle line:84% a little bit, so in this case flip it around, and then add 00:13:09.160 --> 00:13:12.200 align:middle line:84% them together, you end up with something very nice. 00:13:12.200 --> 00:13:12.700 align:middle line:90% Right? 00:13:12.700 --> 00:13:25.420 align:middle line:90% 00:13:25.420 --> 00:13:30.370 align:middle line:84% So he's got two copies of this sum. 00:13:30.370 --> 00:13:34.050 align:middle line:84% So two S is equal to this thing, which means 00:13:34.050 --> 00:13:37.355 align:middle line:90% that S should be equal to that. 00:13:37.355 --> 00:13:40.050 align:middle line:90% 00:13:40.050 --> 00:13:41.620 align:middle line:90% So fairly straightforward. 00:13:41.620 --> 00:13:44.390 align:middle line:84% We can do the same thing with this geometric series here. 00:13:44.390 --> 00:13:50.220 align:middle line:90% 00:13:50.220 --> 00:13:52.840 align:middle line:84% So for simplicity, I'm going to drop the m. 00:13:52.840 --> 00:13:59.820 align:middle line:84% So if instead we have s equals 1 plus x plus x squared 00:13:59.820 --> 00:14:04.830 align:middle line:84% plus x to the n minus 1, and here your colleague 00:14:04.830 --> 00:14:07.170 align:middle line:84% said that we should multiply this by x. 00:14:07.170 --> 00:14:09.780 align:middle line:90% So Sx equals-- 00:14:09.780 --> 00:14:11.380 align:middle line:84% I'm going to shift everything over 00:14:11.380 --> 00:14:14.000 align:middle line:84% so that it's a little bit easier to see what's going on. 00:14:14.000 --> 00:14:22.420 align:middle line:90% 00:14:22.420 --> 00:14:24.300 align:middle line:84% And now this time instead of adding them, 00:14:24.300 --> 00:14:25.633 align:middle line:90% we're going to want to subtract. 00:14:25.633 --> 00:14:27.840 align:middle line:90% 00:14:27.840 --> 00:14:32.670 align:middle line:84% This 1 doesn't have anything to subtract. 00:14:32.670 --> 00:14:36.210 align:middle line:84% And now here we've got plus x minus x 00:14:36.210 --> 00:14:38.050 align:middle line:90% plus x squared minus x squared. 00:14:38.050 --> 00:14:45.580 align:middle line:84% So everything is going to cancel until we get to the end. 00:14:45.580 --> 00:14:49.050 align:middle line:84% And here we've got minus x to the n. 00:14:49.050 --> 00:14:55.200 align:middle line:84% So now we've got S minus xS is equal to 1 00:14:55.200 --> 00:15:01.470 align:middle line:84% minus x to the n which means that S is 1 minus x 00:15:01.470 --> 00:15:04.355 align:middle line:90% to the n over 1 minus x. 00:15:04.355 --> 00:15:08.045 align:middle line:90% 00:15:08.045 --> 00:15:12.660 align:middle line:84% So if we didn't know the formula for a geometric series 00:15:12.660 --> 00:15:15.930 align:middle line:84% off the top of our heads, which I clearly didn't, we 00:15:15.930 --> 00:15:18.210 align:middle line:84% could derive it like this and then 00:15:18.210 --> 00:15:19.920 align:middle line:90% we can check it by induction. 00:15:19.920 --> 00:15:22.150 align:middle line:84% Does that make sense to everybody? 00:15:22.150 --> 00:15:31.860 align:middle line:90% 00:15:31.860 --> 00:15:36.720 align:middle line:84% So if we go back to our annuity example, we had-- 00:15:36.720 --> 00:15:41.010 align:middle line:90% so m was 50k. 00:15:41.010 --> 00:15:46.050 align:middle line:90% n is 20 years. 00:15:46.050 --> 00:15:55.614 align:middle line:84% p was 0.0533 which means that our x here was-- 00:15:55.614 --> 00:15:57.320 align:middle line:90% was it 1 over 1 minus p? 00:15:57.320 --> 00:16:01.200 align:middle line:90% 00:16:01.200 --> 00:16:05.750 align:middle line:90% 1 over 1 plus p. 00:16:05.750 --> 00:16:10.260 align:middle line:90% 00:16:10.260 --> 00:16:14.880 align:middle line:84% So if we plug this in, it turns out that the value 00:16:14.880 --> 00:16:24.852 align:middle line:84% of our annuity is actually about 638,000. 00:16:24.852 --> 00:16:28.963 align:middle line:84% So it's not just less than the $1 million 00:16:28.963 --> 00:16:29.880 align:middle line:90% that we were promised. 00:16:29.880 --> 00:16:31.414 align:middle line:90% It's like substantially less. 00:16:31.414 --> 00:16:31.914 align:middle line:90% Right? 00:16:31.914 --> 00:16:39.320 align:middle line:90% 00:16:39.320 --> 00:16:41.760 align:middle line:84% What if we took an even more extreme example? 00:16:41.760 --> 00:16:44.940 align:middle line:84% Like suppose instead of $50,000 per year for 20 years, 00:16:44.940 --> 00:16:48.050 align:middle line:84% we got $1 per year for million years. 00:16:48.050 --> 00:16:50.420 align:middle line:84% For simplicity, let's just say it's forever. 00:16:50.420 --> 00:16:54.680 align:middle line:84% So suppose we get $1 forever, like every year forever. 00:16:54.680 --> 00:16:55.795 align:middle line:90% How much is that worth? 00:16:55.795 --> 00:17:03.020 align:middle line:90% 00:17:03.020 --> 00:17:05.750 align:middle line:90% Half a million, 100,000? 00:17:05.750 --> 00:17:07.434 align:middle line:90% Yeah? 00:17:07.434 --> 00:17:08.253 align:middle line:90% AUDIENCE: $20. 00:17:08.253 --> 00:17:09.170 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:17:09.170 --> 00:17:09.670 align:middle line:90% About $20. 00:17:09.670 --> 00:17:12.170 align:middle line:90% 00:17:12.170 --> 00:17:15.510 align:middle line:84% So this x to the n just goes to 0. 00:17:15.510 --> 00:17:21.480 align:middle line:84% Turns out that if we have m equals 1, n is infinity. 00:17:21.480 --> 00:17:24.079 align:middle line:90% Same p as we had before. 00:17:24.079 --> 00:17:25.609 align:middle line:90% It's about 20. 00:17:25.609 --> 00:17:29.720 align:middle line:84% So it might be a little bit counterintuitive 00:17:29.720 --> 00:17:35.780 align:middle line:84% that getting money, like, forever, it still 00:17:35.780 --> 00:17:39.890 align:middle line:90% has finite value, right? 00:17:39.890 --> 00:17:43.830 align:middle line:84% But yeah, I don't know, maybe if you think about it, 00:17:43.830 --> 00:17:48.290 align:middle line:90% it kind of makes sense. 00:17:48.290 --> 00:17:51.800 align:middle line:84% Like that money that you're getting like n years 00:17:51.800 --> 00:17:54.155 align:middle line:90% from now, who cares? 00:17:54.155 --> 00:17:56.280 align:middle line:84% You're not even going to be alive to use it, right? 00:17:56.280 --> 00:17:58.370 align:middle line:84% So maybe it also kind of makes sense 00:17:58.370 --> 00:18:01.465 align:middle line:90% that it's not worth very much. 00:18:01.465 --> 00:18:04.100 align:middle line:90% 00:18:04.100 --> 00:18:07.430 align:middle line:84% And in the real world, there are actually 00:18:07.430 --> 00:18:09.540 align:middle line:90% perpetual annuities like this. 00:18:09.540 --> 00:18:11.960 align:middle line:90% They do exist. 00:18:11.960 --> 00:18:15.950 align:middle line:84% I believe there is a surviving one issued by a Dutch water 00:18:15.950 --> 00:18:20.180 align:middle line:90% company in the 17th century. 00:18:20.180 --> 00:18:23.850 align:middle line:84% I think they issued five that are still around today. 00:18:23.850 --> 00:18:29.180 align:middle line:84% And one was acquired by Yale in, let's see, 00:18:29.180 --> 00:18:33.390 align:middle line:84% maybe 30 odd years ago, I think, for $24,000. 00:18:33.390 --> 00:18:38.780 align:middle line:84% And it gets them about $12 annually so only worth it 00:18:38.780 --> 00:18:40.320 align:middle line:90% for the historical value. 00:18:40.320 --> 00:18:42.625 align:middle line:90% The money is a complete wash. 00:18:42.625 --> 00:18:49.331 align:middle line:90% 00:18:49.331 --> 00:18:53.660 align:middle line:84% So are people reasonably happy with the perturbation method? 00:18:53.660 --> 00:18:54.170 align:middle line:90% Yeah? 00:18:54.170 --> 00:18:55.980 align:middle line:90% Some thumbs up? 00:18:55.980 --> 00:18:56.480 align:middle line:90% OK. 00:18:56.480 --> 00:18:59.990 align:middle line:84% So Let's move on to a slightly different method 00:18:59.990 --> 00:19:02.470 align:middle line:90% for evaluating other sums. 00:19:02.470 --> 00:19:05.060 align:middle line:90% 00:19:05.060 --> 00:19:08.812 align:middle line:84% So this is what's called the ansatz method. 00:19:08.812 --> 00:19:10.680 align:middle line:90% Am I spelling this correctly? 00:19:10.680 --> 00:19:11.200 align:middle line:90% I hope so. 00:19:11.200 --> 00:19:17.630 align:middle line:90% 00:19:17.630 --> 00:19:19.790 align:middle line:84% Or if you don't like German, I think 00:19:19.790 --> 00:19:22.845 align:middle line:84% it is, you could also just call it the educated guess method. 00:19:22.845 --> 00:19:25.670 align:middle line:90% 00:19:25.670 --> 00:19:29.920 align:middle line:84% So suppose we have the following sum. 00:19:29.920 --> 00:19:38.650 align:middle line:84% Sum from k equals 1 to n of k squared. 00:19:38.650 --> 00:19:40.930 align:middle line:90% OK. 00:19:40.930 --> 00:19:44.240 align:middle line:84% So first, does anybody already know the answer? 00:19:44.240 --> 00:19:45.900 align:middle line:90% What does the sum evaluate to? 00:19:45.900 --> 00:19:49.190 align:middle line:90% 00:19:49.190 --> 00:19:49.690 align:middle line:90% Oh. 00:19:49.690 --> 00:19:50.120 align:middle line:90% Nobody. 00:19:50.120 --> 00:19:50.620 align:middle line:90% OK. 00:19:50.620 --> 00:19:54.250 align:middle line:84% In that case, next question is open to everyone. 00:19:54.250 --> 00:19:59.350 align:middle line:84% Does anybody have any idea what vaguely the solution 00:19:59.350 --> 00:20:00.120 align:middle line:90% might look like? 00:20:00.120 --> 00:20:04.150 align:middle line:90% 00:20:04.150 --> 00:20:04.970 align:middle line:90% Yeah? 00:20:04.970 --> 00:20:07.523 align:middle line:84% AUDIENCE: It should be some sort of cubic. 00:20:07.523 --> 00:20:08.440 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:20:08.440 --> 00:20:11.770 align:middle line:90% It should be some sort of cubic. 00:20:11.770 --> 00:20:13.820 align:middle line:84% We're summing up a bunch of squares-- 00:20:13.820 --> 00:20:16.480 align:middle line:90% well, n squares. 00:20:16.480 --> 00:20:19.310 align:middle line:84% A bunch of them are about n squared. 00:20:19.310 --> 00:20:22.780 align:middle line:84% And so we should get something that's about n cubed. 00:20:22.780 --> 00:20:35.221 align:middle line:84% So guess S is approximately n cubed. 00:20:35.221 --> 00:20:36.730 align:middle line:84% Now, it won't do us too much good 00:20:36.730 --> 00:20:39.980 align:middle line:84% to put that into our guess and check method. 00:20:39.980 --> 00:20:41.093 align:middle line:90% Yeah? 00:20:41.093 --> 00:20:44.051 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:20:44.051 --> 00:20:47.670 align:middle line:90% 00:20:47.670 --> 00:20:49.910 align:middle line:84% BRYNMOR CHAPMAN: Intuition, basically. 00:20:49.910 --> 00:20:52.960 align:middle line:90% 00:20:52.960 --> 00:20:58.960 align:middle line:84% So one way you could think about it is so half of these terms 00:20:58.960 --> 00:21:02.750 align:middle line:84% are going to be larger than n over 2 squared. 00:21:02.750 --> 00:21:07.060 align:middle line:84% So that's one fourth of n squared. 00:21:07.060 --> 00:21:08.902 align:middle line:90% You've got n over 2 of those. 00:21:08.902 --> 00:21:10.360 align:middle line:84% So you should have something that's 00:21:10.360 --> 00:21:13.195 align:middle line:90% at least one eighth of n cubed. 00:21:13.195 --> 00:21:14.070 align:middle line:90% Does that make sense? 00:21:14.070 --> 00:21:17.860 align:middle line:90% 00:21:17.860 --> 00:21:19.520 align:middle line:84% And then for an upper bound, you've 00:21:19.520 --> 00:21:22.200 align:middle line:84% got n terms which are all at most n squared. 00:21:22.200 --> 00:21:24.500 align:middle line:84% And so it's going to be something between one eighth n 00:21:24.500 --> 00:21:27.590 align:middle line:90% cubed and n cubed itself. 00:21:27.590 --> 00:21:29.900 align:middle line:90% Does that make sense? 00:21:29.900 --> 00:21:31.850 align:middle line:84% We're hinting at something else that we'll 00:21:31.850 --> 00:21:34.430 align:middle line:90% do a little bit later. 00:21:34.430 --> 00:21:39.800 align:middle line:84% If you squint really hard, a sum kind of looks like an integral. 00:21:39.800 --> 00:21:41.810 align:middle line:84% So if you integrate that, you're going 00:21:41.810 --> 00:21:43.820 align:middle line:84% to get something that looks like n cubed. 00:21:43.820 --> 00:21:44.900 align:middle line:90% OK. 00:21:44.900 --> 00:21:46.370 align:middle line:90% Another question? 00:21:46.370 --> 00:21:48.800 align:middle line:84% AUDIENCE: Oh, my question was just if you could write down 00:21:48.800 --> 00:21:51.788 align:middle line:84% [INAUDIBLE] so that it's easier to visualize, like some of them 00:21:51.788 --> 00:21:54.080 align:middle line:84% are greater than n [INAUDIBLE] squared and some of them 00:21:54.080 --> 00:21:54.800 align:middle line:90% are less. 00:21:54.800 --> 00:21:55.940 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:21:55.940 --> 00:21:57.510 align:middle line:90% We could do that if you like. 00:21:57.510 --> 00:22:03.680 align:middle line:84% So we've got 1 plus 2 squared plus 3 squared 00:22:03.680 --> 00:22:14.216 align:middle line:84% plus n over 2 squared plus all the way up to n squared. 00:22:14.216 --> 00:22:20.610 align:middle line:84% So these terms, plus or minus 1 term if it's even or odd, 00:22:20.610 --> 00:22:27.900 align:middle line:84% so these are all greater than or equal to n over 2 squared. 00:22:27.900 --> 00:22:31.230 align:middle line:84% And we've got n over 2 of them or n over 2 plus 1, or whatever. 00:22:31.230 --> 00:22:36.330 align:middle line:84% So the sum is going to be at least n 00:22:36.330 --> 00:22:44.850 align:middle line:84% over 2 squared times n over 2 is one eighth n cubed. 00:22:44.850 --> 00:22:46.260 align:middle line:90% OK. 00:22:46.260 --> 00:22:46.930 align:middle line:90% Yeah? 00:22:46.930 --> 00:22:47.920 align:middle line:90% Question. 00:22:47.920 --> 00:22:51.180 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:22:51.180 --> 00:22:57.937 align:middle line:90% 00:22:57.937 --> 00:22:58.770 align:middle line:90% BRYNMOR CHAPMAN: Ah. 00:22:58.770 --> 00:22:59.270 align:middle line:90% OK. 00:22:59.270 --> 00:23:03.870 align:middle line:84% So we're getting a more refined guess. 00:23:03.870 --> 00:23:07.860 align:middle line:84% So actually, I would argue that n cubed is already good enough 00:23:07.860 --> 00:23:11.580 align:middle line:90% for what we're doing anyway. 00:23:11.580 --> 00:23:13.210 align:middle line:90% And here's why. 00:23:13.210 --> 00:23:17.000 align:middle line:84% So as we said, it's not very precise. 00:23:17.000 --> 00:23:17.500 align:middle line:90% Right? 00:23:17.500 --> 00:23:20.820 align:middle line:84% We don't know, like you said, the leading coefficient should 00:23:20.820 --> 00:23:23.430 align:middle line:90% probably be 1/3. 00:23:23.430 --> 00:23:26.880 align:middle line:84% But we're just going to start from guessing 00:23:26.880 --> 00:23:28.980 align:middle line:90% that it's a cubic of some sort. 00:23:28.980 --> 00:23:31.980 align:middle line:84% We don't know anything about the coefficients, 00:23:31.980 --> 00:23:34.205 align:middle line:84% but it seems like a reasonable guess. 00:23:34.205 --> 00:23:37.920 align:middle line:90% 00:23:37.920 --> 00:23:45.120 align:middle line:84% So we're going to say that S is equal to a times n cubed 00:23:45.120 --> 00:23:51.016 align:middle line:90% plus bn squared plus cn plus d. 00:23:51.016 --> 00:23:53.555 align:middle line:90% 00:23:53.555 --> 00:23:55.180 align:middle line:84% We're just going to take that on faith, 00:23:55.180 --> 00:23:56.975 align:middle line:84% assume it, and see where it leads us. 00:23:56.975 --> 00:24:00.360 align:middle line:90% 00:24:00.360 --> 00:24:04.770 align:middle line:84% So if we assume that, what can we deduce? 00:24:04.770 --> 00:24:05.525 align:middle line:90% Any ideas? 00:24:05.525 --> 00:24:10.410 align:middle line:90% 00:24:10.410 --> 00:24:12.000 align:middle line:84% How might we go about figuring out 00:24:12.000 --> 00:24:13.535 align:middle line:90% what a, b, c, and d should be? 00:24:13.535 --> 00:24:25.200 align:middle line:90% 00:24:25.200 --> 00:24:29.250 align:middle line:84% Well, what if we look at, small n? 00:24:29.250 --> 00:24:31.640 align:middle line:90% So if we have, say, n equals 0? 00:24:31.640 --> 00:24:34.350 align:middle line:90% 00:24:34.350 --> 00:24:36.140 align:middle line:84% What happens if we put in n equals 0? 00:24:36.140 --> 00:24:39.406 align:middle line:90% 00:24:39.406 --> 00:24:41.811 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:24:41.811 --> 00:24:43.920 align:middle line:90% BRYNMOR CHAPMAN: Exactly. 00:24:43.920 --> 00:24:44.850 align:middle line:90% OK. 00:24:44.850 --> 00:24:54.030 align:middle line:84% So then we have a times 0 cubed plus b times 0 squared plus c 00:24:54.030 --> 00:24:58.530 align:middle line:90% times 0 plus d equals what? 00:24:58.530 --> 00:25:00.330 align:middle line:84% What is the value of our sum at 0? 00:25:00.330 --> 00:25:01.108 align:middle line:90% AUDIENCE: 0. 00:25:01.108 --> 00:25:01.900 align:middle line:90% BRYNMOR CHAPMAN: 0. 00:25:01.900 --> 00:25:02.100 align:middle line:90% Yeah. 00:25:02.100 --> 00:25:03.575 align:middle line:90% We've just got the empty sum. 00:25:03.575 --> 00:25:09.870 align:middle line:90% 00:25:09.870 --> 00:25:10.910 align:middle line:90% What else can we do? 00:25:10.910 --> 00:25:16.970 align:middle line:90% 00:25:16.970 --> 00:25:18.500 align:middle line:84% AUDIENCE: You plug in n equals 1. 00:25:18.500 --> 00:25:19.417 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:25:19.417 --> 00:25:19.920 align:middle line:90% n equals 1. 00:25:19.920 --> 00:25:22.910 align:middle line:84% That seems like another decent choice. 00:25:22.910 --> 00:25:25.670 align:middle line:84% And so this time, we'll end up with a plus 00:25:25.670 --> 00:25:31.340 align:middle line:90% b plus c plus d equals what? 00:25:31.340 --> 00:25:34.188 align:middle line:84% What's the value of our sum there? 00:25:34.188 --> 00:25:35.000 align:middle line:90% AUDIENCE: 1. 00:25:35.000 --> 00:25:35.792 align:middle line:90% BRYNMOR CHAPMAN: 1. 00:25:35.792 --> 00:25:38.210 align:middle line:90% 00:25:38.210 --> 00:25:48.650 align:middle line:84% And if we continue like this, we get 8a plus 4b plus 2C plus d 00:25:48.650 --> 00:25:53.930 align:middle line:90% equals 5, so 1 plus 4. 00:25:53.930 --> 00:25:54.868 align:middle line:90% Question. 00:25:54.868 --> 00:25:57.307 align:middle line:84% AUDIENCE: Do we have the values of a, b, c, and d? 00:25:57.307 --> 00:25:58.140 align:middle line:90% BRYNMOR CHAPMAN: No. 00:25:58.140 --> 00:25:59.670 align:middle line:84% We're trying to figure those out. 00:25:59.670 --> 00:26:04.580 align:middle line:84% So we're just taking it on faith that S has this form. 00:26:04.580 --> 00:26:07.922 align:middle line:84% AUDIENCE: But if n equals 1 then we solve one. 00:26:07.922 --> 00:26:09.380 align:middle line:84% BRYNMOR CHAPMAN: So we already know 00:26:09.380 --> 00:26:12.620 align:middle line:84% that because S was defined as the sum of squares. 00:26:12.620 --> 00:26:16.740 align:middle line:84% So if we sum up all of the squares up to n equals 1, 00:26:16.740 --> 00:26:17.760 align:middle line:90% we get 1. 00:26:17.760 --> 00:26:21.435 align:middle line:84% If we sum them up to n equals 2, we get 5, cetera. 00:26:21.435 --> 00:26:24.626 align:middle line:90% 00:26:24.626 --> 00:26:25.400 align:middle line:90% Is this enough? 00:26:25.400 --> 00:26:31.386 align:middle line:90% 00:26:31.386 --> 00:26:32.386 align:middle line:90% AUDIENCE: Need one more. 00:26:32.386 --> 00:26:33.303 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:26:33.303 --> 00:26:34.050 align:middle line:90% We need one more. 00:26:34.050 --> 00:26:37.280 align:middle line:90% So let's put in n equals 3. 00:26:37.280 --> 00:26:48.530 align:middle line:84% Here we've got 27a plus 9b plus 3c plus d equals 14. 00:26:48.530 --> 00:26:51.200 align:middle line:90% Can I arithmetic at least? 00:26:51.200 --> 00:26:52.640 align:middle line:90% I think I can arithmetic. 00:26:52.640 --> 00:26:54.120 align:middle line:90% 1 plus 4 plus 9. 00:26:54.120 --> 00:26:55.590 align:middle line:90% That sounds like 14. 00:26:55.590 --> 00:26:56.090 align:middle line:90% OK. 00:26:56.090 --> 00:27:01.700 align:middle line:84% So now we've got a system of four equations and four 00:27:01.700 --> 00:27:02.820 align:middle line:90% variables. 00:27:02.820 --> 00:27:05.600 align:middle line:84% So we should be able to solve it. 00:27:05.600 --> 00:27:07.688 align:middle line:90% OK. 00:27:07.688 --> 00:27:09.730 align:middle line:84% Has anybody figured out what the solution is yet? 00:27:09.730 --> 00:27:12.500 align:middle line:90% 00:27:12.500 --> 00:27:13.650 align:middle line:90% Well, d equals 0. 00:27:13.650 --> 00:27:16.250 align:middle line:90% That's an easy one. 00:27:16.250 --> 00:27:23.640 align:middle line:84% I think it turns out to be a equals 1/3, b equals 1/2, 00:27:23.640 --> 00:27:26.510 align:middle line:90% c equals 1/6. 00:27:26.510 --> 00:27:27.235 align:middle line:90% Is that right? 00:27:27.235 --> 00:27:30.350 align:middle line:90% 00:27:30.350 --> 00:27:34.180 align:middle line:84% And yeah, d equals 0 because we've got a bunch of zeros here. 00:27:34.180 --> 00:27:37.070 align:middle line:90% 00:27:37.070 --> 00:27:38.774 align:middle line:90% OK. 00:27:38.774 --> 00:27:41.080 align:middle line:90% So are we done? 00:27:41.080 --> 00:27:42.205 align:middle line:90% Have we found our solution? 00:27:42.205 --> 00:27:48.800 align:middle line:90% 00:27:48.800 --> 00:27:49.460 align:middle line:90% Yeah? 00:27:49.460 --> 00:27:50.923 align:middle line:84% AUDIENCE: Got to prove the values. 00:27:50.923 --> 00:27:51.840 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:27:51.840 --> 00:27:56.100 align:middle line:84% So we've maybe found our solution but we don't know. 00:27:56.100 --> 00:28:00.000 align:middle line:84% We just pulled this out of a hat and assumed it. 00:28:00.000 --> 00:28:03.260 align:middle line:84% So we should actually prove that this is correct. 00:28:03.260 --> 00:28:13.570 align:middle line:84% So now we're going to take S equals 1/3 n cubed plus 1/2 n 00:28:13.570 --> 00:28:18.880 align:middle line:84% squared plus 1/6 n and use that as our guess and go back 00:28:18.880 --> 00:28:21.710 align:middle line:84% to the guess and check method and prove this by induction. 00:28:21.710 --> 00:28:26.080 align:middle line:90% 00:28:26.080 --> 00:28:28.600 align:middle line:84% We're actually going to want strong induction. 00:28:28.600 --> 00:28:30.080 align:middle line:90% These are our base cases. 00:28:30.080 --> 00:28:31.840 align:middle line:84% We've already proven that, essentially. 00:28:31.840 --> 00:28:35.320 align:middle line:84% That's what we used to derive a, b, c, and d. 00:28:35.320 --> 00:28:38.500 align:middle line:84% And our inductive step is just going to be a bunch of algebra. 00:28:38.500 --> 00:28:40.580 align:middle line:84% So I'm not going to go into that right now. 00:28:40.580 --> 00:28:46.600 align:middle line:84% But yeah, if you're interested, feel free to prove it. 00:28:46.600 --> 00:28:49.210 align:middle line:90% OK. 00:28:49.210 --> 00:28:49.790 align:middle line:90% Yeah. 00:28:49.790 --> 00:28:52.600 align:middle line:84% And if it works, we'll have a proof 00:28:52.600 --> 00:28:55.370 align:middle line:90% that our formula was correct. 00:28:55.370 --> 00:28:58.750 align:middle line:84% And if it doesn't, we go back to the drawing board. 00:28:58.750 --> 00:29:00.890 align:middle line:84% We know that this form was wrong. 00:29:00.890 --> 00:29:04.210 align:middle line:84% And so maybe we try and figure out a different form 00:29:04.210 --> 00:29:07.420 align:middle line:90% that our answer might take. 00:29:07.420 --> 00:29:09.550 align:middle line:90% Does that make sense? 00:29:09.550 --> 00:29:10.800 align:middle line:90% Are people happy with that? 00:29:10.800 --> 00:29:16.330 align:middle line:90% 00:29:16.330 --> 00:29:17.425 align:middle line:90% Any questions? 00:29:17.425 --> 00:29:19.050 align:middle line:84% Maybe I should phrase it the other way. 00:29:19.050 --> 00:29:25.090 align:middle line:90% 00:29:25.090 --> 00:29:26.530 align:middle line:90% AUDIENCE: What's your name? 00:29:26.530 --> 00:29:27.363 align:middle line:90% BRYNMOR CHAPMAN: Oh. 00:29:27.363 --> 00:29:30.560 align:middle line:84% I'm Brynmor, like the Pennsylvania College. 00:29:30.560 --> 00:29:32.060 align:middle line:90% Same word spelled differently. 00:29:32.060 --> 00:29:35.980 align:middle line:90% 00:29:35.980 --> 00:29:38.450 align:middle line:90% I saw another hand over here. 00:29:38.450 --> 00:29:41.898 align:middle line:90% Was that a question or a-- 00:29:41.898 --> 00:29:44.920 align:middle line:84% AUDIENCE: Your 1/3, 1/2, 1/6, that's just like if you do 00:29:44.920 --> 00:29:45.923 align:middle line:90% the algebra pretty much? 00:29:45.923 --> 00:29:46.840 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:29:46.840 --> 00:29:51.040 align:middle line:84% So that's if you solve this system of equations. 00:29:51.040 --> 00:29:52.400 align:middle line:90% That's what comes out. 00:29:52.400 --> 00:29:54.610 align:middle line:84% You could do Gaussian elimination, 00:29:54.610 --> 00:29:57.057 align:middle line:84% or invert the matrix, or whatever. 00:29:57.057 --> 00:29:58.640 align:middle line:84% AUDIENCE: But if you didn't know that, 00:29:58.640 --> 00:30:01.407 align:middle line:84% did you just know that because it was on your notes or did you 00:30:01.407 --> 00:30:02.990 align:middle line:84% have a way to figure it out instantly, 00:30:02.990 --> 00:30:04.000 align:middle line:90% like the way you just wrote it? 00:30:04.000 --> 00:30:05.000 align:middle line:90% BRYNMOR CHAPMAN: Oh, no. 00:30:05.000 --> 00:30:06.957 align:middle line:90% I'm not that good. 00:30:06.957 --> 00:30:07.790 align:middle line:90% AUDIENCE: All right. 00:30:07.790 --> 00:30:08.610 align:middle line:90% I'm just checking. 00:30:08.610 --> 00:30:10.360 align:middle line:84% BRYNMOR CHAPMAN: Oh, I mean, no, I totally 00:30:10.360 --> 00:30:12.250 align:middle line:90% don't have it on my notes. 00:30:12.250 --> 00:30:13.420 align:middle line:90% Yeah. 00:30:13.420 --> 00:30:14.000 align:middle line:90% Yeah. 00:30:14.000 --> 00:30:14.505 align:middle line:90% Question. 00:30:14.505 --> 00:30:15.880 align:middle line:84% AUDIENCE: Is there any particular 00:30:15.880 --> 00:30:19.342 align:middle line:84% reason you chose this form as your educated guess? 00:30:19.342 --> 00:30:22.043 align:middle line:90% [INAUDIBLE] 00:30:22.043 --> 00:30:22.960 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:30:22.960 --> 00:30:28.180 align:middle line:84% So as we were discussing before we dove into it, 00:30:28.180 --> 00:30:30.040 align:middle line:84% we think that it's probably going 00:30:30.040 --> 00:30:33.760 align:middle line:90% to look something like n cubed. 00:30:33.760 --> 00:30:36.830 align:middle line:84% We don't know exactly how close it is to that, 00:30:36.830 --> 00:30:39.100 align:middle line:84% but it's kind of reasonable to assume 00:30:39.100 --> 00:30:42.970 align:middle line:84% that it's a polynomial like a cubic polynomial. 00:30:42.970 --> 00:30:48.943 align:middle line:84% So maybe we could have started with something like this 00:30:48.943 --> 00:30:49.443 align:middle line:90% instead. 00:30:49.443 --> 00:30:53.140 align:middle line:90% 00:30:53.140 --> 00:30:55.630 align:middle line:84% Like maybe it's some constant times n 00:30:55.630 --> 00:30:58.120 align:middle line:90% cubed plus quadratic term. 00:30:58.120 --> 00:31:03.300 align:middle line:84% And then the ansatz method would have failed. 00:31:03.300 --> 00:31:05.877 align:middle line:84% We would have ended up with suspicious looking guess. 00:31:05.877 --> 00:31:07.710 align:middle line:84% We would have tried to prove it by induction 00:31:07.710 --> 00:31:10.560 align:middle line:90% and it wouldn't have worked. 00:31:10.560 --> 00:31:19.130 align:middle line:84% But yeah, as it turns out, this is a reasonable guess. 00:31:19.130 --> 00:31:19.630 align:middle line:90% Yeah? 00:31:19.630 --> 00:31:21.990 align:middle line:84% AUDIENCE: If we don't know if it's a bad guess, 00:31:21.990 --> 00:31:25.380 align:middle line:84% and we try to prove it by induction, 00:31:25.380 --> 00:31:27.040 align:middle line:84% and we don't see how to prove it, 00:31:27.040 --> 00:31:28.720 align:middle line:84% how do we know that means it's wrong? 00:31:28.720 --> 00:31:32.820 align:middle line:84% What if we just didn't figure out the right way to prove it? 00:31:32.820 --> 00:31:35.700 align:middle line:84% BRYNMOR CHAPMAN: So the question is if we have a bad guess, 00:31:35.700 --> 00:31:37.442 align:middle line:84% and we can't prove it by induction, 00:31:37.442 --> 00:31:39.150 align:middle line:84% how do we know that the guess is actually 00:31:39.150 --> 00:31:41.760 align:middle line:84% wrong and not that we're doing something 00:31:41.760 --> 00:31:44.100 align:middle line:90% suspicious with the proof? 00:31:44.100 --> 00:31:46.710 align:middle line:90% So that's a good question. 00:31:46.710 --> 00:31:50.790 align:middle line:84% So maybe I should digress here a little bit. 00:31:50.790 --> 00:31:54.930 align:middle line:84% If you're trying to figure out a proof or a disproof 00:31:54.930 --> 00:31:57.510 align:middle line:84% for a statement that you're not quite sure whether it's true 00:31:57.510 --> 00:32:01.050 align:middle line:84% or not, often the thing that you'll want to do 00:32:01.050 --> 00:32:03.310 align:middle line:90% is attempt to prove it. 00:32:03.310 --> 00:32:06.312 align:middle line:84% see where your proof seems to be breaking, like 00:32:06.312 --> 00:32:07.770 align:middle line:84% if you find a block, like something 00:32:07.770 --> 00:32:10.170 align:middle line:84% that you can't figure out, and then try and use 00:32:10.170 --> 00:32:12.900 align:middle line:90% that to generate a disproof. 00:32:12.900 --> 00:32:15.250 align:middle line:84% And then you can iterate on that. 00:32:15.250 --> 00:32:19.530 align:middle line:84% And if your disproof gets stuck somewhere, 00:32:19.530 --> 00:32:22.500 align:middle line:84% maybe try and use that block to try and come up 00:32:22.500 --> 00:32:23.850 align:middle line:90% with a different proof. 00:32:23.850 --> 00:32:25.746 align:middle line:90% OK? 00:32:25.746 --> 00:32:29.250 align:middle line:84% And eventually, hopefully, you'll 00:32:29.250 --> 00:32:31.710 align:middle line:84% be able to come up with one or the other. 00:32:31.710 --> 00:32:40.170 align:middle line:84% So in this case, there's a fairly clear path 00:32:40.170 --> 00:32:41.850 align:middle line:90% towards a disproof. 00:32:41.850 --> 00:32:43.600 align:middle line:90% Put in more numbers. 00:32:43.600 --> 00:32:46.310 align:middle line:90% OK. 00:32:46.310 --> 00:32:49.280 align:middle line:90% So we could try n equals 4. 00:32:49.280 --> 00:32:53.130 align:middle line:90% 00:32:53.130 --> 00:33:02.370 align:middle line:84% Then we'd have 64a plus 16b plus 4c plus d should be equal 00:33:02.370 --> 00:33:03.830 align:middle line:90% to, what is it, 30? 00:33:03.830 --> 00:33:06.396 align:middle line:90% 00:33:06.396 --> 00:33:08.370 align:middle line:90% So that's a decent sanity check. 00:33:08.370 --> 00:33:09.220 align:middle line:90% Right? 00:33:09.220 --> 00:33:12.690 align:middle line:84% Like if that doesn't work, then we know that we've 00:33:12.690 --> 00:33:15.360 align:middle line:90% screwed something up. 00:33:15.360 --> 00:33:16.780 align:middle line:84% Does that make sense to everybody? 00:33:16.780 --> 00:33:17.280 align:middle line:90% Yeah. 00:33:17.280 --> 00:33:18.198 align:middle line:90% Question? 00:33:18.198 --> 00:33:25.650 align:middle line:84% AUDIENCE: [INAUDIBLE] How did you decide that those 00:33:25.650 --> 00:33:26.860 align:middle line:90% would be the base cases? 00:33:26.860 --> 00:33:28.920 align:middle line:90% Is it just for simplicity sake? 00:33:28.920 --> 00:33:31.700 align:middle line:84% BRYNMOR CHAPMAN: So that's because we have four variables 00:33:31.700 --> 00:33:32.200 align:middle line:90% here. 00:33:32.200 --> 00:33:33.970 align:middle line:90% We've got a, b, c, and d. 00:33:33.970 --> 00:33:38.156 align:middle line:84% So we needed four equations, like four constraints. 00:33:38.156 --> 00:33:41.700 align:middle line:90% 00:33:41.700 --> 00:33:42.330 align:middle line:90% Yeah. 00:33:42.330 --> 00:33:45.210 align:middle line:84% Like we could equally well have chosen like n equals 1, 00:33:45.210 --> 00:33:48.240 align:middle line:84% n equals 2, n equals 5, n equals 100. 00:33:48.240 --> 00:33:50.110 align:middle line:84% It would have given us the correct answer. 00:33:50.110 --> 00:33:56.270 align:middle line:84% It just would have taken more complicated computation. 00:33:56.270 --> 00:33:56.779 align:middle line:90% Yeah. 00:33:56.779 --> 00:33:57.446 align:middle line:90% AUDIENCE: Sorry. 00:33:57.446 --> 00:34:01.323 align:middle line:84% Will there ever be an n over 2 multiplier that works out 00:34:01.323 --> 00:34:06.020 align:middle line:90% to be n over 2 [INAUDIBLE]? 00:34:06.020 --> 00:34:07.700 align:middle line:90% BRYNMOR CHAPMAN: Oh, here. 00:34:07.700 --> 00:34:09.069 align:middle line:90% Here you mean? 00:34:09.069 --> 00:34:09.949 align:middle line:90% AUDIENCE: Yeah. 00:34:09.949 --> 00:34:11.967 align:middle line:84% Or where the next n over 2 would be. 00:34:11.967 --> 00:34:12.800 align:middle line:90% BRYNMOR CHAPMAN: Oh. 00:34:12.800 --> 00:34:13.300 align:middle line:90% OK. 00:34:13.300 --> 00:34:16.070 align:middle line:90% So the squared term is-- 00:34:16.070 --> 00:34:19.610 align:middle line:84% right, because each of these terms here is at least n over 2 00:34:19.610 --> 00:34:21.080 align:middle line:90% squared. 00:34:21.080 --> 00:34:23.880 align:middle line:84% The last term is because there are n over 2 of them. 00:34:23.880 --> 00:34:27.080 align:middle line:84% And we're trying to evaluate the sum. 00:34:27.080 --> 00:34:29.270 align:middle line:84% So we're summing up n over 2 things that are all 00:34:29.270 --> 00:34:30.505 align:middle line:90% at least n over 2 squared. 00:34:30.505 --> 00:34:34.040 align:middle line:90% 00:34:34.040 --> 00:34:36.600 align:middle line:84% AUDIENCE: What about these 1 plus 2 squared plus 3 squared 00:34:36.600 --> 00:34:37.423 align:middle line:90% [INAUDIBLE]? 00:34:37.423 --> 00:34:39.090 align:middle line:84% BRYNMOR CHAPMAN: They're are at least 0. 00:34:39.090 --> 00:34:40.020 align:middle line:90% We don't care. 00:34:40.020 --> 00:34:41.480 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:34:41.480 --> 00:34:42.397 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:34:42.397 --> 00:34:48.020 align:middle line:90% 00:34:48.020 --> 00:34:53.150 align:middle line:84% OK, are people reasonably happy with this? 00:34:53.150 --> 00:34:54.710 align:middle line:90% Yeah, OK. 00:34:54.710 --> 00:34:58.226 align:middle line:84% So let's move on to a third technique. 00:34:58.226 --> 00:35:14.360 align:middle line:90% 00:35:14.360 --> 00:35:20.790 align:middle line:84% So sometimes, we're going to end up with sums of sums-- 00:35:20.790 --> 00:35:24.290 align:middle line:84% nested summations which we call double sums. 00:35:24.290 --> 00:35:28.770 align:middle line:84% So for instance, what if we had the following? 00:35:28.770 --> 00:35:39.140 align:middle line:84% So I equals 1 to n of the sum from j equals 1 to i of j. 00:35:39.140 --> 00:35:44.420 align:middle line:90% So now, before we had-- 00:35:44.420 --> 00:35:48.830 align:middle line:84% the thing inside our sum was this nice simple expression 00:35:48.830 --> 00:35:50.930 align:middle line:90% k squared. 00:35:50.930 --> 00:35:55.550 align:middle line:84% Now our summand is itself a summation. 00:35:55.550 --> 00:35:56.700 align:middle line:90% What can we do about that? 00:35:56.700 --> 00:35:58.325 align:middle line:84% How would we evaluate things like this? 00:35:58.325 --> 00:36:02.982 align:middle line:90% 00:36:02.982 --> 00:36:06.980 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:36:06.980 --> 00:36:08.840 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so the answer 00:36:08.840 --> 00:36:13.190 align:middle line:84% was to evaluate the inner sum first, figure out a closed 00:36:13.190 --> 00:36:16.310 align:middle line:84% form as a function of i, and then we 00:36:16.310 --> 00:36:17.990 align:middle line:90% can evaluate the outer sum. 00:36:17.990 --> 00:36:23.245 align:middle line:84% So in this case, what do we end up with? 00:36:23.245 --> 00:36:25.120 align:middle line:84% What happens when we evaluate this inner sum? 00:36:25.120 --> 00:36:32.060 align:middle line:90% 00:36:32.060 --> 00:36:34.358 align:middle line:84% I claim that this is something that we already 00:36:34.358 --> 00:36:36.400 align:middle line:84% know how to do, something that we've seen before. 00:36:36.400 --> 00:36:39.770 align:middle line:90% 00:36:39.770 --> 00:36:41.240 align:middle line:90% Yeah? 00:36:41.240 --> 00:36:44.870 align:middle line:84% AUDIENCE: i minus i plus 1 over n? 00:36:44.870 --> 00:36:46.880 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:36:46.880 --> 00:36:53.140 align:middle line:84% So this is the sum from i equals 1 to n of i times i 00:36:53.140 --> 00:36:57.130 align:middle line:90% plus 1, all divided by 2. 00:36:57.130 --> 00:37:02.450 align:middle line:84% Now how could we evaluate this sum? 00:37:02.450 --> 00:37:05.050 align:middle line:84% This expression is a little bit more complicated than the ones 00:37:05.050 --> 00:37:07.542 align:middle line:90% that we had before. 00:37:07.542 --> 00:37:10.060 align:middle line:84% is there a way that we can simplify it? 00:37:10.060 --> 00:37:11.860 align:middle line:90% Yeah? 00:37:11.860 --> 00:37:14.260 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:37:14.260 --> 00:37:18.010 align:middle line:90% 00:37:18.010 --> 00:37:20.920 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so let's distribute this i 00:37:20.920 --> 00:37:22.660 align:middle line:90% and then break the sum up. 00:37:22.660 --> 00:37:29.440 align:middle line:84% So if we distribute this, we end up 00:37:29.440 --> 00:37:35.830 align:middle line:84% with i squared plus i divided by 2. 00:37:35.830 --> 00:37:41.990 align:middle line:84% And now we've got a sum of a sum. 00:37:41.990 --> 00:37:44.296 align:middle line:84% So we can split it into two different summations. 00:37:44.296 --> 00:37:48.130 align:middle line:90% 00:37:48.130 --> 00:37:59.380 align:middle line:84% i equals 1 to n i squared plus sum of i. 00:37:59.380 --> 00:38:05.620 align:middle line:84% So I've taken each of these terms and split it into its own 00:38:05.620 --> 00:38:09.340 align:middle line:84% sum and also pulled this factor of 1/2 out in front. 00:38:09.340 --> 00:38:14.060 align:middle line:84% So this is now something that we know how to evaluate. 00:38:14.060 --> 00:38:16.390 align:middle line:90% We just evaluated it. 00:38:16.390 --> 00:38:19.270 align:middle line:84% This is also something we know how to evaluate. 00:38:19.270 --> 00:38:20.910 align:middle line:90% Also, we just evaluated it. 00:38:20.910 --> 00:38:23.920 align:middle line:90% 00:38:23.920 --> 00:38:28.940 align:middle line:84% So this comes out to the answer that we had there, 00:38:28.940 --> 00:38:51.160 align:middle line:84% so 1/6 n cubed plus 1/4 n squared plus 1/12 n plus 1/4 00:38:51.160 --> 00:38:54.970 align:middle line:90% times n times n plus 1. 00:38:54.970 --> 00:39:01.190 align:middle line:84% Does that make sense to everybody? 00:39:01.190 --> 00:39:17.200 align:middle line:90% 00:39:17.200 --> 00:39:23.630 align:middle line:84% Suppose instead we had something similar but slightly different. 00:39:23.630 --> 00:39:29.080 align:middle line:90% 00:39:29.080 --> 00:39:36.955 align:middle line:84% So sum from i equals 1 to n, sum from j equals i to n. 00:39:36.955 --> 00:39:44.400 align:middle line:90% 00:39:44.400 --> 00:39:48.090 align:middle line:84% So our inner sum now, instead of going from 1 to i, 00:39:48.090 --> 00:39:49.650 align:middle line:90% it's going from i to n. 00:39:49.650 --> 00:40:01.800 align:middle line:90% 00:40:01.800 --> 00:40:07.380 align:middle line:84% Now we could evaluate this in the same way as we just did. 00:40:07.380 --> 00:40:10.800 align:middle line:84% But it turns out that there's a slightly easier way. 00:40:10.800 --> 00:40:13.960 align:middle line:84% We can exchange the order of summation. 00:40:13.960 --> 00:40:33.390 align:middle line:90% 00:40:33.390 --> 00:40:37.620 align:middle line:84% And what I mean by that is that we want the outer sum 00:40:37.620 --> 00:40:41.730 align:middle line:84% to be in terms of j and the inner sum to be in terms of i. 00:40:41.730 --> 00:40:51.500 align:middle line:84% So what are the possible values of j? 00:40:51.500 --> 00:41:01.650 align:middle line:90% 00:41:01.650 --> 00:41:03.090 align:middle line:90% So in our-- yes? 00:41:03.090 --> 00:41:07.440 align:middle line:90% 00:41:07.440 --> 00:41:08.850 align:middle line:90% Yeah? 00:41:08.850 --> 00:41:10.080 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:41:10.080 --> 00:41:10.997 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:41:10.997 --> 00:41:15.610 align:middle line:90% 00:41:15.610 --> 00:41:20.770 align:middle line:84% So if i equals 1 here, j equals i, j equals 1. 00:41:20.770 --> 00:41:23.190 align:middle line:84% So that's the smallest value of j. 00:41:23.190 --> 00:41:27.330 align:middle line:90% And j can clearly go up to n. 00:41:27.330 --> 00:41:37.650 align:middle line:90% 00:41:37.650 --> 00:41:39.875 align:middle line:84% What are the bounds of summation on our inner sum? 00:41:39.875 --> 00:41:45.360 align:middle line:90% 00:41:45.360 --> 00:41:46.670 align:middle line:90% How would we compute this? 00:41:46.670 --> 00:41:53.490 align:middle line:90% 00:41:53.490 --> 00:41:57.810 align:middle line:84% What constraints do we have on i and k originally? 00:41:57.810 --> 00:42:04.650 align:middle line:90% 00:42:04.650 --> 00:42:07.830 align:middle line:84% When we know that 1 is less than or equal to i less than 00:42:07.830 --> 00:42:12.600 align:middle line:84% or equal to n i less than or equal to j less than 00:42:12.600 --> 00:42:13.260 align:middle line:90% or equal to n. 00:42:13.260 --> 00:42:17.070 align:middle line:90% 00:42:17.070 --> 00:42:20.610 align:middle line:84% So we could phrase this as 1 less 00:42:20.610 --> 00:42:23.790 align:middle line:84% than or equal to i less than or equal to j less than 00:42:23.790 --> 00:42:24.485 align:middle line:90% or equal to n. 00:42:24.485 --> 00:42:28.950 align:middle line:90% 00:42:28.950 --> 00:42:32.675 align:middle line:84% And now we've got bounds on i in terms of j. 00:42:32.675 --> 00:42:42.360 align:middle line:90% 00:42:42.360 --> 00:42:44.050 align:middle line:84% Does that make sense to everybody? 00:42:44.050 --> 00:42:46.920 align:middle line:90% 00:42:46.920 --> 00:42:52.260 align:middle line:84% So we still want the sum over the same set of pairs, i and j. 00:42:52.260 --> 00:42:54.340 align:middle line:84% We're just expressing it differently. 00:42:54.340 --> 00:42:57.870 align:middle line:90% 00:42:57.870 --> 00:42:59.375 align:middle line:90% And now our sum n is the same. 00:42:59.375 --> 00:43:02.910 align:middle line:90% 00:43:02.910 --> 00:43:06.090 align:middle line:84% Does it make sense to everybody how we got this? 00:43:06.090 --> 00:43:07.260 align:middle line:90% Yeah? 00:43:07.260 --> 00:43:09.424 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:43:09.424 --> 00:43:14.100 align:middle line:90% 00:43:14.100 --> 00:43:19.620 align:middle line:84% BRYNMOR CHAPMAN: So this just came from the bounds 00:43:19.620 --> 00:43:22.560 align:middle line:90% on our inner sum. 00:43:22.560 --> 00:43:27.390 align:middle line:84% We're saying that j should range from i up to n. 00:43:27.390 --> 00:43:35.250 align:middle line:84% And so then we just stick this pair of inequalities in here. 00:43:35.250 --> 00:43:38.690 align:middle line:84% Does that make sense to everybody? 00:43:38.690 --> 00:43:42.260 align:middle line:84% Are you reasonably happy with this? 00:43:42.260 --> 00:43:46.910 align:middle line:84% So now we've exchanged the order of summation. 00:43:46.910 --> 00:43:52.670 align:middle line:84% We've got j for our outer sum, i for our inner sum. 00:43:52.670 --> 00:43:55.970 align:middle line:84% And now we've got a much simpler sum here 00:43:55.970 --> 00:43:58.130 align:middle line:90% that we can easily evaluate. 00:43:58.130 --> 00:43:59.365 align:middle line:90% What is our inner sum? 00:43:59.365 --> 00:44:08.857 align:middle line:90% 00:44:08.857 --> 00:44:09.690 align:middle line:90% AUDIENCE: j squared. 00:44:09.690 --> 00:44:10.607 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:44:10.607 --> 00:44:20.630 align:middle line:90% 00:44:20.630 --> 00:44:27.020 align:middle line:84% The sum end has no dependence on the index of summation. 00:44:27.020 --> 00:44:28.830 align:middle line:90% So we've just got j terms. 00:44:28.830 --> 00:44:30.110 align:middle line:90% All of them are j. 00:44:30.110 --> 00:44:31.105 align:middle line:90% So it's j squared. 00:44:31.105 --> 00:44:37.016 align:middle line:90% 00:44:37.016 --> 00:44:39.535 align:middle line:84% And now this is just the sum that we had before. 00:44:39.535 --> 00:44:56.150 align:middle line:90% 00:44:56.150 --> 00:44:57.690 align:middle line:84% Does that make sense to everybody? 00:44:57.690 --> 00:45:00.770 align:middle line:90% 00:45:00.770 --> 00:45:01.865 align:middle line:90% Yeah? 00:45:01.865 --> 00:45:04.910 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:45:04.910 --> 00:45:11.330 align:middle line:90% 00:45:11.330 --> 00:45:16.928 align:middle line:90% BRYNMOR CHAPMAN: This one here? 00:45:16.928 --> 00:45:19.916 align:middle line:90% 00:45:19.916 --> 00:45:23.130 align:middle line:84% So that's just an expression that you've already seen. 00:45:23.130 --> 00:45:25.280 align:middle line:84% We've already seen how to evaluate the sum from j 00:45:25.280 --> 00:45:28.550 align:middle line:90% equals 1 to i of j. 00:45:28.550 --> 00:45:30.680 align:middle line:84% So that's just plugging in that closed 00:45:30.680 --> 00:45:32.875 align:middle line:90% formula for that inner sum. 00:45:32.875 --> 00:45:42.500 align:middle line:90% 00:45:42.500 --> 00:45:44.125 align:middle line:84% Are people reasonably happy with this? 00:45:44.125 --> 00:45:46.850 align:middle line:90% 00:45:46.850 --> 00:45:48.023 align:middle line:90% Yeah? 00:45:48.023 --> 00:45:51.105 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:45:51.105 --> 00:45:52.105 align:middle line:90% BRYNMOR CHAPMAN: Did we? 00:45:52.105 --> 00:45:56.660 align:middle line:90% 00:45:56.660 --> 00:45:58.340 align:middle line:90% Oh yeah, here-ish. 00:45:58.340 --> 00:46:00.890 align:middle line:90% 00:46:00.890 --> 00:46:15.950 align:middle line:84% I mean, here it's the particular example where n equals 100. 00:46:15.950 --> 00:46:19.750 align:middle line:84% So why don't we move on to the last topic for today? 00:46:19.750 --> 00:46:30.880 align:middle line:90% 00:46:30.880 --> 00:46:38.745 align:middle line:84% So what happens if we have the following sum? 00:46:38.745 --> 00:46:52.030 align:middle line:90% 00:46:52.030 --> 00:46:55.663 align:middle line:84% Any idea how we might come up with a closed form for this? 00:46:55.663 --> 00:46:57.330 align:middle line:84% Does anybody already know a closed form? 00:46:57.330 --> 00:47:01.100 align:middle line:90% 00:47:01.100 --> 00:47:01.600 align:middle line:90% Nobody? 00:47:01.600 --> 00:47:02.822 align:middle line:90% Really? 00:47:02.822 --> 00:47:04.530 align:middle line:84% That's good because it's an open problem. 00:47:04.530 --> 00:47:07.450 align:middle line:90% 00:47:07.450 --> 00:47:09.200 align:middle line:84% Well, I suppose that doesn't make it good. 00:47:09.200 --> 00:47:11.450 align:middle line:84% If you have a closed form, you could a paper about it. 00:47:11.450 --> 00:47:14.360 align:middle line:90% That would be pretty exciting. 00:47:14.360 --> 00:47:19.070 align:middle line:84% So it turns out that there isn't any nice closed form for this, 00:47:19.070 --> 00:47:21.670 align:middle line:90% as far as we know. 00:47:21.670 --> 00:47:25.645 align:middle line:84% So does that mean we should just-- hands in the air, 00:47:25.645 --> 00:47:26.145 align:middle line:90% give up? 00:47:26.145 --> 00:47:30.110 align:middle line:90% 00:47:30.110 --> 00:47:31.450 align:middle line:90% Getting some nods. 00:47:31.450 --> 00:47:36.940 align:middle line:84% Yeah, I mean, I suppose that's a question of philosophy. 00:47:36.940 --> 00:47:39.710 align:middle line:84% But we can still say something about it, right? 00:47:39.710 --> 00:47:44.080 align:middle line:84% Maybe we don't need to know exactly what this sum is. 00:47:44.080 --> 00:47:47.270 align:middle line:84% But as long as we figure out something, that's close enough, 00:47:47.270 --> 00:47:49.420 align:middle line:90% maybe we're still happy. 00:47:49.420 --> 00:47:51.450 align:middle line:84% So maybe we want to approximate it. 00:47:51.450 --> 00:48:05.544 align:middle line:90% 00:48:05.544 --> 00:48:10.750 align:middle line:84% So can anybody give me a guess as to an approximation 00:48:10.750 --> 00:48:11.340 align:middle line:90% for this sum? 00:48:11.340 --> 00:48:17.920 align:middle line:90% 00:48:17.920 --> 00:48:20.120 align:middle line:84% AUDIENCE: So we were just integrating that as well? 00:48:20.120 --> 00:48:23.515 align:middle line:90% BRYNMOR CHAPMAN: Mm-hmm. 00:48:23.515 --> 00:48:27.880 align:middle line:84% So the answer was, again, if we squint really hard and pretend 00:48:27.880 --> 00:48:31.780 align:middle line:84% that this is an integral, maybe s is approximately equal 00:48:31.780 --> 00:48:39.670 align:middle line:84% to the integral of root x dx, say from 1 to n, 00:48:39.670 --> 00:48:50.920 align:middle line:84% which is going to be about 2/3 n to the 3/2. 00:48:50.920 --> 00:48:53.170 align:middle line:90% Is that about right? 00:48:53.170 --> 00:48:55.480 align:middle line:84% I swear I've done calculus before. 00:48:55.480 --> 00:48:58.060 align:middle line:90% It's been a while. 00:48:58.060 --> 00:49:05.180 align:middle line:84% So the question then becomes, how close is that approximation? 00:49:05.180 --> 00:49:08.410 align:middle line:90% 00:49:08.410 --> 00:49:11.740 align:middle line:84% Do we think we're within a factor of 2? 00:49:11.740 --> 00:49:15.890 align:middle line:84% Do you think we're off by n by root n? 00:49:15.890 --> 00:49:19.000 align:middle line:90% Off by 1 maybe? 00:49:19.000 --> 00:49:21.973 align:middle line:90% It's not entirely clear. 00:49:21.973 --> 00:49:28.380 align:middle line:84% So in order to figure this out, we're 00:49:28.380 --> 00:49:30.635 align:middle line:84% going to use what we call the integral method. 00:49:30.635 --> 00:49:34.560 align:middle line:90% 00:49:34.560 --> 00:49:42.480 align:middle line:84% So give me a moment to draw this. 00:49:42.480 --> 00:49:48.210 align:middle line:84% And redux looks a little bit like that 00:49:48.210 --> 00:49:49.370 align:middle line:90% if you squint really hard. 00:49:49.370 --> 00:49:57.990 align:middle line:90% 00:49:57.990 --> 00:50:08.960 align:middle line:84% So suppose we have a weakly increasing function of x. 00:50:08.960 --> 00:50:18.180 align:middle line:90% 00:50:18.180 --> 00:50:22.445 align:middle line:84% So root x is indeed weakly increasing, so that'll work. 00:50:22.445 --> 00:50:25.320 align:middle line:90% 00:50:25.320 --> 00:50:29.030 align:middle line:84% Now, recall Riemann sums from integral calculus. 00:50:29.030 --> 00:50:32.310 align:middle line:90% 00:50:32.310 --> 00:50:33.490 align:middle line:90% No post-traumatic stress? 00:50:33.490 --> 00:50:35.130 align:middle line:90% Are we OK? 00:50:35.130 --> 00:50:43.250 align:middle line:84% So how could we approximate the integral of f of x? 00:50:43.250 --> 00:50:48.660 align:middle line:90% 00:50:48.660 --> 00:50:50.190 align:middle line:90% Yeah? 00:50:50.190 --> 00:50:53.310 align:middle line:84% AUDIENCE: [INAUDIBLE] the right hand [INAUDIBLE]. 00:50:53.310 --> 00:50:54.270 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:50:54.270 --> 00:51:02.070 align:middle line:84% So if we've got 1, 2, 3, 4, dot, dot, dot, 00:51:02.070 --> 00:51:06.425 align:middle line:84% we can draw a bunch of rectangles, like this. 00:51:06.425 --> 00:51:11.730 align:middle line:90% 00:51:11.730 --> 00:51:15.155 align:middle line:90% And this step function here-- 00:51:15.155 --> 00:51:18.432 align:middle line:90% 00:51:18.432 --> 00:51:20.390 align:middle line:84% actually, maybe I should get a different color. 00:51:20.390 --> 00:51:42.772 align:middle line:90% 00:51:42.772 --> 00:51:44.480 align:middle line:84% For completeness, maybe I should do that. 00:51:44.480 --> 00:51:51.600 align:middle line:90% 00:51:51.600 --> 00:51:55.200 align:middle line:84% So this step function here that I've 00:51:55.200 --> 00:51:57.900 align:middle line:90% drawn is f of the floor of x. 00:51:57.900 --> 00:52:01.350 align:middle line:90% f is weakly increasing. 00:52:01.350 --> 00:52:06.670 align:middle line:84% So that means floor of x is an under approximation for x. 00:52:06.670 --> 00:52:11.540 align:middle line:84% So f of floor of x is an under approximation for f of x. 00:52:11.540 --> 00:52:17.340 align:middle line:90% 00:52:17.340 --> 00:52:22.840 align:middle line:84% Now what happens if we integrate this step function? 00:52:22.840 --> 00:52:35.870 align:middle line:90% 00:52:35.870 --> 00:52:50.780 align:middle line:84% What is the integral from 1 to n of f of floor of x? 00:52:50.780 --> 00:52:51.440 align:middle line:90% Yeah? 00:52:51.440 --> 00:52:53.092 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:52:53.092 --> 00:52:54.050 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:52:54.050 --> 00:52:56.522 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:52:56.522 --> 00:52:57.522 align:middle line:90% BRYNMOR CHAPMAN: Almost. 00:52:57.522 --> 00:53:05.630 align:middle line:90% 00:53:05.630 --> 00:53:09.110 align:middle line:90% So in this case, it actually-- 00:53:09.110 --> 00:53:10.450 align:middle line:90% does it turn out to be s? 00:53:10.450 --> 00:53:13.195 align:middle line:90% 00:53:13.195 --> 00:53:15.560 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:53:15.560 --> 00:53:17.310 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:53:17.310 --> 00:53:19.910 align:middle line:90% We're missing a term here. 00:53:19.910 --> 00:53:28.160 align:middle line:84% So we can break this into n minus 1 integrals. 00:53:28.160 --> 00:53:32.390 align:middle line:84% We've got an interval of width n minus 1. 00:53:32.390 --> 00:53:37.970 align:middle line:84% This is constant on each of those width-1 subintervals. 00:53:37.970 --> 00:53:42.230 align:middle line:84% So we can break it apart into a whole bunch of rectangles. 00:53:42.230 --> 00:53:50.540 align:middle line:84% So this is going to be the sum from i equals 1 to n minus 1 00:53:50.540 --> 00:53:55.160 align:middle line:90% of f of i. 00:53:55.160 --> 00:53:57.890 align:middle line:84% Does that make sense to everybody? 00:53:57.890 --> 00:54:02.510 align:middle line:84% So we've got intervals of width 1. 00:54:02.510 --> 00:54:06.420 align:middle line:84% Our function is constant on each of those intervals. 00:54:06.420 --> 00:54:12.410 align:middle line:84% And it's equal to the left-hand side of that interval. 00:54:12.410 --> 00:54:16.490 align:middle line:84% Does that make sense to everybody. 00:54:16.490 --> 00:54:23.170 align:middle line:84% And this is just going to be s minus f of n. 00:54:23.170 --> 00:54:32.960 align:middle line:90% 00:54:32.960 --> 00:54:34.210 align:middle line:90% Sorry? 00:54:34.210 --> 00:54:36.680 align:middle line:84% AUDIENCE: Why is n minus 1 the [INAUDIBLE]? 00:54:36.680 --> 00:54:39.320 align:middle line:84% BRYNMOR CHAPMAN: So the question is, why are we using n minus 1 00:54:39.320 --> 00:54:40.820 align:middle line:90% as the bound here? 00:54:40.820 --> 00:54:47.630 align:middle line:84% So that's because the width of this interval is n minus 1. 00:54:47.630 --> 00:54:52.820 align:middle line:84% We're splitting into n minus 1 intervals of width 1. 00:54:52.820 --> 00:55:01.730 align:middle line:84% So for instance, if we went up to, say, n equals 5. 00:55:01.730 --> 00:55:07.190 align:middle line:84% So if we were trying to sum f of 1, f of 2, f of 3, f of 4, 00:55:07.190 --> 00:55:13.480 align:middle line:84% f of 5, we would then want to compare to this integral 00:55:13.480 --> 00:55:22.450 align:middle line:84% here, so the integral from 1 to 5. 00:55:22.450 --> 00:55:24.546 align:middle line:90% And this gives us 4 rectangles. 00:55:24.546 --> 00:55:27.730 align:middle line:90% 00:55:27.730 --> 00:55:34.490 align:middle line:84% We've got the integral from 1 to 2 from 2 to 3, 3 to 4, 4 to 5. 00:55:34.490 --> 00:55:43.330 align:middle line:84% And the integral, say, from 2 to 3 has width 1 and height f of 2. 00:55:43.330 --> 00:55:51.940 align:middle line:84% So maybe we could phrase this as 1 times f of i. 00:55:51.940 --> 00:55:52.860 align:middle line:90% Does that make sense? 00:55:52.860 --> 00:56:06.370 align:middle line:90% 00:56:06.370 --> 00:56:08.530 align:middle line:84% So we've got an underapproximation 00:56:08.530 --> 00:56:10.330 align:middle line:90% for our integral now. 00:56:10.330 --> 00:56:15.160 align:middle line:84% So I haven't actually written the integral. 00:56:15.160 --> 00:56:24.030 align:middle line:84% So let's let i equal the integral from 1 to n f of x dx. 00:56:24.030 --> 00:56:27.318 align:middle line:90% 00:56:27.318 --> 00:56:33.650 align:middle line:84% So we said that this function here underapproximates f. 00:56:33.650 --> 00:56:38.860 align:middle line:84% So this integral under approximates i. 00:56:38.860 --> 00:56:43.090 align:middle line:84% So that tells us that i is greater than 00:56:43.090 --> 00:56:47.230 align:middle line:90% or equal to s minus f of n. 00:56:47.230 --> 00:56:51.580 align:middle line:90% 00:56:51.580 --> 00:56:55.270 align:middle line:84% Now equivalently, we could phrase this as a bound on s 00:56:55.270 --> 00:56:58.330 align:middle line:84% in terms of i because s was what we originally 00:56:58.330 --> 00:57:01.420 align:middle line:90% wanted to figure out. 00:57:01.420 --> 00:57:04.870 align:middle line:84% So instead, we could write this as s is less than 00:57:04.870 --> 00:57:10.555 align:middle line:90% or equal to i plus f of n. 00:57:10.555 --> 00:57:11.430 align:middle line:90% Does that make sense? 00:57:11.430 --> 00:57:23.980 align:middle line:90% 00:57:23.980 --> 00:57:29.176 align:middle line:84% What if, instead, we overapproximated? 00:57:29.176 --> 00:57:38.920 align:middle line:90% 00:57:38.920 --> 00:57:45.585 align:middle line:84% So now we've got f of the ceiling of x. 00:57:45.585 --> 00:57:56.350 align:middle line:90% 00:57:56.350 --> 00:57:58.060 align:middle line:84% So the new step function that I've just 00:57:58.060 --> 00:58:03.890 align:middle line:84% drawn in green is f of ceiling of x, 00:58:03.890 --> 00:58:08.970 align:middle line:84% which is an overapproximation for f of x. 00:58:08.970 --> 00:58:15.000 align:middle line:84% So if we integrate it the integral from 1 to n 00:58:15.000 --> 00:58:23.960 align:middle line:84% of f of ceiling of x dx should be greater than or equal to i. 00:58:23.960 --> 00:58:28.050 align:middle line:90% 00:58:28.050 --> 00:58:35.340 align:middle line:84% Now, doing exactly the same thing as we had before, 00:58:35.340 --> 00:58:36.575 align:middle line:90% what is this integral? 00:58:36.575 --> 00:58:46.170 align:middle line:90% 00:58:46.170 --> 00:58:47.580 align:middle line:90% Yeah? 00:58:47.580 --> 00:58:49.818 align:middle line:84% AUDIENCE: Sum from 2 to n of f of i? 00:58:49.818 --> 00:58:50.735 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:58:50.735 --> 00:58:53.265 align:middle line:90% 00:58:53.265 --> 00:58:59.370 align:middle line:84% So sum from i equals 2 to n of f of i-- 00:58:59.370 --> 00:59:07.985 align:middle line:84% say times 1-- and this is just s minus f of 1. 00:59:07.985 --> 00:59:14.070 align:middle line:90% 00:59:14.070 --> 00:59:19.680 align:middle line:90% So s minus f of 1 is at least i. 00:59:19.680 --> 00:59:27.840 align:middle line:84% Once again, we can rephrase this as a bound on s in terms of i. 00:59:27.840 --> 00:59:39.020 align:middle line:84% So s is greater than or equal to i plus f of 1. 00:59:39.020 --> 00:59:53.490 align:middle line:90% 00:59:53.490 --> 00:59:56.050 align:middle line:84% So does that make sense to everybody? 00:59:56.050 --> 01:00:02.520 align:middle line:90% 01:00:02.520 --> 01:00:12.150 align:middle line:84% So putting these together now, we have i plus f of 1 less than 01:00:12.150 --> 01:00:18.180 align:middle line:84% or equal to s less than or equal to i plus f of n. 01:00:18.180 --> 01:00:25.860 align:middle line:90% 01:00:25.860 --> 01:00:29.460 align:middle line:84% So this is what we call the integral bound for increasing 01:00:29.460 --> 01:00:30.460 align:middle line:90% functions. 01:00:30.460 --> 01:01:03.512 align:middle line:90% 01:01:03.512 --> 01:01:05.095 align:middle line:84% Are people reasonably happy with this? 01:01:05.095 --> 01:01:07.800 align:middle line:90% 01:01:07.800 --> 01:01:12.410 align:middle line:84% So if we go back to our original question where f of x equals 01:01:12.410 --> 01:01:15.535 align:middle line:84% root x, what are the bounds we get? 01:01:15.535 --> 01:01:18.050 align:middle line:90% 01:01:18.050 --> 01:01:19.420 align:middle line:90% What is i first? 01:01:19.420 --> 01:01:25.130 align:middle line:90% 01:01:25.130 --> 01:01:27.330 align:middle line:84% It wasn't quite what I wrote down before. 01:01:27.330 --> 01:01:28.290 align:middle line:90% Yeah? 01:01:28.290 --> 01:01:30.610 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:01:30.610 --> 01:01:31.370 align:middle line:90% 01:01:31.370 --> 01:01:32.910 align:middle line:84% BRYNMOR CHAPMAN: It is from 1 to n. 01:01:32.910 --> 01:01:37.910 align:middle line:84% Remember what I wrote originally was a 2/3 n to the 3/2, 01:01:37.910 --> 01:01:40.630 align:middle line:84% which is, I believe, the integral from 0 to n. 01:01:40.630 --> 01:01:43.190 align:middle line:90% 01:01:43.190 --> 01:01:50.930 align:middle line:84% So what we actually have here is i equals the integral from 1 01:01:50.930 --> 01:02:05.030 align:middle line:84% to n of x to the 1/2 dx, which is equal to 2/3 x to the 3/2 01:02:05.030 --> 01:02:15.860 align:middle line:84% evaluated at 1 and n, which is going to be 2/3 times n 01:02:15.860 --> 01:02:23.980 align:middle line:84% to the 3/2 minus 1, so almost what we had before. 01:02:23.980 --> 01:02:28.640 align:middle line:90% 01:02:28.640 --> 01:02:35.570 align:middle line:84% So our lower bound, then, is this value plus f of 1. 01:02:35.570 --> 01:02:37.440 align:middle line:90% f of 1 is just root 1. 01:02:37.440 --> 01:02:40.100 align:middle line:90% It's 1. 01:02:40.100 --> 01:02:44.360 align:middle line:84% And our upper bound is i plus root n. 01:02:44.360 --> 01:02:47.240 align:middle line:90% 01:02:47.240 --> 01:02:51.920 align:middle line:84% So it turns out that 2/3 n to the 3/2 is a pretty good 01:02:51.920 --> 01:02:54.210 align:middle line:90% approximation. 01:02:54.210 --> 01:03:00.370 align:middle line:84% It's going to be a little bit larger than that but not much. 01:03:00.370 --> 01:03:05.360 align:middle line:84% We're going to be within an additive root n. 01:03:05.360 --> 01:03:12.040 align:middle line:84% And given that our approximation is the cube of that, 01:03:12.040 --> 01:03:15.570 align:middle line:90% that's actually pretty good. 01:03:15.570 --> 01:03:19.793 align:middle line:84% Does that make sense to everybody? 01:03:19.793 --> 01:03:21.335 align:middle line:84% Everybody reasonably happy with that? 01:03:21.335 --> 01:03:49.560 align:middle line:90% 01:03:49.560 --> 01:03:56.650 align:middle line:84% What if, instead, we had s equals sum from i equals 1 to n 01:03:56.650 --> 01:03:58.650 align:middle line:90% of, say, 1 over i? 01:03:58.650 --> 01:04:04.948 align:middle line:90% 01:04:04.948 --> 01:04:06.615 align:middle line:84% Can we apply our integral bound to this? 01:04:06.615 --> 01:04:12.100 align:middle line:90% 01:04:12.100 --> 01:04:15.280 align:middle line:90% Who thinks yes? 01:04:15.280 --> 01:04:17.860 align:middle line:90% Who thinks no? 01:04:17.860 --> 01:04:21.380 align:middle line:84% Can anybody explain why in either direction? 01:04:21.380 --> 01:04:21.880 align:middle line:90% Yeah? 01:04:21.880 --> 01:04:25.120 align:middle line:90% 01:04:25.120 --> 01:04:30.680 align:middle line:84% AUDIENCE: [INAUDIBLE] but it's going to be [INAUDIBLE], 01:04:30.680 --> 01:04:33.370 align:middle line:84% but after 1, it's going to be a convergent [INAUDIBLE] 01:04:33.370 --> 01:04:36.550 align:middle line:90% between 1 and 0. 01:04:36.550 --> 01:04:41.020 align:middle line:84% BRYNMOR CHAPMAN: So the answer was after 1, 01:04:41.020 --> 01:04:42.293 align:middle line:90% it's going to converge. 01:04:42.293 --> 01:04:44.920 align:middle line:90% 01:04:44.920 --> 01:04:46.230 align:middle line:90% I would dispute that. 01:04:46.230 --> 01:04:49.630 align:middle line:90% 01:04:49.630 --> 01:04:51.370 align:middle line:90% Yeah? 01:04:51.370 --> 01:04:53.290 align:middle line:84% AUDIENCE: If I recall correctly, the integral 01:04:53.290 --> 01:04:55.990 align:middle line:90% is natural log [INAUDIBLE]. 01:04:55.990 --> 01:05:00.100 align:middle line:90% 01:05:00.100 --> 01:05:02.500 align:middle line:84% BRYNMOR CHAPMAN: OK, fair enough. 01:05:02.500 --> 01:05:04.725 align:middle line:84% So even if we could apply the integral bound, 01:05:04.725 --> 01:05:05.850 align:middle line:90% it may not get us very far. 01:05:05.850 --> 01:05:09.730 align:middle line:90% 01:05:09.730 --> 01:05:11.510 align:middle line:84% I think there's a bigger problem here. 01:05:11.510 --> 01:05:12.040 align:middle line:90% Yeah? 01:05:12.040 --> 01:05:13.470 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:13.470 --> 01:05:14.720 align:middle line:90% BRYNMOR CHAPMAN: That's right. 01:05:14.720 --> 01:05:19.000 align:middle line:84% So remember that our assumption was that f 01:05:19.000 --> 01:05:21.310 align:middle line:90% should be weakly increasing. 01:05:21.310 --> 01:05:25.220 align:middle line:84% 1 over i is not, or 1 over x is not. 01:05:25.220 --> 01:05:25.780 align:middle line:90% Yeah? 01:05:25.780 --> 01:05:31.840 align:middle line:84% AUDIENCE: Well, at least it's weakly decreasing [INAUDIBLE]. 01:05:31.840 --> 01:05:34.960 align:middle line:84% BRYNMOR CHAPMAN: So the observation here 01:05:34.960 --> 01:05:39.640 align:middle line:84% is that if we take f of x equals 1 over x, 01:05:39.640 --> 01:05:41.216 align:middle line:90% this is weakly decreasing. 01:05:41.216 --> 01:05:48.160 align:middle line:90% 01:05:48.160 --> 01:05:52.390 align:middle line:90% So we can do something similar. 01:05:52.390 --> 01:05:54.140 align:middle line:90% What exactly are we going to do? 01:05:54.140 --> 01:06:10.380 align:middle line:84% Well, let's take g of x to be equal to f of n plus 1 minus x. 01:06:10.380 --> 01:06:15.040 align:middle line:90% 01:06:15.040 --> 01:06:19.455 align:middle line:90% Now what do we know about g? 01:06:19.455 --> 01:06:34.360 align:middle line:90% 01:06:34.360 --> 01:06:37.870 align:middle line:84% So basically, what we've done here, 01:06:37.870 --> 01:06:41.180 align:middle line:84% we are essentially negating the argument. 01:06:41.180 --> 01:06:45.200 align:middle line:84% So we're kind of flipping it horizontally. 01:06:45.200 --> 01:06:48.430 align:middle line:84% We started with a weakly decreasing function. 01:06:48.430 --> 01:06:50.627 align:middle line:84% So when we flip it like this, we're 01:06:50.627 --> 01:06:52.710 align:middle line:84% going to end up with a weakly increasing function. 01:06:52.710 --> 01:07:05.710 align:middle line:90% 01:07:05.710 --> 01:07:06.700 align:middle line:90% Yeah? 01:07:06.700 --> 01:07:07.910 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:07:07.910 --> 01:07:12.700 align:middle line:90% 01:07:12.700 --> 01:07:14.950 align:middle line:84% BRYNMOR CHAPMAN: So are you worried about the n plus 1 01:07:14.950 --> 01:07:16.425 align:middle line:90% as opposed to the n? 01:07:16.425 --> 01:07:17.800 align:middle line:84% AUDIENCE: Yeah, I'm just confused 01:07:17.800 --> 01:07:21.165 align:middle line:84% on why you can't do that or how that flips it. 01:07:21.165 --> 01:07:22.540 align:middle line:84% BRYNMOR CHAPMAN: So basically, we 01:07:22.540 --> 01:07:27.070 align:middle line:84% are mapping 1 to n and vice versa. 01:07:27.070 --> 01:07:29.230 align:middle line:84% And then everything in between should just 01:07:29.230 --> 01:07:31.660 align:middle line:90% interpolate linearly. 01:07:31.660 --> 01:07:36.020 align:middle line:84% So if we put 1 into here, we end up with f of n. 01:07:36.020 --> 01:07:38.710 align:middle line:84% If we put n into here, we end up with f of 1. 01:07:38.710 --> 01:07:40.210 align:middle line:90% Does that make sense? 01:07:40.210 --> 01:07:42.070 align:middle line:84% So the question is, why do we have 01:07:42.070 --> 01:07:45.670 align:middle line:84% the n plus 1 there instead of just an n or a minus x 01:07:45.670 --> 01:07:49.890 align:middle line:84% or something like that, which kind of gets 01:07:49.890 --> 01:07:53.285 align:middle line:84% into the next question-- what is s in terms of g? 01:07:53.285 --> 01:08:12.090 align:middle line:90% 01:08:12.090 --> 01:08:16.529 align:middle line:84% So maybe it would be better to write it using ellipses 01:08:16.529 --> 01:08:18.569 align:middle line:90% instead of sigma notation. 01:08:18.569 --> 01:08:24.479 align:middle line:84% So we defined it as f of 1 plus f of 2, 01:08:24.479 --> 01:08:28.234 align:middle line:90% plus dot, dot, dot plus f of n. 01:08:28.234 --> 01:08:31.170 align:middle line:90% 01:08:31.170 --> 01:08:35.074 align:middle line:84% Now how can we rewrite this in terms of g? 01:08:35.074 --> 01:08:40.740 align:middle line:90% 01:08:40.740 --> 01:08:42.870 align:middle line:84% AUDIENCE: p over n plus p over n minus 1. 01:08:42.870 --> 01:08:43.787 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:08:43.787 --> 01:08:54.029 align:middle line:90% 01:08:54.029 --> 01:08:58.630 align:middle line:84% So this is actually equal to the sum of g of i-- 01:08:58.630 --> 01:08:59.464 align:middle line:90% same bounds. 01:08:59.464 --> 01:09:03.090 align:middle line:90% 01:09:03.090 --> 01:09:10.625 align:middle line:90% So now what can we do? 01:09:10.625 --> 01:09:17.694 align:middle line:90% 01:09:17.694 --> 01:09:19.140 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:09:19.140 --> 01:09:21.270 align:middle line:84% BRYNMOR CHAPMAN: Yeah, g satisfies 01:09:21.270 --> 01:09:26.010 align:middle line:84% all of the requirements that we needed for our integral bound. 01:09:26.010 --> 01:09:30.300 align:middle line:84% So we can apply the integral bound now to g instead of 01:09:30.300 --> 01:09:31.410 align:middle line:90% to our original f. 01:09:31.410 --> 01:09:33.960 align:middle line:90% 01:09:33.960 --> 01:09:46.310 align:middle line:84% So now we know that i plus g of 1 less than or equal to s 01:09:46.310 --> 01:09:51.740 align:middle line:84% less than or equal to i plus g of n. 01:09:51.740 --> 01:09:57.350 align:middle line:84% So the integral is exactly the same. 01:09:57.350 --> 01:09:59.310 align:middle line:90% We've just flipped it. 01:09:59.310 --> 01:10:03.200 align:middle line:84% But the area under it is going to be the same. 01:10:03.200 --> 01:10:05.430 align:middle line:90% And what is this g of 1? 01:10:05.430 --> 01:10:08.605 align:middle line:84% Well, this g of 1 is just f of n. 01:10:08.605 --> 01:10:12.980 align:middle line:90% 01:10:12.980 --> 01:10:14.965 align:middle line:90% g of n is just f of 1. 01:10:14.965 --> 01:10:24.886 align:middle line:90% 01:10:24.886 --> 01:10:29.800 align:middle line:84% So this is the integral bound for weakly decreasing f. 01:10:29.800 --> 01:10:40.040 align:middle line:90% 01:10:40.040 --> 01:10:41.610 align:middle line:84% Does that make sense to everybody? 01:10:41.610 --> 01:10:46.100 align:middle line:90% 01:10:46.100 --> 01:10:47.256 align:middle line:90% Yeah? 01:10:47.256 --> 01:10:49.400 align:middle line:84% AUDIENCE: Could this also [INAUDIBLE] a negative 01:10:49.400 --> 01:10:50.260 align:middle line:90% [INAUDIBLE]? 01:10:50.260 --> 01:10:53.480 align:middle line:90% 01:10:53.480 --> 01:10:55.390 align:middle line:84% BRYNMOR CHAPMAN: Strictly speaking-- 01:10:55.390 --> 01:10:58.610 align:middle line:90% 01:10:58.610 --> 01:11:01.769 align:middle line:84% AUDIENCE: Do we need it to be positive or negative? 01:11:01.769 --> 01:11:07.520 align:middle line:90% 01:11:07.520 --> 01:11:10.070 align:middle line:84% BRYNMOR CHAPMAN: So you end up with a slightly 01:11:10.070 --> 01:11:19.000 align:middle line:84% different problem if you just negate it, which is-- 01:11:19.000 --> 01:11:23.120 align:middle line:90% 01:11:23.120 --> 01:11:26.690 align:middle line:84% well, I'll hint at it in a moment. 01:11:26.690 --> 01:11:32.200 align:middle line:84% What happens if you take limits as n goes to infinity? 01:11:32.200 --> 01:11:34.160 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:11:34.160 --> 01:11:35.360 align:middle line:90% BRYNMOR CHAPMAN: Pardon? 01:11:35.360 --> 01:11:38.540 align:middle line:84% AUDIENCE: Oh, it's n goes to infinity. 01:11:38.540 --> 01:11:41.630 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so you have to-- 01:11:41.630 --> 01:11:45.300 align:middle line:84% you can do that, but you'd have to adjust for it, 01:11:45.300 --> 01:11:48.500 align:middle line:84% and it doesn't work quite as nicely with the limits. 01:11:48.500 --> 01:11:51.170 align:middle line:84% So the answer is, strictly speaking, yes, you 01:11:51.170 --> 01:11:55.310 align:middle line:90% could negate it if you want to. 01:11:55.310 --> 01:11:59.165 align:middle line:90% But flipping it, and-- 01:11:59.165 --> 01:12:04.730 align:middle line:90% 01:12:04.730 --> 01:12:06.630 align:middle line:84% I personally find it easier to think about. 01:12:06.630 --> 01:12:13.320 align:middle line:90% 01:12:13.320 --> 01:12:17.930 align:middle line:84% So that was the idea, to negate f instead of x. 01:12:17.930 --> 01:12:24.530 align:middle line:84% But segue into the last thing-- we've 01:12:24.530 --> 01:12:27.090 align:middle line:84% got a couple of minutes, or eight minutes. 01:12:27.090 --> 01:12:27.655 align:middle line:90% Oh, nice. 01:12:27.655 --> 01:12:30.350 align:middle line:90% 01:12:30.350 --> 01:12:38.570 align:middle line:84% So what about s equals sum from I equals 1 to infinity of, 01:12:38.570 --> 01:12:40.510 align:middle line:90% let's say-- 01:12:40.510 --> 01:12:42.490 align:middle line:84% because as somebody pointed out, we do not 01:12:42.490 --> 01:12:44.360 align:middle line:90% want to be working with logs. 01:12:44.360 --> 01:12:47.250 align:middle line:90% So let's say 1 over i squared. 01:12:47.250 --> 01:12:51.400 align:middle line:90% 01:12:51.400 --> 01:12:59.490 align:middle line:84% So how could we evaluate this infinite sum? 01:12:59.490 --> 01:13:14.380 align:middle line:90% 01:13:14.380 --> 01:13:17.520 align:middle line:84% Well, we can basically do what we were just doing right. 01:13:17.520 --> 01:13:21.220 align:middle line:90% 01:13:21.220 --> 01:13:23.530 align:middle line:84% Well, does anybody remember how we actually 01:13:23.530 --> 01:13:25.605 align:middle line:90% define what an infinite sum is? 01:13:25.605 --> 01:13:29.800 align:middle line:90% 01:13:29.800 --> 01:13:32.235 align:middle line:84% It's just going to be a limit of partial sums. 01:13:32.235 --> 01:13:43.300 align:middle line:90% 01:13:43.300 --> 01:13:52.540 align:middle line:84% Now all of these, we know how to evaluate. 01:13:52.540 --> 01:14:02.740 align:middle line:84% So each of these, if we take i sub 01:14:02.740 --> 01:14:09.490 align:middle line:84% n to be equal to the integral from 1 to n of 1 over i 01:14:09.490 --> 01:14:10.780 align:middle line:90% squared-- 01:14:10.780 --> 01:14:14.180 align:middle line:90% sorry, 1 over x squared dx-- 01:14:14.180 --> 01:14:16.780 align:middle line:90% 01:14:16.780 --> 01:14:18.615 align:middle line:90% what is this integral? 01:14:18.615 --> 01:14:53.200 align:middle line:90% 01:14:53.200 --> 01:14:54.920 align:middle line:90% Anybody want to evaluate that? 01:14:54.920 --> 01:14:57.730 align:middle line:90% No, we don't like calculus? 01:14:57.730 --> 01:14:58.250 align:middle line:90% Me too. 01:14:58.250 --> 01:14:59.542 align:middle line:90% That's why I'm asking for help. 01:14:59.542 --> 01:15:00.301 align:middle line:90% Yeah? 01:15:00.301 --> 01:15:03.130 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:15:03.130 --> 01:15:06.225 align:middle line:84% BRYNMOR CHAPMAN: Yeah, 1 minus 1 over n. 01:15:06.225 --> 01:15:10.090 align:middle line:90% 01:15:10.090 --> 01:15:11.480 align:middle line:90% Why are you laughing at me? 01:15:11.480 --> 01:15:13.210 align:middle line:90% That's so rude. 01:15:13.210 --> 01:15:15.580 align:middle line:90% Oh, come on, guys. 01:15:15.580 --> 01:15:18.290 align:middle line:84% So we know what these partial sums are, 01:15:18.290 --> 01:15:21.670 align:middle line:84% or we have approximations for these partial sums. 01:15:21.670 --> 01:15:27.023 align:middle line:84% So the partial sum s sub n is going to be bounded on one side 01:15:27.023 --> 01:15:27.523 align:middle line:90% by-- 01:15:27.523 --> 01:15:31.383 align:middle line:90% 01:15:31.383 --> 01:15:32.925 align:middle line:84% oops, why don't I write it like this? 01:15:32.925 --> 01:15:36.090 align:middle line:90% 01:15:36.090 --> 01:15:42.000 align:middle line:84% So i sub n plus f of n less than or equal to s 01:15:42.000 --> 01:15:47.070 align:middle line:84% sub n less than or equal to i sub n plus f of 1. 01:15:47.070 --> 01:15:54.335 align:middle line:84% Now what happens when we take limits? 01:15:54.335 --> 01:15:57.390 align:middle line:90% 01:15:57.390 --> 01:16:02.050 align:middle line:84% So weak inequalities are preserved by limits. 01:16:02.050 --> 01:16:15.060 align:middle line:84% So the limit of i sub n plus f of n is going to be less than 01:16:15.060 --> 01:16:20.940 align:middle line:84% or equal to s, less than or equal to limit 01:16:20.940 --> 01:16:24.725 align:middle line:90% of i sub n plus f of 1. 01:16:24.725 --> 01:16:37.661 align:middle line:90% 01:16:37.661 --> 01:16:38.930 align:middle line:90% Now what does the limit do? 01:16:38.930 --> 01:16:46.680 align:middle line:90% 01:16:46.680 --> 01:16:48.280 align:middle line:90% Well, we can separate these. 01:16:48.280 --> 01:16:50.345 align:middle line:90% What's the limit of i sub n? 01:16:50.345 --> 01:16:54.510 align:middle line:90% 01:16:54.510 --> 01:16:57.090 align:middle line:90% Yeah? 01:16:57.090 --> 01:16:59.730 align:middle line:84% AUDIENCE: 1 minus infinity or [INAUDIBLE]. 01:16:59.730 --> 01:17:03.840 align:middle line:84% BRYNMOR CHAPMAN: 1 minus 1 divided by infinity, basically. 01:17:03.840 --> 01:17:09.570 align:middle line:84% So as n goes to infinity, this 1 over n term disappears. 01:17:09.570 --> 01:17:15.570 align:middle line:84% So this part is just going to be 1. 01:17:15.570 --> 01:17:17.375 align:middle line:90% What's the limit of f of n? 01:17:17.375 --> 01:17:21.630 align:middle line:90% 01:17:21.630 --> 01:17:23.130 align:middle line:90% Yeah? 01:17:23.130 --> 01:17:24.630 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:17:24.630 --> 01:17:26.626 align:middle line:90% BRYNMOR CHAPMAN: Also infinity. 01:17:26.626 --> 01:17:27.600 align:middle line:90% AUDIENCE: It's also 0. 01:17:27.600 --> 01:17:29.040 align:middle line:90% BRYNMOR CHAPMAN: Yeah, 0. 01:17:29.040 --> 01:17:33.360 align:middle line:90% 01:17:33.360 --> 01:17:35.190 align:middle line:84% Less than or equal to s less than 01:17:35.190 --> 01:17:39.600 align:middle line:84% or equal to-- we just established that this is 1. 01:17:39.600 --> 01:17:40.520 align:middle line:90% What's f of 1? 01:17:40.520 --> 01:17:43.360 align:middle line:90% 01:17:43.360 --> 01:17:43.860 align:middle line:90% 1. 01:17:43.860 --> 01:17:49.470 align:middle line:90% 01:17:49.470 --> 01:17:53.083 align:middle line:90% So we have an approximation. 01:17:53.083 --> 01:17:53.750 align:middle line:90% It's a good one. 01:17:53.750 --> 01:17:59.910 align:middle line:90% 01:17:59.910 --> 01:18:01.870 align:middle line:84% I'm seeing some people like, yeah, good enough, 01:18:01.870 --> 01:18:03.540 align:middle line:90% let's just go home. 01:18:03.540 --> 01:18:07.560 align:middle line:84% Some people like, what the hell, we're off by a factor of 2. 01:18:07.560 --> 01:18:12.672 align:middle line:84% Yeah, it's between 1 and 2-- not so great. 01:18:12.672 --> 01:18:14.255 align:middle line:84% Any ideas how we could make it better? 01:18:14.255 --> 01:18:21.090 align:middle line:90% 01:18:21.090 --> 01:18:22.880 align:middle line:84% How can we make our bounds more precise? 01:18:22.880 --> 01:18:38.060 align:middle line:90% 01:18:38.060 --> 01:18:40.470 align:middle line:84% Maybe if we go back to the picture, 01:18:40.470 --> 01:18:42.820 align:middle line:84% it'll be a little bit easier to see. 01:18:42.820 --> 01:18:54.980 align:middle line:90% 01:18:54.980 --> 01:18:57.590 align:middle line:84% This lack of precision is coming from the fact 01:18:57.590 --> 01:19:02.540 align:middle line:84% that we're looking at the difference between these two 01:19:02.540 --> 01:19:04.110 align:middle line:90% step functions. 01:19:04.110 --> 01:19:09.740 align:middle line:84% So the farther apart they are, the worse our approximation 01:19:09.740 --> 01:19:10.910 align:middle line:90% is going to be. 01:19:10.910 --> 01:19:18.020 align:middle line:84% So if we just forget about the first few terms, 01:19:18.020 --> 01:19:22.730 align:middle line:84% compute those exactly, and then just approximate the rest, 01:19:22.730 --> 01:19:26.090 align:middle line:84% we're going to end up with a more precise bound. 01:19:26.090 --> 01:19:43.190 align:middle line:84% So if, instead, we say, s equals 1 plus 1/4 plus 1/9 plus the sum 01:19:43.190 --> 01:19:53.755 align:middle line:84% from i equals 1 to infinity of 1 over i plus 3 squared-- 01:19:53.755 --> 01:19:58.910 align:middle line:90% 01:19:58.910 --> 01:20:00.085 align:middle line:90% sorry, it's a bit crowded. 01:20:00.085 --> 01:20:06.420 align:middle line:90% 01:20:06.420 --> 01:20:08.350 align:middle line:84% Now, what happens if we approximate this sum? 01:20:08.350 --> 01:20:19.520 align:middle line:90% 01:20:19.520 --> 01:20:20.770 align:middle line:90% How close are we going to get? 01:20:20.770 --> 01:20:56.300 align:middle line:90% 01:20:56.300 --> 01:21:04.820 align:middle line:84% Well, our error term, remember, was just 01:21:04.820 --> 01:21:08.570 align:middle line:90% the first term in our series. 01:21:08.570 --> 01:21:14.960 align:middle line:84% We are going to be between the integral and the integral plus 01:21:14.960 --> 01:21:17.840 align:middle line:90% the first term. 01:21:17.840 --> 01:21:23.860 align:middle line:84% Here, the first term is much smaller. 01:21:23.860 --> 01:21:26.530 align:middle line:90% So this time, we've got-- 01:21:26.530 --> 01:21:27.440 align:middle line:90% what is it? 01:21:27.440 --> 01:21:30.010 align:middle line:90% 1 over 16. 01:21:30.010 --> 01:21:35.090 align:middle line:84% So instead of being between i and i plus 1, 01:21:35.090 --> 01:21:42.310 align:middle line:84% we're now between i and i plus 1 over 16 for a different i-- 01:21:42.310 --> 01:21:43.540 align:middle line:90% same i? 01:21:43.540 --> 01:21:44.810 align:middle line:90% No, different i. 01:21:44.810 --> 01:21:49.180 align:middle line:90% 01:21:49.180 --> 01:21:51.610 align:middle line:84% Does that make sense to everybody? 01:21:51.610 --> 01:21:53.330 align:middle line:84% So if you want a better approximation, 01:21:53.330 --> 01:21:55.540 align:middle line:84% just strip some terms of the front 01:21:55.540 --> 01:22:01.630 align:middle line:84% and approximate the tail of the sum instead of the entire thing. 01:22:01.630 --> 01:22:05.200 align:middle line:84% So that's all we have time for today. 01:22:05.200 --> 01:22:05.980 align:middle line:90% Let's see. 01:22:05.980 --> 01:22:08.110 align:middle line:84% I'm not sure if we we'll see you next week. 01:22:08.110 --> 01:22:09.830 align:middle line:90% Somebody will see you next week. 01:22:09.830 --> 01:22:12.910 align:middle line:90% Zach we'll see you next week. 01:22:12.910 --> 01:22:14.930 align:middle line:84% Feel free to come up if you have any questions. 01:22:14.930 --> 01:22:16.590 align:middle line:90% Yeah? 01:22:16.590 --> 01:22:21.000 align:middle line:90%