WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.217 Commons license. 00:00:03.217 --> 00:00:05.770 Your support will help MIT OpenCourseWare 00:00:05.770 --> 00:00:10.050 continue to offer high quality educational resources for free. 00:00:10.050 --> 00:00:12.265 To make a donation or to view additional materials 00:00:12.265 --> 00:00:16.649 for hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.649 --> 00:00:18.530 at ocw.mit.edu. 00:00:21.510 --> 00:00:24.520 PROFESSOR: Last time we left off with a question 00:00:24.520 --> 00:00:26.185 having to do with playing with blocks. 00:00:26.185 --> 00:00:30.450 And this is supposed to give us a visceral feel for something 00:00:30.450 --> 00:00:32.820 anyway, having to do with series. 00:00:32.820 --> 00:00:35.650 And the question was whether I could stack these blocks, 00:00:35.650 --> 00:00:39.460 build up a stack so that-- I'm going to try here. 00:00:39.460 --> 00:00:42.920 I'm already off balance here, see. 00:00:42.920 --> 00:00:46.150 The question is can I build this so 00:00:46.150 --> 00:00:51.350 that the-- let's draw a picture of it, 00:00:51.350 --> 00:00:54.870 so that the first block is like this. 00:00:54.870 --> 00:00:56.567 The next block is like this. 00:00:56.567 --> 00:00:58.150 And maybe the next block is like this. 00:00:58.150 --> 00:01:00.240 And notice there is no visible means 00:01:00.240 --> 00:01:02.410 of support for this block. 00:01:02.410 --> 00:01:08.310 It's completely to the left of the first block. 00:01:08.310 --> 00:01:11.055 And the question is, will this fall down? 00:01:13.730 --> 00:01:21.610 Or at least, or more precisely, eventually we'll ask, 00:01:21.610 --> 00:01:24.650 you know, how far can we go? 00:01:24.650 --> 00:01:28.760 Now before you answer this question, 00:01:28.760 --> 00:01:30.900 the claim is that this is a kind of 00:01:30.900 --> 00:01:33.480 a natural, physical question, which 00:01:33.480 --> 00:01:37.330 involves some important answer. 00:01:37.330 --> 00:01:40.290 No matter whether the answer is you can do it or you can't. 00:01:40.290 --> 00:01:42.090 So this is a good kind of math question 00:01:42.090 --> 00:01:44.440 where no matter what the answer is, when you figure out 00:01:44.440 --> 00:01:46.850 the answer, you're going to get something interesting out 00:01:46.850 --> 00:01:47.350 of it. 00:01:47.350 --> 00:01:48.766 Because they're two possibilities. 00:01:48.766 --> 00:01:53.055 Either there is a limit to how far to the left we can go -- 00:01:53.055 --> 00:01:56.400 in which case that's a very interesting number -- 00:01:56.400 --> 00:01:58.810 or else there is no limit. 00:01:58.810 --> 00:02:00.370 You can go arbitrarily far. 00:02:00.370 --> 00:02:03.240 And that's also interesting and curious. 00:02:03.240 --> 00:02:05.400 And that's the difference between convergence 00:02:05.400 --> 00:02:08.190 and divergence, the thing that we 00:02:08.190 --> 00:02:11.380 were talking about up to now concerning series. 00:02:11.380 --> 00:02:13.890 So my first question is, do you think 00:02:13.890 --> 00:02:19.810 that I can get it so that this thing doesn't fall down 00:02:19.810 --> 00:02:24.020 with-- well you see I have about eight blocks here or so. 00:02:24.020 --> 00:02:25.570 So you can vote now. 00:02:25.570 --> 00:02:27.750 How many in favor that I can succeed 00:02:27.750 --> 00:02:30.450 in doing this sort of thing with maybe more than three blocks. 00:02:30.450 --> 00:02:32.040 How many in favor? 00:02:32.040 --> 00:02:33.860 All right somebody is voting twice. 00:02:33.860 --> 00:02:34.560 That's good. 00:02:34.560 --> 00:02:35.980 I like that. 00:02:35.980 --> 00:02:37.005 How about opposed? 00:02:39.930 --> 00:02:43.450 So that was really close to a tie. 00:02:43.450 --> 00:02:44.780 All right. 00:02:44.780 --> 00:02:49.029 But I think the there was slightly more opposed. 00:02:49.029 --> 00:02:49.570 I don't know. 00:02:49.570 --> 00:02:52.000 You guys who are in the back maybe could tell. 00:02:52.000 --> 00:02:53.750 Anyway it was pretty close. 00:02:53.750 --> 00:02:54.250 All right. 00:02:54.250 --> 00:02:57.117 So now I'm going-- because this is a real life thing, 00:02:57.117 --> 00:02:58.200 I'm going to try to do it. 00:02:58.200 --> 00:02:59.476 All right? 00:02:59.476 --> 00:03:00.550 All right. 00:03:00.550 --> 00:03:04.240 So now I'm going to tell you what the trick is. 00:03:04.240 --> 00:03:08.890 The trick is to do it backwards. 00:03:08.890 --> 00:03:11.230 When most people are playing with blocks, 00:03:11.230 --> 00:03:14.420 they decide to build it from the bottom up. 00:03:14.420 --> 00:03:15.670 Right? 00:03:15.670 --> 00:03:19.280 But we're going to build it from the top down, 00:03:19.280 --> 00:03:20.581 from the top down. 00:03:20.581 --> 00:03:22.080 And that's going to make it possible 00:03:22.080 --> 00:03:25.080 for us to do the optimal thing at each stage. 00:03:25.080 --> 00:03:28.400 So when I build it from the top down, the best I can do 00:03:28.400 --> 00:03:30.520 is well, it'll fall off. 00:03:30.520 --> 00:03:34.040 I need to have it you know, halfway across. 00:03:34.040 --> 00:03:35.470 That's the best I can do. 00:03:35.470 --> 00:03:38.000 So the top one I'm going to build like that. 00:03:38.000 --> 00:03:41.730 I'm going to take it as far to the left as I can. 00:03:41.730 --> 00:03:43.910 And then I'm going to put the next one down 00:03:43.910 --> 00:03:46.160 as far to the left as I can. 00:03:46.160 --> 00:03:50.450 And then the next one as far to the left as I can. 00:03:50.450 --> 00:03:51.740 That was a little too far. 00:03:51.740 --> 00:03:55.350 And then I'm going to do the next one as far to the left 00:03:55.350 --> 00:03:56.382 as I can. 00:03:56.382 --> 00:03:58.250 And then I'm going to do the next one -- 00:03:58.250 --> 00:04:04.450 well let's line it up first -- as far to the left as I can. 00:04:04.450 --> 00:04:05.150 OK? 00:04:05.150 --> 00:04:09.036 And then the next one as far to the left as I can. 00:04:09.036 --> 00:04:09.920 All right. 00:04:09.920 --> 00:04:15.700 Now those of you who are in this line can see, all right, 00:04:15.700 --> 00:04:16.650 I succeeded. 00:04:16.650 --> 00:04:18.740 All right, that's over the edge. 00:04:18.740 --> 00:04:20.050 All right? 00:04:20.050 --> 00:04:21.604 So it can be done. 00:04:21.604 --> 00:04:22.390 All right. 00:04:25.450 --> 00:04:26.590 All right. 00:04:26.590 --> 00:04:32.750 So now we know that we can get farther than you know, 00:04:32.750 --> 00:04:34.500 we can make it overflow. 00:04:34.500 --> 00:04:37.390 So the question now is, how far can I get? 00:04:37.390 --> 00:04:38.240 OK. 00:04:38.240 --> 00:04:40.630 Do you think I can get to here? 00:04:40.630 --> 00:04:43.260 Can I get to the end over here? 00:04:43.260 --> 00:04:50.022 So how many people think I can get this far over to here? 00:04:50.022 --> 00:04:51.730 How many people think I can get this far? 00:04:51.730 --> 00:04:53.525 Well you know, remember, I'm going 00:04:53.525 --> 00:04:55.519 to have to use more than just this one 00:04:55.519 --> 00:04:56.560 more block that I've got. 00:04:56.560 --> 00:04:58.280 I don't, right? 00:04:58.280 --> 00:05:00.900 Obviously I'm thinking, actually I do 00:05:00.900 --> 00:05:02.330 have some more blocks at home. 00:05:02.330 --> 00:05:03.294 But, OK. 00:05:03.294 --> 00:05:04.085 We're not going to. 00:05:04.085 --> 00:05:06.510 But anyway, do you think I can get over to here? 00:05:06.510 --> 00:05:09.710 How many people say yes? 00:05:09.710 --> 00:05:13.060 And how many people say no? 00:05:13.060 --> 00:05:15.180 More people said no then yes. 00:05:15.180 --> 00:05:15.680 All right. 00:05:15.680 --> 00:05:18.480 So maybe the stopping place is some mysterious number 00:05:18.480 --> 00:05:19.995 in between here. 00:05:19.995 --> 00:05:20.690 All right? 00:05:20.690 --> 00:05:21.720 Well OK. 00:05:21.720 --> 00:05:24.840 So now we're going to do the arithmetic. 00:05:24.840 --> 00:05:28.870 And we're going to figure out what happens with this problem. 00:05:28.870 --> 00:05:29.500 OK? 00:05:29.500 --> 00:05:32.175 So let's do it. 00:05:32.175 --> 00:05:35.760 All right, so now again the idea is, 00:05:35.760 --> 00:05:46.710 the idea is we're going to start with the top, the top block. 00:05:46.710 --> 00:05:48.450 We'll call that block number one. 00:05:53.070 --> 00:05:56.295 And then the farthest, if you like, to the right 00:05:56.295 --> 00:05:57.920 that you can put a block underneath it, 00:05:57.920 --> 00:05:59.070 is exactly halfway. 00:06:01.800 --> 00:06:04.610 All right, well, that's the best job I can do. 00:06:04.610 --> 00:06:08.270 Now in order to make my units work out easily, 00:06:08.270 --> 00:06:13.160 I'm going to decide to call the length of the block 2. 00:06:13.160 --> 00:06:14.090 All right? 00:06:14.090 --> 00:06:17.000 And that means if I start at location 0, 00:06:17.000 --> 00:06:22.550 then the first place where I am is supposed to be halfway. 00:06:22.550 --> 00:06:24.070 And that will be 1. 00:06:26.650 --> 00:06:31.660 OK so the first step in the process is 1 more to the right. 00:06:31.660 --> 00:06:33.610 Or if you like, if I were building up -- 00:06:33.610 --> 00:06:36.068 which is what you would actually have to do in real life -- 00:06:36.068 --> 00:06:39.280 it would be 1 to the left. 00:06:39.280 --> 00:06:41.610 OK now the next one. 00:06:41.610 --> 00:06:44.294 Now here is the way that you start 00:06:44.294 --> 00:06:45.460 figuring out the arithmetic. 00:06:45.460 --> 00:06:48.190 The next one is based on a physical principle. 00:06:48.190 --> 00:06:53.350 Which is that the farthest I can stick this next block 00:06:53.350 --> 00:06:57.950 underneath is what's called the center of mass of these two, 00:06:57.950 --> 00:07:00.136 which is exactly halfway here. 00:07:00.136 --> 00:07:01.760 That is, there's a quarter of this guy, 00:07:01.760 --> 00:07:04.400 and a quarter of that guy balancing each other. 00:07:04.400 --> 00:07:04.900 Right? 00:07:04.900 --> 00:07:06.160 So that's as far as I can go. 00:07:06.160 --> 00:07:08.625 If I go farther than that, it'll fall over. 00:07:08.625 --> 00:07:10.600 So that's the absolute farthest I can do. 00:07:10.600 --> 00:07:14.630 So the next block is going to be over here. 00:07:14.630 --> 00:07:18.210 And a quarter of 2 is 1/2. 00:07:18.210 --> 00:07:21.552 So this is 3/2 here. 00:07:21.552 --> 00:07:24.430 All right so we went to 1. 00:07:24.430 --> 00:07:27.960 We went to 3/2 here. 00:07:27.960 --> 00:07:31.300 And then I'm going to keep on going with this eventually. 00:07:31.300 --> 00:07:33.140 All right so we're going to figure out 00:07:33.140 --> 00:07:36.770 what happens with this stack. 00:07:36.770 --> 00:07:37.774 Question? 00:07:37.774 --> 00:07:40.065 AUDIENCE: How do you know that this 00:07:40.065 --> 00:07:42.670 is the best way to optimize? 00:07:42.670 --> 00:07:44.515 PROFESSOR: The question is how do I 00:07:44.515 --> 00:07:46.870 know that this is the best way to optimize? 00:07:46.870 --> 00:07:49.099 I can't answer that question. 00:07:49.099 --> 00:07:50.640 But I can tell you that it's the best 00:07:50.640 --> 00:07:52.735 way if I start with a top like this, 00:07:52.735 --> 00:07:53.860 and the next one like this. 00:07:53.860 --> 00:07:56.100 Right, because I'm doing the farthest 00:07:56.100 --> 00:07:57.470 possible at each stage. 00:07:57.470 --> 00:07:59.840 That actually has a name in computer science, that's 00:07:59.840 --> 00:08:01.610 called the greedy algorithm. 00:08:01.610 --> 00:08:04.520 I'm trying to do the best possible at each stage. 00:08:04.520 --> 00:08:07.040 The greedy algorithm starting from the bottom 00:08:07.040 --> 00:08:09.420 is an extremely bad strategy. 00:08:09.420 --> 00:08:12.790 Because when you do that, you stack it this way, 00:08:12.790 --> 00:08:14.120 and it almost falls over. 00:08:14.120 --> 00:08:16.050 And then the next time you can't do anything. 00:08:16.050 --> 00:08:19.030 So the greedy algorithm is terrible from the bottom. 00:08:19.030 --> 00:08:21.225 This is the greedy algorithm starting from the top, 00:08:21.225 --> 00:08:23.455 and it turns out to do much better 00:08:23.455 --> 00:08:25.580 then the greedy algorithm starting from the bottom. 00:08:25.580 --> 00:08:27.913 But of course I'm not addressing whether there might not 00:08:27.913 --> 00:08:31.500 be some other incredibly clever strategy where I wiggle around 00:08:31.500 --> 00:08:34.260 and make it go up. 00:08:34.260 --> 00:08:35.676 I'm not addressing that question. 00:08:35.676 --> 00:08:36.130 All right? 00:08:36.130 --> 00:08:37.838 It turns out this is the best you can do. 00:08:37.838 --> 00:08:40.790 But that's not clear. 00:08:40.790 --> 00:08:44.650 All right so now, here we have this thing. 00:08:44.650 --> 00:08:47.600 And now I have to figure out what the arithmetic pattern is, 00:08:47.600 --> 00:08:50.340 so that I can figure out what I was doing with those shapes. 00:08:53.850 --> 00:08:58.132 So let's figure out a thought experiment here. 00:08:58.132 --> 00:08:59.526 All right? 00:08:59.526 --> 00:09:00.900 Now the thought experiment I want 00:09:00.900 --> 00:09:03.450 to imagine for you is, you've got 00:09:03.450 --> 00:09:10.070 a stack of a bunch of blocks, and this is the first N blocks. 00:09:14.150 --> 00:09:15.310 All right? 00:09:15.310 --> 00:09:19.010 And now we're going to put one underneath it. 00:09:19.010 --> 00:09:21.650 And what we're going to figure out 00:09:21.650 --> 00:09:26.340 is the center of mass of those N blocks, 00:09:26.340 --> 00:09:30.960 which I'm going to call C sub N. OK. 00:09:30.960 --> 00:09:32.615 And that's the place where I'm going 00:09:32.615 --> 00:09:34.320 to put this very next block. 00:09:34.320 --> 00:09:36.480 I'll put it in a different color here. 00:09:36.480 --> 00:09:39.560 Here's the new-- the next block over. 00:09:39.560 --> 00:09:42.360 And the next block over is the (N+1)st block. 00:09:49.900 --> 00:09:54.380 And now I want you to think about what's going on here. 00:09:54.380 --> 00:09:56.880 If the center of mass of the first N blocks 00:09:56.880 --> 00:10:01.700 is this number, this new one, it's of length 2. 00:10:01.700 --> 00:10:04.470 And its center of mass is 1 further 00:10:04.470 --> 00:10:07.850 to the right than the center of mass that we had before. 00:10:07.850 --> 00:10:10.930 So in other words, I've added to this configuration of N blocks 00:10:10.930 --> 00:10:13.220 one more block, which is shifted. 00:10:13.220 --> 00:10:15.650 Whose center mass is not lined up 00:10:15.650 --> 00:10:17.630 with the center of mass of this, but actually 00:10:17.630 --> 00:10:20.200 over farther to the right. 00:10:20.200 --> 00:10:23.170 All right so the new center of mass 00:10:23.170 --> 00:10:27.060 of this new block-- And this is the extra piece of information 00:10:27.060 --> 00:10:29.670 that I want to observe, is that this thing has 00:10:29.670 --> 00:10:36.740 a center of mass at C_N + 1. 00:10:36.740 --> 00:10:39.870 It's 1 unit over because this total length is 2. 00:10:39.870 --> 00:10:45.561 So right in the middle there is 1 over, according to my units. 00:10:45.561 --> 00:10:48.240 All right now this is going to make it possible 00:10:48.240 --> 00:10:51.970 for me to figure out what the new center of mass is. 00:10:51.970 --> 00:11:04.860 So C_(N+1) is the center of mass of N+1 blocks. 00:11:04.860 --> 00:11:08.910 Now this is really only in the horizontal variable, right? 00:11:08.910 --> 00:11:11.260 I'm not keeping track of the center of mass-- Actually 00:11:11.260 --> 00:11:13.510 this thing is hard to build because the center of mass 00:11:13.510 --> 00:11:14.240 is also rising. 00:11:14.240 --> 00:11:15.830 It's getting higher and higher. 00:11:15.830 --> 00:11:19.510 But I'm only keeping track of its left-right characteristic. 00:11:19.510 --> 00:11:21.590 So this is the x-coordinate of it. 00:11:27.760 --> 00:11:31.590 All right so now here's the idea. 00:11:31.590 --> 00:11:34.640 I'm combining the white ones, the N blocks, 00:11:34.640 --> 00:11:37.630 with the pink one, which is the one on the bottom. 00:11:37.630 --> 00:11:41.020 And there are N of the white ones. 00:11:41.020 --> 00:11:42.520 And there's 1 of the pink one. 00:11:42.520 --> 00:11:44.811 And so in order to get the center of mass of the whole, 00:11:44.811 --> 00:11:47.700 I have to take the weighted average of the two. 00:11:47.700 --> 00:11:56.090 That's N*C_N plus 1 times the center of mass of the pink one, 00:11:56.090 --> 00:11:59.665 which is C_N + 1. 00:11:59.665 --> 00:12:02.040 And then I have to divide -- if it's the weighted average 00:12:02.040 --> 00:12:06.560 of the total of N + 1 blocks -- by N + 1. 00:12:06.560 --> 00:12:09.090 This is going to give me the new center of mass 00:12:09.090 --> 00:12:11.855 of my configuration at the (N+1)st stage. 00:12:14.800 --> 00:12:16.330 And now I can just do the arithmetic 00:12:16.330 --> 00:12:19.340 and figure out what this is. 00:12:19.340 --> 00:12:21.810 And the two C_Ns combine. 00:12:21.810 --> 00:12:27.560 I get (N+1)C_N + 1, divided by N+1. 00:12:30.250 --> 00:12:33.680 And if I combine these two things and do the cancellation, 00:12:33.680 --> 00:12:36.140 that gives me this recurrence formula, 00:12:36.140 --> 00:12:41.040 C_(N+1) is equal to C_N plus-- There's a little extra. 00:12:41.040 --> 00:12:42.010 These two cancel. 00:12:42.010 --> 00:12:43.380 That gives me the C_N. 00:12:43.380 --> 00:12:46.740 But then I also have 1/(N+1). 00:12:54.230 --> 00:12:57.270 Well that's how much gain I can get in the center of mass 00:12:57.270 --> 00:12:58.620 by adding one more block. 00:12:58.620 --> 00:13:01.520 That's how much I can shift things over, 00:13:01.520 --> 00:13:03.550 depending on how we're thinking of things, 00:13:03.550 --> 00:13:05.700 to the left or the right, depending on which 00:13:05.700 --> 00:13:07.137 direction we're building them. 00:13:10.960 --> 00:13:13.840 All right, so now I'm going to work out the formulas. 00:13:13.840 --> 00:13:17.680 First of all C_1, that was the center of the first block. 00:13:17.680 --> 00:13:20.490 I put its left end at 0; the center of the first block 00:13:20.490 --> 00:13:22.250 is at 1. 00:13:22.250 --> 00:13:25.910 That means that C_1 is 1. 00:13:25.910 --> 00:13:27.310 OK? 00:13:27.310 --> 00:13:31.090 C_2 according to this formula-- And actually I've 00:13:31.090 --> 00:13:35.530 worked it out, we'll check it in a-- C_2 is C_1 + 1/2. 00:13:35.530 --> 00:13:37.380 All right, so that's the case N = 1. 00:13:37.380 --> 00:13:40.130 So this is 1 + 1/2. 00:13:40.130 --> 00:13:43.420 That's what we already did. 00:13:43.420 --> 00:13:45.790 That's the 3/2 number. 00:13:45.790 --> 00:13:51.320 Now the next one is C_2 + 1/3. 00:13:51.320 --> 00:13:53.530 That's the formula again. 00:13:53.530 --> 00:13:59.590 And so that comes out to be 1 + 1/2 + 1/3. 00:13:59.590 --> 00:14:03.100 And now you can see what the pattern is. 00:14:03.100 --> 00:14:06.810 C_N-- If you just keep on going here, 00:14:06.810 --> 00:14:14.130 C_N is going to be 1 + 1/2 + 1/3 + 1/4... 00:14:14.130 --> 00:14:21.410 plus 1/N. 00:14:21.410 --> 00:14:25.600 So now I would like you to vote again. 00:14:25.600 --> 00:14:27.790 Do you think I can-- Now that we have the formula, 00:14:27.790 --> 00:14:30.830 do you think I can get over to here? 00:14:30.830 --> 00:14:33.180 How many people think I can get over to here? 00:14:36.160 --> 00:14:40.410 How many people think I can't get over to here? 00:14:40.410 --> 00:14:42.700 There's still a lot of people who do. 00:14:42.700 --> 00:14:46.420 So it's still almost 50/50. 00:14:46.420 --> 00:14:47.470 That's amazing. 00:14:47.470 --> 00:14:49.800 Well so we'll address that in a few minutes. 00:14:49.800 --> 00:14:52.690 So now let me tell you what's going on. 00:14:52.690 --> 00:14:56.050 This C_N of course, is the same as what we called last time 00:14:56.050 --> 00:14:57.530 S_N. 00:14:57.530 --> 00:15:01.230 And remember that we actually estimated the size of this guy. 00:15:01.230 --> 00:15:04.620 This is related to what's called the harmonic series. 00:15:04.620 --> 00:15:08.120 And what we showed was that log N 00:15:08.120 --> 00:15:14.690 is less than S_N, which is less than S_N + 1. 00:15:14.690 --> 00:15:15.335 All right? 00:15:17.910 --> 00:15:20.360 Now I'm going to call your attention 00:15:20.360 --> 00:15:24.260 to the red part, which is the divergence 00:15:24.260 --> 00:15:28.500 part of this estimate, which is this one for the time being, 00:15:28.500 --> 00:15:30.290 all right. 00:15:30.290 --> 00:15:32.780 Just saying that this thing is growing. 00:15:32.780 --> 00:15:41.270 And what this is saying is that as N goes to infinity, 00:15:41.270 --> 00:15:48.810 log N goes to infinity, So that means 00:15:48.810 --> 00:15:57.220 that S_N goes to infinity, because of this inequality 00:15:57.220 --> 00:15:57.980 here. 00:15:57.980 --> 00:16:01.780 It's bigger than log N. And so if N is big enough, 00:16:01.780 --> 00:16:04.002 we can get as far as we like. 00:16:04.002 --> 00:16:06.030 All right? 00:16:06.030 --> 00:16:08.312 So I can get to here. 00:16:08.312 --> 00:16:10.270 And at least half of you, at least the ones who 00:16:10.270 --> 00:16:12.224 voted, that was-- I don't know. 00:16:12.224 --> 00:16:13.890 We have a quorum here, but I'm not sure. 00:16:13.890 --> 00:16:16.470 We certainly didn't have a majority on either side. 00:16:16.470 --> 00:16:19.420 Anyway this thing does go to infinity. 00:16:19.420 --> 00:16:21.330 So in principle, if I had enough blocks, 00:16:21.330 --> 00:16:25.216 I could get it over to here. 00:16:25.216 --> 00:16:26.590 All right, and that's the meaning 00:16:26.590 --> 00:16:28.300 of divergence in this case. 00:16:32.430 --> 00:16:36.670 On the other hand, I want to discuss with you-- 00:16:36.670 --> 00:16:38.600 And the reason why I use this example, 00:16:38.600 --> 00:16:40.390 is I want to discuss with you also what's 00:16:40.390 --> 00:16:45.460 going on with this other inequality here, 00:16:45.460 --> 00:16:49.360 and what its significance is. 00:16:49.360 --> 00:16:52.970 Which is that it's going to take us a lot of numbers N, 00:16:52.970 --> 00:16:57.050 a lot of blocks, to get up to a certain level. 00:16:57.050 --> 00:17:00.180 In other words, I can't do it with just eight blocks or nine 00:17:00.180 --> 00:17:00.800 blocks. 00:17:00.800 --> 00:17:02.258 In order to get over here, I'd have 00:17:02.258 --> 00:17:06.000 to use quite a few of them. 00:17:06.000 --> 00:17:11.120 So let's just see how many it is. 00:17:11.120 --> 00:17:14.230 So I worked this out carefully. 00:17:14.230 --> 00:17:15.645 And let's see what I got. 00:17:18.150 --> 00:17:31.760 So to get across the lab tables, all right. 00:17:31.760 --> 00:17:35.780 This distance here, I already did this secretly. 00:17:35.780 --> 00:17:38.985 But I don't actually even have enough of these to show you. 00:17:38.985 --> 00:17:43.834 But, well 1, 2, 3, 4, 5, 6, and 1/2. 00:17:43.834 --> 00:17:44.750 I guess that's enough. 00:17:44.750 --> 00:17:46.240 So it's 6 and a half. 00:17:46.240 --> 00:17:50.000 So it's two lab tables is 13 of these blocks. 00:17:50.000 --> 00:17:51.190 All right. 00:17:51.190 --> 00:18:00.310 So there are 13 blocks, which is equal to 26 units. 00:18:00.310 --> 00:18:04.020 OK, that's how far to get across I need. 00:18:04.020 --> 00:18:06.090 And the first one is already 2. 00:18:06.090 --> 00:18:09.470 So it's really 26 minus 2, which is 24. 00:18:09.470 --> 00:18:11.450 Which that's what I need. 00:18:11.450 --> 00:18:13.110 OK. 00:18:13.110 --> 00:18:22.910 So I need log N to be equal to 24, roughly speaking, 00:18:22.910 --> 00:18:25.090 in order to get that far. 00:18:25.090 --> 00:18:28.195 So let's just see how big that is. 00:18:28.195 --> 00:18:30.300 All right. 00:18:30.300 --> 00:18:31.760 I think I worked this out. 00:18:42.010 --> 00:18:43.170 So let's see. 00:18:43.170 --> 00:18:48.020 That means that N is equal to e^24-- 00:18:48.020 --> 00:18:57.160 and if you realize that these blocks are 3 centimeters high-- 00:18:57.160 --> 00:19:00.450 OK let's see how many that we would need here. 00:19:00.450 --> 00:19:02.210 That's kind of a lot. 00:19:02.210 --> 00:19:09.440 Let's see, it's 3 centimeters times e^24, 00:19:09.440 --> 00:19:15.660 which is about 8*10^8 meters. 00:19:15.660 --> 00:19:17.600 OK. 00:19:17.600 --> 00:19:20.810 And that is twice the distance to the moon. 00:19:32.310 --> 00:19:35.190 So OK, so I could do it maybe. 00:19:35.190 --> 00:19:37.936 But I would need a lot of blocks. 00:19:37.936 --> 00:19:38.435 Right? 00:19:38.435 --> 00:19:42.190 So that's not very plausible here, all right. 00:19:42.190 --> 00:19:44.380 So those of you who voted against this 00:19:44.380 --> 00:19:47.080 were actually sort of half right. 00:19:47.080 --> 00:19:48.630 And in fact, if you wanted to get it 00:19:48.630 --> 00:19:52.300 to the wall over there, which is over 30 feet, 00:19:52.300 --> 00:19:55.330 the height would be about the diameter 00:19:55.330 --> 00:19:57.360 of the observable universe. 00:19:57.360 --> 00:20:01.580 That's kind of a long way. 00:20:01.580 --> 00:20:05.780 There's one other thing that I wanted to point out 00:20:05.780 --> 00:20:09.340 to you about this shape here. 00:20:09.340 --> 00:20:13.516 Which is that if you lean to the left, right, 00:20:13.516 --> 00:20:14.890 if you put your head like this -- 00:20:14.890 --> 00:20:17.510 of course you have to be on your side to look at it -- 00:20:17.510 --> 00:20:25.810 this curve is the shape of a logarithmic curve. 00:20:25.810 --> 00:20:29.240 So in other words, if you think of the vertical as the x-axis, 00:20:29.240 --> 00:20:32.010 and the horizontal that way is the vertical, 00:20:32.010 --> 00:20:34.610 is the up direction, then this thing 00:20:34.610 --> 00:20:38.510 is growing very, very, very, very slowly. 00:20:38.510 --> 00:20:43.030 If you send the x-axis all the way up to the moon, 00:20:43.030 --> 00:20:47.540 the graph still hasn't gotten across the lab tables here. 00:20:47.540 --> 00:20:48.905 It's only partway there. 00:20:48.905 --> 00:20:52.110 If you go twice the distance to the moon up that way, 00:20:52.110 --> 00:20:54.340 it's gotten finally to that end. 00:20:54.340 --> 00:20:56.990 All right so that's how slowly the logarithm grows. 00:20:56.990 --> 00:20:58.189 It grows very, very slowly. 00:20:58.189 --> 00:21:00.730 And if you look at it another way, if you stand on your head, 00:21:00.730 --> 00:21:05.420 you can see an exponential curve. 00:21:05.420 --> 00:21:08.470 So you get some sense as to the growth properties 00:21:08.470 --> 00:21:10.420 of these functions. 00:21:10.420 --> 00:21:14.620 And fortunately these are protecting us 00:21:14.620 --> 00:21:18.550 from all kinds of stuff that would 00:21:18.550 --> 00:21:20.200 happen if there weren't exponentially 00:21:20.200 --> 00:21:21.750 small tails in the world. 00:21:21.750 --> 00:21:23.670 Like you know, I could walk through this wall 00:21:23.670 --> 00:21:27.000 which I wouldn't like doing. 00:21:27.000 --> 00:21:32.410 OK, now so this is our last example. 00:21:32.410 --> 00:21:35.000 And the important number, unfortunately we 00:21:35.000 --> 00:21:36.790 didn't discover another important number. 00:21:36.790 --> 00:21:40.250 There wasn't an amazing number place where this stopped. 00:21:40.250 --> 00:21:44.160 All we discovered again is some property of infinity. 00:21:44.160 --> 00:21:46.150 So infinity is still a nice number. 00:21:46.150 --> 00:21:50.940 And the theme here is just that infinity isn't just one thing, 00:21:50.940 --> 00:21:54.009 it has a character which is a rate of growth. 00:21:54.009 --> 00:21:55.550 And you shouldn't just think of there 00:21:55.550 --> 00:21:57.072 being one order of infinity. 00:21:57.072 --> 00:21:58.530 There are lots of different orders. 00:21:58.530 --> 00:22:01.280 And some of them have different meaning from others. 00:22:01.280 --> 00:22:03.880 All right so that's the theme I wanted 00:22:03.880 --> 00:22:07.900 to do, and just have a visceral example of infinity. 00:22:07.900 --> 00:22:13.570 Now, we're going to move on now to some other kinds 00:22:13.570 --> 00:22:16.210 of techniques. 00:22:16.210 --> 00:22:20.790 And this is going to be our last subject. 00:22:20.790 --> 00:22:23.820 What we're going to talk about is 00:22:23.820 --> 00:22:27.320 what are known as power series. 00:22:27.320 --> 00:22:29.920 And we've already seen our first power series. 00:22:32.710 --> 00:22:35.240 And I'm going to remind you of that. 00:22:43.980 --> 00:22:45.490 Here we are with power series. 00:22:51.490 --> 00:22:53.980 Our first series was this one. 00:22:58.870 --> 00:23:05.200 And we mentioned last time that it was equal to 1/(1-x), 00:23:05.200 --> 00:23:06.430 for x less than 1. 00:23:09.770 --> 00:23:12.040 Well this one is known as the geometric series. 00:23:12.040 --> 00:23:15.050 You didn't use the letter x last time, I used the letter a. 00:23:15.050 --> 00:23:16.910 But this is known as the geometric series. 00:23:24.950 --> 00:23:31.550 Now I'm going to show you one reason why this is true, 00:23:31.550 --> 00:23:33.900 why the formula holds. 00:23:33.900 --> 00:23:36.050 And it's just the kind of manipulation 00:23:36.050 --> 00:23:41.270 that was done when these things were first introduced. 00:23:41.270 --> 00:23:46.250 And here's the idea of a proof. 00:23:46.250 --> 00:23:52.670 So suppose that this sum is equal to some number 00:23:52.670 --> 00:23:57.360 S, which is the sum of all of these numbers here. 00:24:00.540 --> 00:24:02.130 The first thing that I'm going to do 00:24:02.130 --> 00:24:05.500 is I'm going to multiply by x. 00:24:05.500 --> 00:24:08.326 OK, so if I multiply by x. 00:24:08.326 --> 00:24:09.560 Let's think about that. 00:24:09.560 --> 00:24:13.810 I multiply by x on both the left and the right-hand side. 00:24:13.810 --> 00:24:20.990 Then on the left side, I get x + x^2 + x^3 plus, and so forth. 00:24:20.990 --> 00:24:22.810 And on the right side, I get S x. 00:24:27.380 --> 00:24:29.990 And now I'm going to subtract the two 00:24:29.990 --> 00:24:32.910 equations, one from the other. 00:24:32.910 --> 00:24:36.230 And there's a very, very substantial cancellation. 00:24:36.230 --> 00:24:39.180 This whole tail here gets canceled off. 00:24:39.180 --> 00:24:41.030 And the only thing that's left is the 1. 00:24:41.030 --> 00:24:45.950 So when I subtract, I get 1 on the left-hand side. 00:24:45.950 --> 00:24:53.000 And on the right-hand side, I get S - S x. 00:24:53.000 --> 00:24:53.500 All right? 00:24:58.900 --> 00:25:03.850 And now that can be rewritten as S(1-x). 00:25:03.850 --> 00:25:06.180 And so I've got my formula here. 00:25:06.180 --> 00:25:13.553 This is 1/(1-x) = S. All right. 00:25:17.850 --> 00:25:26.340 Now this reasoning has one flaw. 00:25:26.340 --> 00:25:28.120 It's not complete. 00:25:28.120 --> 00:25:33.080 And this reasoning is basically correct. 00:25:33.080 --> 00:25:45.680 But it's incomplete because it requires that S exists. 00:25:50.270 --> 00:25:55.100 For example, it doesn't make any sense in the case x = 1. 00:25:55.100 --> 00:25:57.950 So for example in the case x = 1, 00:25:57.950 --> 00:26:01.710 we have 1 + 1 + 1 plus et cetera, 00:26:01.710 --> 00:26:04.830 equals whatever we call S. And then when we multiply through 00:26:04.830 --> 00:26:08.798 by 1, we get 1 + 1 + 1 plus... 00:26:08.798 --> 00:26:10.640 equals S*1. 00:26:10.640 --> 00:26:12.560 And now you see that the subtraction gives us 00:26:12.560 --> 00:26:16.330 infinity minus infinity is equal to infinity minus infinity. 00:26:16.330 --> 00:26:19.790 That's what's really going on in the argument in this context. 00:26:19.790 --> 00:26:21.340 So it's just nonsense. 00:26:21.340 --> 00:26:24.380 I mean it doesn't give us anything meaningful. 00:26:24.380 --> 00:26:27.090 So this argument, it's great. 00:26:27.090 --> 00:26:30.000 And it gives us the right answer, but not always. 00:26:30.000 --> 00:26:33.130 And the times when it gives us the answer, the correct answer, 00:26:33.130 --> 00:26:36.800 is when the series is convergent. 00:26:36.800 --> 00:26:38.800 And that's why we care about convergence. 00:26:38.800 --> 00:26:42.060 Because we want manipulations like this to be allowed. 00:26:47.780 --> 00:26:50.200 So the good case, this is the red case 00:26:50.200 --> 00:26:52.730 that we were describing last time. 00:26:52.730 --> 00:26:54.790 That's the bad case. 00:26:54.790 --> 00:26:58.820 But what we want is the good case, the convergent case. 00:26:58.820 --> 00:27:01.770 And that is the case when x is less than 1. 00:27:01.770 --> 00:27:03.182 So this is the convergent case. 00:27:11.290 --> 00:27:11.960 Yep. 00:27:11.960 --> 00:27:14.140 OK, so they're much more detailed things 00:27:14.140 --> 00:27:15.710 to check exactly what's going on. 00:27:15.710 --> 00:27:18.090 But I'm going to just say general words about how 00:27:18.090 --> 00:27:20.099 you recognize convergence. 00:27:20.099 --> 00:27:22.140 And then we're not going to worry about-- so much 00:27:22.140 --> 00:27:24.740 about convergence, because it works very, very well. 00:27:24.740 --> 00:27:27.480 And it's always easy to diagnose when 00:27:27.480 --> 00:27:31.590 there's convergence with a power series. 00:27:31.590 --> 00:27:33.450 All right so here's the general setup. 00:27:44.700 --> 00:27:48.510 The general setup is that we have 00:27:48.510 --> 00:27:55.600 not just the coefficients 1 all the time, but any numbers here, 00:27:55.600 --> 00:27:56.660 dot, dot, dot. 00:27:56.660 --> 00:27:59.310 And we abbreviate that with the summation notation. 00:27:59.310 --> 00:28:05.360 This is the sum a_n x^n, n equals 0 to infinity. 00:28:05.360 --> 00:28:07.260 And that's what's known as a power series. 00:28:12.370 --> 00:28:16.370 Fortunately there is a very simple rule 00:28:16.370 --> 00:28:20.260 about how power series converge. 00:28:20.260 --> 00:28:23.200 And it's the following. 00:28:23.200 --> 00:28:28.480 There's a magic number R which depends on these numbers here 00:28:28.480 --> 00:28:30.160 such that-- And this thing is known 00:28:30.160 --> 00:28:31.460 as a radius of convergence. 00:28:37.020 --> 00:28:40.090 In the problem that we had, it's this number 1 here. 00:28:40.090 --> 00:28:43.030 This thing works for x less than 1. 00:28:43.030 --> 00:28:46.190 In our case, it's maybe x less than R. So 00:28:46.190 --> 00:28:47.960 that's some symmetric interval, right? 00:28:47.960 --> 00:28:53.630 That's the same as minus R less than x less than R, 00:28:53.630 --> 00:28:58.310 and so where there's convergence. 00:28:58.310 --> 00:29:00.310 OK, where the series converges. 00:29:00.310 --> 00:29:00.810 Converges. 00:29:07.210 --> 00:29:12.150 And then there's the region where 00:29:12.150 --> 00:29:16.500 every computation that you give will give you nonsense. 00:29:16.500 --> 00:29:24.740 So x greater than R is the sum a_n x^n diverges. 00:29:28.710 --> 00:29:42.030 And x equals R is very delicate, borderline, 00:29:42.030 --> 00:29:44.990 and will not be used by us. 00:29:50.060 --> 00:29:54.750 OK, we're going to stick inside the radius of convergence. 00:29:54.750 --> 00:29:57.620 Now the way you'll be able to recognize this, 00:29:57.620 --> 00:29:58.920 is the following. 00:29:58.920 --> 00:30:03.110 What always happens is that these numbers 00:30:03.110 --> 00:30:15.370 tend to 0 exponentially fast, fast for x in R, 00:30:15.370 --> 00:30:26.600 and doesn't even tend to 0 at all for x greater than R. 00:30:26.600 --> 00:30:30.200 All right so it'll be totally obvious. 00:30:30.200 --> 00:30:32.210 When you look at this series here, 00:30:32.210 --> 00:30:34.040 what's happening when x less than R 00:30:34.040 --> 00:30:37.400 is that the numbers are getting smaller and smaller, less 00:30:37.400 --> 00:30:37.990 than 1. 00:30:37.990 --> 00:30:39.290 When x is bigger than 1, the numbers 00:30:39.290 --> 00:30:40.539 are getting bigger and bigger. 00:30:40.539 --> 00:30:42.850 There's no chance that the series converges. 00:30:42.850 --> 00:30:45.870 So that's going to be the case with all power series. 00:30:45.870 --> 00:30:47.680 There's going to be a cutoff. 00:30:47.680 --> 00:30:49.369 And it'll be one particular number. 00:30:49.369 --> 00:30:51.785 And below that it'll be obvious that you have convergence, 00:30:51.785 --> 00:30:53.368 and you'll be able to do computations. 00:30:53.368 --> 00:30:55.690 And above that every formula will be wrong 00:30:55.690 --> 00:30:57.220 and won't make sense. 00:30:57.220 --> 00:30:59.370 So it's a very clean thing. 00:30:59.370 --> 00:31:01.660 There is this very subtle borderline, 00:31:01.660 --> 00:31:04.350 but we're not going to discuss that in this class. 00:31:04.350 --> 00:31:10.383 And it's actually not used in direct studies of power series. 00:31:10.383 --> 00:31:13.820 AUDIENCE: How can you tell when the numbers are declining 00:31:13.820 --> 00:31:16.780 exponentially fast, whereas just-- In other words 00:31:16.780 --> 00:31:18.030 1/x [INAUDIBLE]? 00:31:18.030 --> 00:31:20.470 PROFESSOR: OK so, the question is 00:31:20.470 --> 00:31:23.070 why was I able to tell you this word here? 00:31:23.070 --> 00:31:25.510 Why was I able to tell you not only is it going to 0, 00:31:25.510 --> 00:31:27.550 but it's going exponentially fast? 00:31:27.550 --> 00:31:29.000 I'm telling you extra information. 00:31:29.000 --> 00:31:32.380 I'm telling you it always goes exponentially fast. 00:31:32.380 --> 00:31:33.556 You can identify it. 00:31:36.150 --> 00:31:37.620 In other words, you'll see it. 00:31:37.620 --> 00:31:39.490 And it will happen every single time. 00:31:39.490 --> 00:31:41.600 I'm just promising you that it works that way. 00:31:41.600 --> 00:31:43.900 And it's really for the same reason 00:31:43.900 --> 00:31:46.840 that it works that way here, that these are powers. 00:31:46.840 --> 00:31:48.760 And what's going on over here is there are, 00:31:48.760 --> 00:31:52.181 it's close to powers, with these a_n's. 00:31:52.181 --> 00:31:54.640 All right? 00:31:54.640 --> 00:31:56.960 There's a long discussion of radius of convergence 00:31:56.960 --> 00:31:58.350 in many textbooks. 00:31:58.350 --> 00:32:04.670 But really it's not necessary, all right, for this purpose? 00:32:04.670 --> 00:32:05.418 Yeah? 00:32:05.418 --> 00:32:07.090 AUDIENCE: How do you find R? 00:32:07.090 --> 00:32:08.400 PROFESSOR: The question was how do you find R? 00:32:08.400 --> 00:32:10.316 Yes, so I just said, there's a long discussion 00:32:10.316 --> 00:32:12.660 for how you find the radius of convergence in textbooks. 00:32:12.660 --> 00:32:15.800 But we will not be discussing that here. 00:32:15.800 --> 00:32:17.680 And it won't be necessary for you. 00:32:17.680 --> 00:32:20.890 Because it will be obvious in any given series what the R is. 00:32:20.890 --> 00:32:23.650 It will always either 1 or infinity. 00:32:23.650 --> 00:32:25.850 It will always work for all x, or maybe it'll 00:32:25.850 --> 00:32:26.820 stop at some point. 00:32:26.820 --> 00:32:29.280 But it'll be very clear where it stops, 00:32:29.280 --> 00:32:33.201 as it is for the geometric series. 00:32:33.201 --> 00:32:36.070 All right? 00:32:36.070 --> 00:32:39.639 OK, so now I need to give you the basic facts, 00:32:39.639 --> 00:32:40.805 and give you a few examples. 00:32:44.400 --> 00:32:46.420 So why are we looking at these series? 00:32:51.240 --> 00:32:55.010 Well the answer is we're looking at these series 00:32:55.010 --> 00:32:59.270 because the role that they play is exactly 00:32:59.270 --> 00:33:03.420 the reverse of this equation here. 00:33:03.420 --> 00:33:06.540 That is -- and this is a theme which I have tried to emphasize 00:33:06.540 --> 00:33:09.730 throughout this course -- you can read equalities in two 00:33:09.730 --> 00:33:11.870 directions. 00:33:11.870 --> 00:33:15.610 Both are interesting, typically. 00:33:15.610 --> 00:33:18.116 You can think, I don't know what the value of this is. 00:33:18.116 --> 00:33:19.240 Here's a way of evaluating. 00:33:19.240 --> 00:33:21.540 And in other words, the right side 00:33:21.540 --> 00:33:23.090 is a formula for the left side. 00:33:23.090 --> 00:33:25.400 Or you can think of the left side 00:33:25.400 --> 00:33:27.140 as being a formula for the right side. 00:33:30.720 --> 00:33:34.230 And the idea of series is that they're flexible enough 00:33:34.230 --> 00:33:35.780 to represent all of the functions 00:33:35.780 --> 00:33:39.300 that we've encountered in this course. 00:33:39.300 --> 00:33:42.069 This is the tool which is very much like the decimal expansion 00:33:42.069 --> 00:33:43.610 which allows you to represent numbers 00:33:43.610 --> 00:33:45.130 like the square root of 2. 00:33:45.130 --> 00:33:47.520 Now we're going to be representing all the numbers, 00:33:47.520 --> 00:33:52.550 all the functions that we know: e^x, arctangent, sine, cosine. 00:33:52.550 --> 00:33:55.280 All of those functions become completely flexible, 00:33:55.280 --> 00:33:57.970 and completely available to us, and computationally available 00:33:57.970 --> 00:33:59.900 to us directly. 00:33:59.900 --> 00:34:01.440 So that's what this is a tool for. 00:34:01.440 --> 00:34:03.580 And it's just like decimal expansions 00:34:03.580 --> 00:34:05.240 giving you handle on all real numbers. 00:34:08.920 --> 00:34:12.110 So here's how it works. 00:34:12.110 --> 00:34:28.170 The rules for convergent power series 00:34:28.170 --> 00:34:33.920 are just like polynomials. 00:34:41.280 --> 00:34:43.660 All of the manipulations that you do for power series 00:34:43.660 --> 00:34:46.120 are essentially the same as for polynomials. 00:34:46.120 --> 00:34:49.190 So what kinds of things do we do with polynomials? 00:34:49.190 --> 00:34:50.130 We add them. 00:34:53.280 --> 00:34:54.445 We multiply them together. 00:34:57.630 --> 00:35:00.790 We do substitutions. 00:35:00.790 --> 00:35:01.290 Right? 00:35:01.290 --> 00:35:04.970 We take one function of another function. 00:35:04.970 --> 00:35:06.030 We divide them. 00:35:10.110 --> 00:35:11.590 OK. 00:35:11.590 --> 00:35:15.040 And these are all really not very surprising operations. 00:35:15.040 --> 00:35:17.710 And we will be able to do them with power series too. 00:35:17.710 --> 00:35:20.370 The ones that are interesting, really interesting 00:35:20.370 --> 00:35:24.360 for calculus, are the last two. 00:35:24.360 --> 00:35:29.790 We differentiate them, and we integrate them. 00:35:34.070 --> 00:35:36.060 And all of these operations we'll 00:35:36.060 --> 00:35:38.110 be able to do for power series as well. 00:35:42.950 --> 00:35:49.120 So now let's explain the high points of this. 00:35:49.120 --> 00:35:51.530 Which is mainly just the differentiation 00:35:51.530 --> 00:35:53.110 and the integration part. 00:35:53.110 --> 00:36:03.200 So if I take a series like this and so forth, 00:36:03.200 --> 00:36:08.054 the formula for its derivative is just like polynomials. 00:36:08.054 --> 00:36:10.220 That's what I just said, it's just like polynomials. 00:36:10.220 --> 00:36:12.570 So the derivative of the constant is 0. 00:36:12.570 --> 00:36:15.850 The derivative of this term is a_1. 00:36:15.850 --> 00:36:19.340 This one is plus 2 a_2 x 2 x. 00:36:19.340 --> 00:36:23.920 This one is 3 a_3 x^2, et cetera. 00:36:23.920 --> 00:36:27.070 That's the formula. 00:36:27.070 --> 00:36:40.672 Similarly if I integrate, well there's 00:36:40.672 --> 00:36:42.130 an unknown constant which I'm going 00:36:42.130 --> 00:36:44.930 to put first rather than last. 00:36:44.930 --> 00:36:46.927 Which corresponds sort of to the a_0 term which 00:36:46.927 --> 00:36:48.010 is going to get wiped out. 00:36:48.010 --> 00:36:50.770 That a_0 term suddenly becomes a_0 x. 00:36:50.770 --> 00:36:56.980 And the anti-derivative of this next term is a_1 x^2 / 2. 00:36:56.980 --> 00:37:02.680 And the next term is a_2 x^3 / 3, and so forth. 00:37:05.200 --> 00:37:06.456 Yeah, question? 00:37:06.456 --> 00:37:08.357 AUDIENCE: Is that a series or a polynomial? 00:37:08.357 --> 00:37:10.190 PROFESSOR: Is this a series or a polynomial? 00:37:10.190 --> 00:37:11.180 Good question. 00:37:11.180 --> 00:37:14.700 It's a polynomial if it ends. 00:37:14.700 --> 00:37:19.490 If it goes on infinitely far, then it's a series. 00:37:19.490 --> 00:37:22.420 They look practically the same, polynomials and series. 00:37:22.420 --> 00:37:26.286 There's this little dot, dot, dot here. 00:37:26.286 --> 00:37:27.660 Is this a series or a polynomial? 00:37:27.660 --> 00:37:28.780 It's the same rule. 00:37:28.780 --> 00:37:30.460 If it stops at a finite stage, this one 00:37:30.460 --> 00:37:32.210 stops at a finite stage. 00:37:32.210 --> 00:37:35.074 If it goes on forever, it goes on forever. 00:37:35.074 --> 00:37:37.980 AUDIENCE: So I thought that the series add up finite numbers. 00:37:37.980 --> 00:37:41.735 You can add up terms of x in series? 00:37:45.520 --> 00:37:47.380 PROFESSOR: So an interesting question. 00:37:47.380 --> 00:37:49.435 So the question that was just asked 00:37:49.435 --> 00:37:53.060 is I thought that a series added up finite numbers. 00:37:53.060 --> 00:37:55.140 You could add up x? 00:37:55.140 --> 00:37:56.900 That was what you said, right? 00:37:56.900 --> 00:38:01.140 OK now notice that I pulled that off 00:38:01.140 --> 00:38:03.720 on you by changing the letter a to the letter 00:38:03.720 --> 00:38:10.170 x at the very beginning of this commentary here. 00:38:10.170 --> 00:38:11.760 This is a series. 00:38:11.760 --> 00:38:15.630 For each individual value of x, it's a number. 00:38:15.630 --> 00:38:17.790 So in other words, if I plug in here x = 1/2, 00:38:17.790 --> 00:38:20.650 I'm going to add 1 + 1/2 + 1/4 + 1/8, 00:38:20.650 --> 00:38:22.470 and I'll get a number which is 2. 00:38:22.470 --> 00:38:25.030 And I'll plug in a number over here, and I'll get a number. 00:38:25.030 --> 00:38:27.820 On the other hand, I can do this for each value of x. 00:38:27.820 --> 00:38:31.770 So the interpretation of this is that it's a function of x. 00:38:31.770 --> 00:38:34.380 And similarly this is a function of x. 00:38:34.380 --> 00:38:37.400 It works when you plug in the possible values 00:38:37.400 --> 00:38:42.330 x between -1 and 1. 00:38:42.330 --> 00:38:44.760 So there's really no distinction there, it's 00:38:44.760 --> 00:38:47.010 just I slipped it passed you. 00:38:47.010 --> 00:38:48.130 These are functions of x. 00:38:50.790 --> 00:38:53.090 And the notion of a power series is this idea 00:38:53.090 --> 00:38:54.860 that you put coefficients on a series, 00:38:54.860 --> 00:38:57.040 but then you allow yourself the flexibility 00:38:57.040 --> 00:38:58.990 to stick powers here. 00:38:58.990 --> 00:39:01.884 And that's exactly what we're doing. 00:39:01.884 --> 00:39:03.300 OK there are other kinds of series 00:39:03.300 --> 00:39:05.210 where you stick other interesting functions in here 00:39:05.210 --> 00:39:06.168 like sines and cosines. 00:39:06.168 --> 00:39:08.620 There are lots of other series that people study. 00:39:08.620 --> 00:39:10.465 And these are the simplest ones. 00:39:10.465 --> 00:39:12.360 And all those examples are extremely 00:39:12.360 --> 00:39:14.515 helpful for representing functions. 00:39:14.515 --> 00:39:18.920 But we're only going to do this example here. 00:39:18.920 --> 00:39:23.610 All right, so here are the two rules. 00:39:23.610 --> 00:39:29.880 And now there's only one other complication here 00:39:29.880 --> 00:39:35.350 which I have to explain to you before giving you 00:39:35.350 --> 00:39:41.270 a bunch of examples to show you that this works extremely well. 00:39:41.270 --> 00:39:43.960 And the last thing that I have to do 00:39:43.960 --> 00:39:45.460 for you is explain to you something 00:39:45.460 --> 00:39:46.580 called Taylor's formula. 00:39:55.280 --> 00:40:00.730 Taylor's formula is the way you get from the representations 00:40:00.730 --> 00:40:04.430 that we're used to of functions, to a representation in the form 00:40:04.430 --> 00:40:06.180 of these coefficients. 00:40:06.180 --> 00:40:08.240 When I gave you the function e^x, 00:40:08.240 --> 00:40:11.280 it didn't look like a polynomial. 00:40:11.280 --> 00:40:14.010 And we have to figure out which of these guys 00:40:14.010 --> 00:40:19.090 it is, if it's going to fall into our category here. 00:40:19.090 --> 00:40:20.370 And here's the formula. 00:40:20.370 --> 00:40:23.430 I'll explain to you how it works in a second. 00:40:23.430 --> 00:40:25.650 So the formula is f(x) turns out-- 00:40:25.650 --> 00:40:28.560 There's a formula in terms of the derivatives of f. 00:40:28.560 --> 00:40:31.260 Namely, you differentiate n times, 00:40:31.260 --> 00:40:34.260 and you evaluate it at 0, and you divide by n factorial, 00:40:34.260 --> 00:40:37.640 and multiply by x^n. 00:40:37.640 --> 00:40:40.510 So here's Taylor's formula. 00:40:40.510 --> 00:40:44.270 This tells you what the Taylor series is. 00:40:44.270 --> 00:40:48.330 Now about half of our job for the next few minutes 00:40:48.330 --> 00:40:51.630 is going to be to give examples of this. 00:40:51.630 --> 00:40:56.810 But let me just explain to you why this has to be. 00:40:56.810 --> 00:41:00.830 If you pick out this number here, this is the a_n, 00:41:00.830 --> 00:41:03.290 the magic number a_n here. 00:41:03.290 --> 00:41:05.450 So let's just illustrate it. 00:41:05.450 --> 00:41:15.090 If f(x) happens to be a_0 + a_1 x + a_2 x^2 + a_3 x^3 plus dot, 00:41:15.090 --> 00:41:15.940 dot, dot. 00:41:15.940 --> 00:41:19.810 And now I differentiate it, right? 00:41:19.810 --> 00:41:24.416 I get a 1 + 2 a_2 x + 3 a_3 x. 00:41:26.930 --> 00:41:33.120 If I differentiate it another time, I get 2 a_2 plus 3-- 00:41:33.120 --> 00:41:38.560 sorry, 3*2*a_3 x plus dot, dot, dot. 00:41:38.560 --> 00:41:47.360 And now a third time, I get 3*2 a_3 plus et cetera. 00:41:47.360 --> 00:41:54.000 So this next term is really in disguise, 4*3*2 x a-- sorry, 00:41:54.000 --> 00:41:56.410 a_4 x. 00:41:56.410 --> 00:41:59.410 That's what really comes down if I kept track of the fourth term 00:41:59.410 --> 00:42:00.980 there. 00:42:00.980 --> 00:42:04.660 So now here is my function. 00:42:04.660 --> 00:42:07.980 But now you see if I plug in x = 0, 00:42:07.980 --> 00:42:13.750 I can pick off the third term. 00:42:13.750 --> 00:42:21.079 f triple prime of 0 is equal to 3*2 a_3. 00:42:21.079 --> 00:42:23.620 Right, because all the rest of those terms, when I plug in 0, 00:42:23.620 --> 00:42:25.080 are just 0. 00:42:25.080 --> 00:42:26.730 Here's the formula. 00:42:26.730 --> 00:42:30.610 And so the pattern here is this. 00:42:30.610 --> 00:42:33.860 And what's really going on here is this is really 3*2*1 a_3. 00:42:37.110 --> 00:42:48.480 And in general a_n is equal to f, nth derivative, 00:42:48.480 --> 00:42:50.410 divided by n!. 00:42:50.410 --> 00:42:56.224 And of course, n!, I remind you, is n times n-1 times 00:42:56.224 --> 00:42:57.570 n-2, all the way down to 1. 00:43:02.600 --> 00:43:05.980 Now there's one more crazy convention 00:43:05.980 --> 00:43:08.220 which is always used. 00:43:08.220 --> 00:43:12.300 Which is that there's something very strange here down at 0, 00:43:12.300 --> 00:43:16.990 which is that 0 factorial turns out, has to be set equal to 1. 00:43:16.990 --> 00:43:19.270 All right, so that's what you do in order 00:43:19.270 --> 00:43:20.540 to make this formula work out. 00:43:20.540 --> 00:43:22.623 And that's one of the reasons for this convention. 00:43:29.470 --> 00:43:30.600 All right. 00:43:30.600 --> 00:43:36.680 So my next goal is to give you some examples. 00:43:36.680 --> 00:43:45.280 And let's do a couple. 00:43:48.130 --> 00:43:51.880 So here's, well you know, I'm going 00:43:51.880 --> 00:43:56.120 to have to let you see a few of them next time. 00:43:56.120 --> 00:43:58.960 But let me just tell you this one, which 00:43:58.960 --> 00:44:00.820 is by far the most impressive. 00:44:06.370 --> 00:44:11.840 So what happens with e^x -- if the function is f(x) = e^x -- 00:44:11.840 --> 00:44:18.190 is that its derivative is also e^x. 00:44:18.190 --> 00:44:21.390 And its second derivative is also e^x. 00:44:21.390 --> 00:44:23.010 And it just keeps on going that way. 00:44:23.010 --> 00:44:25.180 They're all the same. 00:44:25.180 --> 00:44:30.010 So that means that these numbers in Taylor's formula, 00:44:30.010 --> 00:44:33.020 in the numerator-- The nth derivative 00:44:33.020 --> 00:44:37.090 is very easy to evaluate. 00:44:37.090 --> 00:44:39.280 It's just e^x. 00:44:39.280 --> 00:44:43.890 And if I evaluate it at x = 0, I just get 1. 00:44:43.890 --> 00:44:46.990 So all of those numerators are 1. 00:44:46.990 --> 00:44:54.721 So the formula here, is the sum n equals 0 to infinity, of 1/n! 00:44:54.721 --> 00:44:55.220 x^n. 00:45:00.960 --> 00:45:04.730 In particular, we now have an honest formula for e 00:45:04.730 --> 00:45:06.800 to the first power. 00:45:06.800 --> 00:45:08.330 Which is just e. 00:45:08.330 --> 00:45:13.790 Which if I plug it in, x = 1, I get 1, this is the n = 0 term. 00:45:13.790 --> 00:45:16.590 Plus 1, This is the n = 1 term. 00:45:16.590 --> 00:45:18.870 Plus 1/2! 00:45:18.870 --> 00:45:20.550 plus 1/3! 00:45:20.550 --> 00:45:21.150 plus 1/4! 00:45:27.040 --> 00:45:27.540 Right? 00:45:27.540 --> 00:45:31.360 So this is our first honest formula for e. 00:45:31.360 --> 00:45:33.960 And also, this is how you compute 00:45:33.960 --> 00:45:35.340 the exponential function. 00:45:41.750 --> 00:45:49.490 Finally if you take a function like sin x, 00:45:49.490 --> 00:45:52.130 what you'll discover is that we can complete the sort 00:45:52.130 --> 00:45:55.957 of strange business that we did at the beginning of the course 00:45:55.957 --> 00:46:03.480 -- or cos x -- where we took the linear and quadratic 00:46:03.480 --> 00:46:04.950 approximations. 00:46:04.950 --> 00:46:10.060 Now we're going to get complete formulas for these functions. 00:46:10.060 --> 00:46:16.297 sin x turns out to be equal to x - x^3 / 3! 00:46:16.297 --> 00:46:19.233 + x^5 / 5! 00:46:19.233 --> 00:46:24.860 - x^7 / 7!, et cetera. 00:46:24.860 --> 00:46:30.680 And cos x = 1 - x^2 / 2! 00:46:30.680 --> 00:46:36.950 -- that's the same as this 2 here -- plus x^4 / 4! 00:46:36.950 --> 00:46:42.590 minus x^6 / 6!, plus et cetera. 00:46:42.590 --> 00:46:48.030 Now these may feel like they're hard to memorize because I've 00:46:48.030 --> 00:46:49.425 just pulled them out of a hat. 00:46:51.990 --> 00:46:55.630 I do expect you to know them. 00:46:55.630 --> 00:46:58.990 They're actually extremely similar formulas. 00:46:58.990 --> 00:47:02.230 The exponential here just has this collection of factorials. 00:47:02.230 --> 00:47:06.860 The sine is all the odd powers with alternating signs. 00:47:06.860 --> 00:47:10.260 And the cosine is all the even powers with alternating signs. 00:47:10.260 --> 00:47:14.130 So all three of them form part of the same family. 00:47:14.130 --> 00:47:16.140 So this will actually make it easier 00:47:16.140 --> 00:47:18.150 for you to remember, rather than harder. 00:47:21.490 --> 00:47:24.100 And so with that, I'll leave the practice 00:47:24.100 --> 00:47:26.210 on differentiation for next time. 00:47:26.210 --> 00:47:27.570 And good luck, everybody. 00:47:27.570 --> 00:47:29.940 I'll talk to you individually.