1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,217 Commons license. 3 00:00:03,217 --> 00:00:05,770 Your support will help MIT OpenCourseWare 4 00:00:05,770 --> 00:00:10,050 continue to offer high quality educational resources for free. 5 00:00:10,050 --> 00:00:12,265 To make a donation or to view additional materials 6 00:00:12,265 --> 00:00:16,649 for hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,649 --> 00:00:18,530 at ocw.mit.edu. 8 00:00:21,510 --> 00:00:24,520 PROFESSOR: Last time we left off with a question 9 00:00:24,520 --> 00:00:26,185 having to do with playing with blocks. 10 00:00:26,185 --> 00:00:30,450 And this is supposed to give us a visceral feel for something 11 00:00:30,450 --> 00:00:32,820 anyway, having to do with series. 12 00:00:32,820 --> 00:00:35,650 And the question was whether I could stack these blocks, 13 00:00:35,650 --> 00:00:39,460 build up a stack so that-- I'm going to try here. 14 00:00:39,460 --> 00:00:42,920 I'm already off balance here, see. 15 00:00:42,920 --> 00:00:46,150 The question is can I build this so 16 00:00:46,150 --> 00:00:51,350 that the-- let's draw a picture of it, 17 00:00:51,350 --> 00:00:54,870 so that the first block is like this. 18 00:00:54,870 --> 00:00:56,567 The next block is like this. 19 00:00:56,567 --> 00:00:58,150 And maybe the next block is like this. 20 00:00:58,150 --> 00:01:00,240 And notice there is no visible means 21 00:01:00,240 --> 00:01:02,410 of support for this block. 22 00:01:02,410 --> 00:01:08,310 It's completely to the left of the first block. 23 00:01:08,310 --> 00:01:11,055 And the question is, will this fall down? 24 00:01:13,730 --> 00:01:21,610 Or at least, or more precisely, eventually we'll ask, 25 00:01:21,610 --> 00:01:24,650 you know, how far can we go? 26 00:01:24,650 --> 00:01:28,760 Now before you answer this question, 27 00:01:28,760 --> 00:01:30,900 the claim is that this is a kind of 28 00:01:30,900 --> 00:01:33,480 a natural, physical question, which 29 00:01:33,480 --> 00:01:37,330 involves some important answer. 30 00:01:37,330 --> 00:01:40,290 No matter whether the answer is you can do it or you can't. 31 00:01:40,290 --> 00:01:42,090 So this is a good kind of math question 32 00:01:42,090 --> 00:01:44,440 where no matter what the answer is, when you figure out 33 00:01:44,440 --> 00:01:46,850 the answer, you're going to get something interesting out 34 00:01:46,850 --> 00:01:47,350 of it. 35 00:01:47,350 --> 00:01:48,766 Because they're two possibilities. 36 00:01:48,766 --> 00:01:53,055 Either there is a limit to how far to the left we can go -- 37 00:01:53,055 --> 00:01:56,400 in which case that's a very interesting number -- 38 00:01:56,400 --> 00:01:58,810 or else there is no limit. 39 00:01:58,810 --> 00:02:00,370 You can go arbitrarily far. 40 00:02:00,370 --> 00:02:03,240 And that's also interesting and curious. 41 00:02:03,240 --> 00:02:05,400 And that's the difference between convergence 42 00:02:05,400 --> 00:02:08,190 and divergence, the thing that we 43 00:02:08,190 --> 00:02:11,380 were talking about up to now concerning series. 44 00:02:11,380 --> 00:02:13,890 So my first question is, do you think 45 00:02:13,890 --> 00:02:19,810 that I can get it so that this thing doesn't fall down 46 00:02:19,810 --> 00:02:24,020 with-- well you see I have about eight blocks here or so. 47 00:02:24,020 --> 00:02:25,570 So you can vote now. 48 00:02:25,570 --> 00:02:27,750 How many in favor that I can succeed 49 00:02:27,750 --> 00:02:30,450 in doing this sort of thing with maybe more than three blocks. 50 00:02:30,450 --> 00:02:32,040 How many in favor? 51 00:02:32,040 --> 00:02:33,860 All right somebody is voting twice. 52 00:02:33,860 --> 00:02:34,560 That's good. 53 00:02:34,560 --> 00:02:35,980 I like that. 54 00:02:35,980 --> 00:02:37,005 How about opposed? 55 00:02:39,930 --> 00:02:43,450 So that was really close to a tie. 56 00:02:43,450 --> 00:02:44,780 All right. 57 00:02:44,780 --> 00:02:49,029 But I think the there was slightly more opposed. 58 00:02:49,029 --> 00:02:49,570 I don't know. 59 00:02:49,570 --> 00:02:52,000 You guys who are in the back maybe could tell. 60 00:02:52,000 --> 00:02:53,750 Anyway it was pretty close. 61 00:02:53,750 --> 00:02:54,250 All right. 62 00:02:54,250 --> 00:02:57,117 So now I'm going-- because this is a real life thing, 63 00:02:57,117 --> 00:02:58,200 I'm going to try to do it. 64 00:02:58,200 --> 00:02:59,476 All right? 65 00:02:59,476 --> 00:03:00,550 All right. 66 00:03:00,550 --> 00:03:04,240 So now I'm going to tell you what the trick is. 67 00:03:04,240 --> 00:03:08,890 The trick is to do it backwards. 68 00:03:08,890 --> 00:03:11,230 When most people are playing with blocks, 69 00:03:11,230 --> 00:03:14,420 they decide to build it from the bottom up. 70 00:03:14,420 --> 00:03:15,670 Right? 71 00:03:15,670 --> 00:03:19,280 But we're going to build it from the top down, 72 00:03:19,280 --> 00:03:20,581 from the top down. 73 00:03:20,581 --> 00:03:22,080 And that's going to make it possible 74 00:03:22,080 --> 00:03:25,080 for us to do the optimal thing at each stage. 75 00:03:25,080 --> 00:03:28,400 So when I build it from the top down, the best I can do 76 00:03:28,400 --> 00:03:30,520 is well, it'll fall off. 77 00:03:30,520 --> 00:03:34,040 I need to have it you know, halfway across. 78 00:03:34,040 --> 00:03:35,470 That's the best I can do. 79 00:03:35,470 --> 00:03:38,000 So the top one I'm going to build like that. 80 00:03:38,000 --> 00:03:41,730 I'm going to take it as far to the left as I can. 81 00:03:41,730 --> 00:03:43,910 And then I'm going to put the next one down 82 00:03:43,910 --> 00:03:46,160 as far to the left as I can. 83 00:03:46,160 --> 00:03:50,450 And then the next one as far to the left as I can. 84 00:03:50,450 --> 00:03:51,740 That was a little too far. 85 00:03:51,740 --> 00:03:55,350 And then I'm going to do the next one as far to the left 86 00:03:55,350 --> 00:03:56,382 as I can. 87 00:03:56,382 --> 00:03:58,250 And then I'm going to do the next one -- 88 00:03:58,250 --> 00:04:04,450 well let's line it up first -- as far to the left as I can. 89 00:04:04,450 --> 00:04:05,150 OK? 90 00:04:05,150 --> 00:04:09,036 And then the next one as far to the left as I can. 91 00:04:09,036 --> 00:04:09,920 All right. 92 00:04:09,920 --> 00:04:15,700 Now those of you who are in this line can see, all right, 93 00:04:15,700 --> 00:04:16,650 I succeeded. 94 00:04:16,650 --> 00:04:18,740 All right, that's over the edge. 95 00:04:18,740 --> 00:04:20,050 All right? 96 00:04:20,050 --> 00:04:21,604 So it can be done. 97 00:04:21,604 --> 00:04:22,390 All right. 98 00:04:25,450 --> 00:04:26,590 All right. 99 00:04:26,590 --> 00:04:32,750 So now we know that we can get farther than you know, 100 00:04:32,750 --> 00:04:34,500 we can make it overflow. 101 00:04:34,500 --> 00:04:37,390 So the question now is, how far can I get? 102 00:04:37,390 --> 00:04:38,240 OK. 103 00:04:38,240 --> 00:04:40,630 Do you think I can get to here? 104 00:04:40,630 --> 00:04:43,260 Can I get to the end over here? 105 00:04:43,260 --> 00:04:50,022 So how many people think I can get this far over to here? 106 00:04:50,022 --> 00:04:51,730 How many people think I can get this far? 107 00:04:51,730 --> 00:04:53,525 Well you know, remember, I'm going 108 00:04:53,525 --> 00:04:55,519 to have to use more than just this one 109 00:04:55,519 --> 00:04:56,560 more block that I've got. 110 00:04:56,560 --> 00:04:58,280 I don't, right? 111 00:04:58,280 --> 00:05:00,900 Obviously I'm thinking, actually I do 112 00:05:00,900 --> 00:05:02,330 have some more blocks at home. 113 00:05:02,330 --> 00:05:03,294 But, OK. 114 00:05:03,294 --> 00:05:04,085 We're not going to. 115 00:05:04,085 --> 00:05:06,510 But anyway, do you think I can get over to here? 116 00:05:06,510 --> 00:05:09,710 How many people say yes? 117 00:05:09,710 --> 00:05:13,060 And how many people say no? 118 00:05:13,060 --> 00:05:15,180 More people said no then yes. 119 00:05:15,180 --> 00:05:15,680 All right. 120 00:05:15,680 --> 00:05:18,480 So maybe the stopping place is some mysterious number 121 00:05:18,480 --> 00:05:19,995 in between here. 122 00:05:19,995 --> 00:05:20,690 All right? 123 00:05:20,690 --> 00:05:21,720 Well OK. 124 00:05:21,720 --> 00:05:24,840 So now we're going to do the arithmetic. 125 00:05:24,840 --> 00:05:28,870 And we're going to figure out what happens with this problem. 126 00:05:28,870 --> 00:05:29,500 OK? 127 00:05:29,500 --> 00:05:32,175 So let's do it. 128 00:05:32,175 --> 00:05:35,760 All right, so now again the idea is, 129 00:05:35,760 --> 00:05:46,710 the idea is we're going to start with the top, the top block. 130 00:05:46,710 --> 00:05:48,450 We'll call that block number one. 131 00:05:53,070 --> 00:05:56,295 And then the farthest, if you like, to the right 132 00:05:56,295 --> 00:05:57,920 that you can put a block underneath it, 133 00:05:57,920 --> 00:05:59,070 is exactly halfway. 134 00:06:01,800 --> 00:06:04,610 All right, well, that's the best job I can do. 135 00:06:04,610 --> 00:06:08,270 Now in order to make my units work out easily, 136 00:06:08,270 --> 00:06:13,160 I'm going to decide to call the length of the block 2. 137 00:06:13,160 --> 00:06:14,090 All right? 138 00:06:14,090 --> 00:06:17,000 And that means if I start at location 0, 139 00:06:17,000 --> 00:06:22,550 then the first place where I am is supposed to be halfway. 140 00:06:22,550 --> 00:06:24,070 And that will be 1. 141 00:06:26,650 --> 00:06:31,660 OK so the first step in the process is 1 more to the right. 142 00:06:31,660 --> 00:06:33,610 Or if you like, if I were building up -- 143 00:06:33,610 --> 00:06:36,068 which is what you would actually have to do in real life -- 144 00:06:36,068 --> 00:06:39,280 it would be 1 to the left. 145 00:06:39,280 --> 00:06:41,610 OK now the next one. 146 00:06:41,610 --> 00:06:44,294 Now here is the way that you start 147 00:06:44,294 --> 00:06:45,460 figuring out the arithmetic. 148 00:06:45,460 --> 00:06:48,190 The next one is based on a physical principle. 149 00:06:48,190 --> 00:06:53,350 Which is that the farthest I can stick this next block 150 00:06:53,350 --> 00:06:57,950 underneath is what's called the center of mass of these two, 151 00:06:57,950 --> 00:07:00,136 which is exactly halfway here. 152 00:07:00,136 --> 00:07:01,760 That is, there's a quarter of this guy, 153 00:07:01,760 --> 00:07:04,400 and a quarter of that guy balancing each other. 154 00:07:04,400 --> 00:07:04,900 Right? 155 00:07:04,900 --> 00:07:06,160 So that's as far as I can go. 156 00:07:06,160 --> 00:07:08,625 If I go farther than that, it'll fall over. 157 00:07:08,625 --> 00:07:10,600 So that's the absolute farthest I can do. 158 00:07:10,600 --> 00:07:14,630 So the next block is going to be over here. 159 00:07:14,630 --> 00:07:18,210 And a quarter of 2 is 1/2. 160 00:07:18,210 --> 00:07:21,552 So this is 3/2 here. 161 00:07:21,552 --> 00:07:24,430 All right so we went to 1. 162 00:07:24,430 --> 00:07:27,960 We went to 3/2 here. 163 00:07:27,960 --> 00:07:31,300 And then I'm going to keep on going with this eventually. 164 00:07:31,300 --> 00:07:33,140 All right so we're going to figure out 165 00:07:33,140 --> 00:07:36,770 what happens with this stack. 166 00:07:36,770 --> 00:07:37,774 Question? 167 00:07:37,774 --> 00:07:40,065 AUDIENCE: How do you know that this 168 00:07:40,065 --> 00:07:42,670 is the best way to optimize? 169 00:07:42,670 --> 00:07:44,515 PROFESSOR: The question is how do I 170 00:07:44,515 --> 00:07:46,870 know that this is the best way to optimize? 171 00:07:46,870 --> 00:07:49,099 I can't answer that question. 172 00:07:49,099 --> 00:07:50,640 But I can tell you that it's the best 173 00:07:50,640 --> 00:07:52,735 way if I start with a top like this, 174 00:07:52,735 --> 00:07:53,860 and the next one like this. 175 00:07:53,860 --> 00:07:56,100 Right, because I'm doing the farthest 176 00:07:56,100 --> 00:07:57,470 possible at each stage. 177 00:07:57,470 --> 00:07:59,840 That actually has a name in computer science, that's 178 00:07:59,840 --> 00:08:01,610 called the greedy algorithm. 179 00:08:01,610 --> 00:08:04,520 I'm trying to do the best possible at each stage. 180 00:08:04,520 --> 00:08:07,040 The greedy algorithm starting from the bottom 181 00:08:07,040 --> 00:08:09,420 is an extremely bad strategy. 182 00:08:09,420 --> 00:08:12,790 Because when you do that, you stack it this way, 183 00:08:12,790 --> 00:08:14,120 and it almost falls over. 184 00:08:14,120 --> 00:08:16,050 And then the next time you can't do anything. 185 00:08:16,050 --> 00:08:19,030 So the greedy algorithm is terrible from the bottom. 186 00:08:19,030 --> 00:08:21,225 This is the greedy algorithm starting from the top, 187 00:08:21,225 --> 00:08:23,455 and it turns out to do much better 188 00:08:23,455 --> 00:08:25,580 then the greedy algorithm starting from the bottom. 189 00:08:25,580 --> 00:08:27,913 But of course I'm not addressing whether there might not 190 00:08:27,913 --> 00:08:31,500 be some other incredibly clever strategy where I wiggle around 191 00:08:31,500 --> 00:08:34,260 and make it go up. 192 00:08:34,260 --> 00:08:35,676 I'm not addressing that question. 193 00:08:35,676 --> 00:08:36,130 All right? 194 00:08:36,130 --> 00:08:37,838 It turns out this is the best you can do. 195 00:08:37,838 --> 00:08:40,790 But that's not clear. 196 00:08:40,790 --> 00:08:44,650 All right so now, here we have this thing. 197 00:08:44,650 --> 00:08:47,600 And now I have to figure out what the arithmetic pattern is, 198 00:08:47,600 --> 00:08:50,340 so that I can figure out what I was doing with those shapes. 199 00:08:53,850 --> 00:08:58,132 So let's figure out a thought experiment here. 200 00:08:58,132 --> 00:08:59,526 All right? 201 00:08:59,526 --> 00:09:00,900 Now the thought experiment I want 202 00:09:00,900 --> 00:09:03,450 to imagine for you is, you've got 203 00:09:03,450 --> 00:09:10,070 a stack of a bunch of blocks, and this is the first N blocks. 204 00:09:14,150 --> 00:09:15,310 All right? 205 00:09:15,310 --> 00:09:19,010 And now we're going to put one underneath it. 206 00:09:19,010 --> 00:09:21,650 And what we're going to figure out 207 00:09:21,650 --> 00:09:26,340 is the center of mass of those N blocks, 208 00:09:26,340 --> 00:09:30,960 which I'm going to call C sub N. OK. 209 00:09:30,960 --> 00:09:32,615 And that's the place where I'm going 210 00:09:32,615 --> 00:09:34,320 to put this very next block. 211 00:09:34,320 --> 00:09:36,480 I'll put it in a different color here. 212 00:09:36,480 --> 00:09:39,560 Here's the new-- the next block over. 213 00:09:39,560 --> 00:09:42,360 And the next block over is the (N+1)st block. 214 00:09:49,900 --> 00:09:54,380 And now I want you to think about what's going on here. 215 00:09:54,380 --> 00:09:56,880 If the center of mass of the first N blocks 216 00:09:56,880 --> 00:10:01,700 is this number, this new one, it's of length 2. 217 00:10:01,700 --> 00:10:04,470 And its center of mass is 1 further 218 00:10:04,470 --> 00:10:07,850 to the right than the center of mass that we had before. 219 00:10:07,850 --> 00:10:10,930 So in other words, I've added to this configuration of N blocks 220 00:10:10,930 --> 00:10:13,220 one more block, which is shifted. 221 00:10:13,220 --> 00:10:15,650 Whose center mass is not lined up 222 00:10:15,650 --> 00:10:17,630 with the center of mass of this, but actually 223 00:10:17,630 --> 00:10:20,200 over farther to the right. 224 00:10:20,200 --> 00:10:23,170 All right so the new center of mass 225 00:10:23,170 --> 00:10:27,060 of this new block-- And this is the extra piece of information 226 00:10:27,060 --> 00:10:29,670 that I want to observe, is that this thing has 227 00:10:29,670 --> 00:10:36,740 a center of mass at C_N + 1. 228 00:10:36,740 --> 00:10:39,870 It's 1 unit over because this total length is 2. 229 00:10:39,870 --> 00:10:45,561 So right in the middle there is 1 over, according to my units. 230 00:10:45,561 --> 00:10:48,240 All right now this is going to make it possible 231 00:10:48,240 --> 00:10:51,970 for me to figure out what the new center of mass is. 232 00:10:51,970 --> 00:11:04,860 So C_(N+1) is the center of mass of N+1 blocks. 233 00:11:04,860 --> 00:11:08,910 Now this is really only in the horizontal variable, right? 234 00:11:08,910 --> 00:11:11,260 I'm not keeping track of the center of mass-- Actually 235 00:11:11,260 --> 00:11:13,510 this thing is hard to build because the center of mass 236 00:11:13,510 --> 00:11:14,240 is also rising. 237 00:11:14,240 --> 00:11:15,830 It's getting higher and higher. 238 00:11:15,830 --> 00:11:19,510 But I'm only keeping track of its left-right characteristic. 239 00:11:19,510 --> 00:11:21,590 So this is the x-coordinate of it. 240 00:11:27,760 --> 00:11:31,590 All right so now here's the idea. 241 00:11:31,590 --> 00:11:34,640 I'm combining the white ones, the N blocks, 242 00:11:34,640 --> 00:11:37,630 with the pink one, which is the one on the bottom. 243 00:11:37,630 --> 00:11:41,020 And there are N of the white ones. 244 00:11:41,020 --> 00:11:42,520 And there's 1 of the pink one. 245 00:11:42,520 --> 00:11:44,811 And so in order to get the center of mass of the whole, 246 00:11:44,811 --> 00:11:47,700 I have to take the weighted average of the two. 247 00:11:47,700 --> 00:11:56,090 That's N*C_N plus 1 times the center of mass of the pink one, 248 00:11:56,090 --> 00:11:59,665 which is C_N + 1. 249 00:11:59,665 --> 00:12:02,040 And then I have to divide -- if it's the weighted average 250 00:12:02,040 --> 00:12:06,560 of the total of N + 1 blocks -- by N + 1. 251 00:12:06,560 --> 00:12:09,090 This is going to give me the new center of mass 252 00:12:09,090 --> 00:12:11,855 of my configuration at the (N+1)st stage. 253 00:12:14,800 --> 00:12:16,330 And now I can just do the arithmetic 254 00:12:16,330 --> 00:12:19,340 and figure out what this is. 255 00:12:19,340 --> 00:12:21,810 And the two C_Ns combine. 256 00:12:21,810 --> 00:12:27,560 I get (N+1)C_N + 1, divided by N+1. 257 00:12:30,250 --> 00:12:33,680 And if I combine these two things and do the cancellation, 258 00:12:33,680 --> 00:12:36,140 that gives me this recurrence formula, 259 00:12:36,140 --> 00:12:41,040 C_(N+1) is equal to C_N plus-- There's a little extra. 260 00:12:41,040 --> 00:12:42,010 These two cancel. 261 00:12:42,010 --> 00:12:43,380 That gives me the C_N. 262 00:12:43,380 --> 00:12:46,740 But then I also have 1/(N+1). 263 00:12:54,230 --> 00:12:57,270 Well that's how much gain I can get in the center of mass 264 00:12:57,270 --> 00:12:58,620 by adding one more block. 265 00:12:58,620 --> 00:13:01,520 That's how much I can shift things over, 266 00:13:01,520 --> 00:13:03,550 depending on how we're thinking of things, 267 00:13:03,550 --> 00:13:05,700 to the left or the right, depending on which 268 00:13:05,700 --> 00:13:07,137 direction we're building them. 269 00:13:10,960 --> 00:13:13,840 All right, so now I'm going to work out the formulas. 270 00:13:13,840 --> 00:13:17,680 First of all C_1, that was the center of the first block. 271 00:13:17,680 --> 00:13:20,490 I put its left end at 0; the center of the first block 272 00:13:20,490 --> 00:13:22,250 is at 1. 273 00:13:22,250 --> 00:13:25,910 That means that C_1 is 1. 274 00:13:25,910 --> 00:13:27,310 OK? 275 00:13:27,310 --> 00:13:31,090 C_2 according to this formula-- And actually I've 276 00:13:31,090 --> 00:13:35,530 worked it out, we'll check it in a-- C_2 is C_1 + 1/2. 277 00:13:35,530 --> 00:13:37,380 All right, so that's the case N = 1. 278 00:13:37,380 --> 00:13:40,130 So this is 1 + 1/2. 279 00:13:40,130 --> 00:13:43,420 That's what we already did. 280 00:13:43,420 --> 00:13:45,790 That's the 3/2 number. 281 00:13:45,790 --> 00:13:51,320 Now the next one is C_2 + 1/3. 282 00:13:51,320 --> 00:13:53,530 That's the formula again. 283 00:13:53,530 --> 00:13:59,590 And so that comes out to be 1 + 1/2 + 1/3. 284 00:13:59,590 --> 00:14:03,100 And now you can see what the pattern is. 285 00:14:03,100 --> 00:14:06,810 C_N-- If you just keep on going here, 286 00:14:06,810 --> 00:14:14,130 C_N is going to be 1 + 1/2 + 1/3 + 1/4... 287 00:14:14,130 --> 00:14:21,410 plus 1/N. 288 00:14:21,410 --> 00:14:25,600 So now I would like you to vote again. 289 00:14:25,600 --> 00:14:27,790 Do you think I can-- Now that we have the formula, 290 00:14:27,790 --> 00:14:30,830 do you think I can get over to here? 291 00:14:30,830 --> 00:14:33,180 How many people think I can get over to here? 292 00:14:36,160 --> 00:14:40,410 How many people think I can't get over to here? 293 00:14:40,410 --> 00:14:42,700 There's still a lot of people who do. 294 00:14:42,700 --> 00:14:46,420 So it's still almost 50/50. 295 00:14:46,420 --> 00:14:47,470 That's amazing. 296 00:14:47,470 --> 00:14:49,800 Well so we'll address that in a few minutes. 297 00:14:49,800 --> 00:14:52,690 So now let me tell you what's going on. 298 00:14:52,690 --> 00:14:56,050 This C_N of course, is the same as what we called last time 299 00:14:56,050 --> 00:14:57,530 S_N. 300 00:14:57,530 --> 00:15:01,230 And remember that we actually estimated the size of this guy. 301 00:15:01,230 --> 00:15:04,620 This is related to what's called the harmonic series. 302 00:15:04,620 --> 00:15:08,120 And what we showed was that log N 303 00:15:08,120 --> 00:15:14,690 is less than S_N, which is less than S_N + 1. 304 00:15:14,690 --> 00:15:15,335 All right? 305 00:15:17,910 --> 00:15:20,360 Now I'm going to call your attention 306 00:15:20,360 --> 00:15:24,260 to the red part, which is the divergence 307 00:15:24,260 --> 00:15:28,500 part of this estimate, which is this one for the time being, 308 00:15:28,500 --> 00:15:30,290 all right. 309 00:15:30,290 --> 00:15:32,780 Just saying that this thing is growing. 310 00:15:32,780 --> 00:15:41,270 And what this is saying is that as N goes to infinity, 311 00:15:41,270 --> 00:15:48,810 log N goes to infinity, So that means 312 00:15:48,810 --> 00:15:57,220 that S_N goes to infinity, because of this inequality 313 00:15:57,220 --> 00:15:57,980 here. 314 00:15:57,980 --> 00:16:01,780 It's bigger than log N. And so if N is big enough, 315 00:16:01,780 --> 00:16:04,002 we can get as far as we like. 316 00:16:04,002 --> 00:16:06,030 All right? 317 00:16:06,030 --> 00:16:08,312 So I can get to here. 318 00:16:08,312 --> 00:16:10,270 And at least half of you, at least the ones who 319 00:16:10,270 --> 00:16:12,224 voted, that was-- I don't know. 320 00:16:12,224 --> 00:16:13,890 We have a quorum here, but I'm not sure. 321 00:16:13,890 --> 00:16:16,470 We certainly didn't have a majority on either side. 322 00:16:16,470 --> 00:16:19,420 Anyway this thing does go to infinity. 323 00:16:19,420 --> 00:16:21,330 So in principle, if I had enough blocks, 324 00:16:21,330 --> 00:16:25,216 I could get it over to here. 325 00:16:25,216 --> 00:16:26,590 All right, and that's the meaning 326 00:16:26,590 --> 00:16:28,300 of divergence in this case. 327 00:16:32,430 --> 00:16:36,670 On the other hand, I want to discuss with you-- 328 00:16:36,670 --> 00:16:38,600 And the reason why I use this example, 329 00:16:38,600 --> 00:16:40,390 is I want to discuss with you also what's 330 00:16:40,390 --> 00:16:45,460 going on with this other inequality here, 331 00:16:45,460 --> 00:16:49,360 and what its significance is. 332 00:16:49,360 --> 00:16:52,970 Which is that it's going to take us a lot of numbers N, 333 00:16:52,970 --> 00:16:57,050 a lot of blocks, to get up to a certain level. 334 00:16:57,050 --> 00:17:00,180 In other words, I can't do it with just eight blocks or nine 335 00:17:00,180 --> 00:17:00,800 blocks. 336 00:17:00,800 --> 00:17:02,258 In order to get over here, I'd have 337 00:17:02,258 --> 00:17:06,000 to use quite a few of them. 338 00:17:06,000 --> 00:17:11,120 So let's just see how many it is. 339 00:17:11,120 --> 00:17:14,230 So I worked this out carefully. 340 00:17:14,230 --> 00:17:15,645 And let's see what I got. 341 00:17:18,150 --> 00:17:31,760 So to get across the lab tables, all right. 342 00:17:31,760 --> 00:17:35,780 This distance here, I already did this secretly. 343 00:17:35,780 --> 00:17:38,985 But I don't actually even have enough of these to show you. 344 00:17:38,985 --> 00:17:43,834 But, well 1, 2, 3, 4, 5, 6, and 1/2. 345 00:17:43,834 --> 00:17:44,750 I guess that's enough. 346 00:17:44,750 --> 00:17:46,240 So it's 6 and a half. 347 00:17:46,240 --> 00:17:50,000 So it's two lab tables is 13 of these blocks. 348 00:17:50,000 --> 00:17:51,190 All right. 349 00:17:51,190 --> 00:18:00,310 So there are 13 blocks, which is equal to 26 units. 350 00:18:00,310 --> 00:18:04,020 OK, that's how far to get across I need. 351 00:18:04,020 --> 00:18:06,090 And the first one is already 2. 352 00:18:06,090 --> 00:18:09,470 So it's really 26 minus 2, which is 24. 353 00:18:09,470 --> 00:18:11,450 Which that's what I need. 354 00:18:11,450 --> 00:18:13,110 OK. 355 00:18:13,110 --> 00:18:22,910 So I need log N to be equal to 24, roughly speaking, 356 00:18:22,910 --> 00:18:25,090 in order to get that far. 357 00:18:25,090 --> 00:18:28,195 So let's just see how big that is. 358 00:18:28,195 --> 00:18:30,300 All right. 359 00:18:30,300 --> 00:18:31,760 I think I worked this out. 360 00:18:42,010 --> 00:18:43,170 So let's see. 361 00:18:43,170 --> 00:18:48,020 That means that N is equal to e^24-- 362 00:18:48,020 --> 00:18:57,160 and if you realize that these blocks are 3 centimeters high-- 363 00:18:57,160 --> 00:19:00,450 OK let's see how many that we would need here. 364 00:19:00,450 --> 00:19:02,210 That's kind of a lot. 365 00:19:02,210 --> 00:19:09,440 Let's see, it's 3 centimeters times e^24, 366 00:19:09,440 --> 00:19:15,660 which is about 8*10^8 meters. 367 00:19:15,660 --> 00:19:17,600 OK. 368 00:19:17,600 --> 00:19:20,810 And that is twice the distance to the moon. 369 00:19:32,310 --> 00:19:35,190 So OK, so I could do it maybe. 370 00:19:35,190 --> 00:19:37,936 But I would need a lot of blocks. 371 00:19:37,936 --> 00:19:38,435 Right? 372 00:19:38,435 --> 00:19:42,190 So that's not very plausible here, all right. 373 00:19:42,190 --> 00:19:44,380 So those of you who voted against this 374 00:19:44,380 --> 00:19:47,080 were actually sort of half right. 375 00:19:47,080 --> 00:19:48,630 And in fact, if you wanted to get it 376 00:19:48,630 --> 00:19:52,300 to the wall over there, which is over 30 feet, 377 00:19:52,300 --> 00:19:55,330 the height would be about the diameter 378 00:19:55,330 --> 00:19:57,360 of the observable universe. 379 00:19:57,360 --> 00:20:01,580 That's kind of a long way. 380 00:20:01,580 --> 00:20:05,780 There's one other thing that I wanted to point out 381 00:20:05,780 --> 00:20:09,340 to you about this shape here. 382 00:20:09,340 --> 00:20:13,516 Which is that if you lean to the left, right, 383 00:20:13,516 --> 00:20:14,890 if you put your head like this -- 384 00:20:14,890 --> 00:20:17,510 of course you have to be on your side to look at it -- 385 00:20:17,510 --> 00:20:25,810 this curve is the shape of a logarithmic curve. 386 00:20:25,810 --> 00:20:29,240 So in other words, if you think of the vertical as the x-axis, 387 00:20:29,240 --> 00:20:32,010 and the horizontal that way is the vertical, 388 00:20:32,010 --> 00:20:34,610 is the up direction, then this thing 389 00:20:34,610 --> 00:20:38,510 is growing very, very, very, very slowly. 390 00:20:38,510 --> 00:20:43,030 If you send the x-axis all the way up to the moon, 391 00:20:43,030 --> 00:20:47,540 the graph still hasn't gotten across the lab tables here. 392 00:20:47,540 --> 00:20:48,905 It's only partway there. 393 00:20:48,905 --> 00:20:52,110 If you go twice the distance to the moon up that way, 394 00:20:52,110 --> 00:20:54,340 it's gotten finally to that end. 395 00:20:54,340 --> 00:20:56,990 All right so that's how slowly the logarithm grows. 396 00:20:56,990 --> 00:20:58,189 It grows very, very slowly. 397 00:20:58,189 --> 00:21:00,730 And if you look at it another way, if you stand on your head, 398 00:21:00,730 --> 00:21:05,420 you can see an exponential curve. 399 00:21:05,420 --> 00:21:08,470 So you get some sense as to the growth properties 400 00:21:08,470 --> 00:21:10,420 of these functions. 401 00:21:10,420 --> 00:21:14,620 And fortunately these are protecting us 402 00:21:14,620 --> 00:21:18,550 from all kinds of stuff that would 403 00:21:18,550 --> 00:21:20,200 happen if there weren't exponentially 404 00:21:20,200 --> 00:21:21,750 small tails in the world. 405 00:21:21,750 --> 00:21:23,670 Like you know, I could walk through this wall 406 00:21:23,670 --> 00:21:27,000 which I wouldn't like doing. 407 00:21:27,000 --> 00:21:32,410 OK, now so this is our last example. 408 00:21:32,410 --> 00:21:35,000 And the important number, unfortunately we 409 00:21:35,000 --> 00:21:36,790 didn't discover another important number. 410 00:21:36,790 --> 00:21:40,250 There wasn't an amazing number place where this stopped. 411 00:21:40,250 --> 00:21:44,160 All we discovered again is some property of infinity. 412 00:21:44,160 --> 00:21:46,150 So infinity is still a nice number. 413 00:21:46,150 --> 00:21:50,940 And the theme here is just that infinity isn't just one thing, 414 00:21:50,940 --> 00:21:54,009 it has a character which is a rate of growth. 415 00:21:54,009 --> 00:21:55,550 And you shouldn't just think of there 416 00:21:55,550 --> 00:21:57,072 being one order of infinity. 417 00:21:57,072 --> 00:21:58,530 There are lots of different orders. 418 00:21:58,530 --> 00:22:01,280 And some of them have different meaning from others. 419 00:22:01,280 --> 00:22:03,880 All right so that's the theme I wanted 420 00:22:03,880 --> 00:22:07,900 to do, and just have a visceral example of infinity. 421 00:22:07,900 --> 00:22:13,570 Now, we're going to move on now to some other kinds 422 00:22:13,570 --> 00:22:16,210 of techniques. 423 00:22:16,210 --> 00:22:20,790 And this is going to be our last subject. 424 00:22:20,790 --> 00:22:23,820 What we're going to talk about is 425 00:22:23,820 --> 00:22:27,320 what are known as power series. 426 00:22:27,320 --> 00:22:29,920 And we've already seen our first power series. 427 00:22:32,710 --> 00:22:35,240 And I'm going to remind you of that. 428 00:22:43,980 --> 00:22:45,490 Here we are with power series. 429 00:22:51,490 --> 00:22:53,980 Our first series was this one. 430 00:22:58,870 --> 00:23:05,200 And we mentioned last time that it was equal to 1/(1-x), 431 00:23:05,200 --> 00:23:06,430 for x less than 1. 432 00:23:09,770 --> 00:23:12,040 Well this one is known as the geometric series. 433 00:23:12,040 --> 00:23:15,050 You didn't use the letter x last time, I used the letter a. 434 00:23:15,050 --> 00:23:16,910 But this is known as the geometric series. 435 00:23:24,950 --> 00:23:31,550 Now I'm going to show you one reason why this is true, 436 00:23:31,550 --> 00:23:33,900 why the formula holds. 437 00:23:33,900 --> 00:23:36,050 And it's just the kind of manipulation 438 00:23:36,050 --> 00:23:41,270 that was done when these things were first introduced. 439 00:23:41,270 --> 00:23:46,250 And here's the idea of a proof. 440 00:23:46,250 --> 00:23:52,670 So suppose that this sum is equal to some number 441 00:23:52,670 --> 00:23:57,360 S, which is the sum of all of these numbers here. 442 00:24:00,540 --> 00:24:02,130 The first thing that I'm going to do 443 00:24:02,130 --> 00:24:05,500 is I'm going to multiply by x. 444 00:24:05,500 --> 00:24:08,326 OK, so if I multiply by x. 445 00:24:08,326 --> 00:24:09,560 Let's think about that. 446 00:24:09,560 --> 00:24:13,810 I multiply by x on both the left and the right-hand side. 447 00:24:13,810 --> 00:24:20,990 Then on the left side, I get x + x^2 + x^3 plus, and so forth. 448 00:24:20,990 --> 00:24:22,810 And on the right side, I get S x. 449 00:24:27,380 --> 00:24:29,990 And now I'm going to subtract the two 450 00:24:29,990 --> 00:24:32,910 equations, one from the other. 451 00:24:32,910 --> 00:24:36,230 And there's a very, very substantial cancellation. 452 00:24:36,230 --> 00:24:39,180 This whole tail here gets canceled off. 453 00:24:39,180 --> 00:24:41,030 And the only thing that's left is the 1. 454 00:24:41,030 --> 00:24:45,950 So when I subtract, I get 1 on the left-hand side. 455 00:24:45,950 --> 00:24:53,000 And on the right-hand side, I get S - S x. 456 00:24:53,000 --> 00:24:53,500 All right? 457 00:24:58,900 --> 00:25:03,850 And now that can be rewritten as S(1-x). 458 00:25:03,850 --> 00:25:06,180 And so I've got my formula here. 459 00:25:06,180 --> 00:25:13,553 This is 1/(1-x) = S. All right. 460 00:25:17,850 --> 00:25:26,340 Now this reasoning has one flaw. 461 00:25:26,340 --> 00:25:28,120 It's not complete. 462 00:25:28,120 --> 00:25:33,080 And this reasoning is basically correct. 463 00:25:33,080 --> 00:25:45,680 But it's incomplete because it requires that S exists. 464 00:25:50,270 --> 00:25:55,100 For example, it doesn't make any sense in the case x = 1. 465 00:25:55,100 --> 00:25:57,950 So for example in the case x = 1, 466 00:25:57,950 --> 00:26:01,710 we have 1 + 1 + 1 plus et cetera, 467 00:26:01,710 --> 00:26:04,830 equals whatever we call S. And then when we multiply through 468 00:26:04,830 --> 00:26:08,798 by 1, we get 1 + 1 + 1 plus... 469 00:26:08,798 --> 00:26:10,640 equals S*1. 470 00:26:10,640 --> 00:26:12,560 And now you see that the subtraction gives us 471 00:26:12,560 --> 00:26:16,330 infinity minus infinity is equal to infinity minus infinity. 472 00:26:16,330 --> 00:26:19,790 That's what's really going on in the argument in this context. 473 00:26:19,790 --> 00:26:21,340 So it's just nonsense. 474 00:26:21,340 --> 00:26:24,380 I mean it doesn't give us anything meaningful. 475 00:26:24,380 --> 00:26:27,090 So this argument, it's great. 476 00:26:27,090 --> 00:26:30,000 And it gives us the right answer, but not always. 477 00:26:30,000 --> 00:26:33,130 And the times when it gives us the answer, the correct answer, 478 00:26:33,130 --> 00:26:36,800 is when the series is convergent. 479 00:26:36,800 --> 00:26:38,800 And that's why we care about convergence. 480 00:26:38,800 --> 00:26:42,060 Because we want manipulations like this to be allowed. 481 00:26:47,780 --> 00:26:50,200 So the good case, this is the red case 482 00:26:50,200 --> 00:26:52,730 that we were describing last time. 483 00:26:52,730 --> 00:26:54,790 That's the bad case. 484 00:26:54,790 --> 00:26:58,820 But what we want is the good case, the convergent case. 485 00:26:58,820 --> 00:27:01,770 And that is the case when x is less than 1. 486 00:27:01,770 --> 00:27:03,182 So this is the convergent case. 487 00:27:11,290 --> 00:27:11,960 Yep. 488 00:27:11,960 --> 00:27:14,140 OK, so they're much more detailed things 489 00:27:14,140 --> 00:27:15,710 to check exactly what's going on. 490 00:27:15,710 --> 00:27:18,090 But I'm going to just say general words about how 491 00:27:18,090 --> 00:27:20,099 you recognize convergence. 492 00:27:20,099 --> 00:27:22,140 And then we're not going to worry about-- so much 493 00:27:22,140 --> 00:27:24,740 about convergence, because it works very, very well. 494 00:27:24,740 --> 00:27:27,480 And it's always easy to diagnose when 495 00:27:27,480 --> 00:27:31,590 there's convergence with a power series. 496 00:27:31,590 --> 00:27:33,450 All right so here's the general setup. 497 00:27:44,700 --> 00:27:48,510 The general setup is that we have 498 00:27:48,510 --> 00:27:55,600 not just the coefficients 1 all the time, but any numbers here, 499 00:27:55,600 --> 00:27:56,660 dot, dot, dot. 500 00:27:56,660 --> 00:27:59,310 And we abbreviate that with the summation notation. 501 00:27:59,310 --> 00:28:05,360 This is the sum a_n x^n, n equals 0 to infinity. 502 00:28:05,360 --> 00:28:07,260 And that's what's known as a power series. 503 00:28:12,370 --> 00:28:16,370 Fortunately there is a very simple rule 504 00:28:16,370 --> 00:28:20,260 about how power series converge. 505 00:28:20,260 --> 00:28:23,200 And it's the following. 506 00:28:23,200 --> 00:28:28,480 There's a magic number R which depends on these numbers here 507 00:28:28,480 --> 00:28:30,160 such that-- And this thing is known 508 00:28:30,160 --> 00:28:31,460 as a radius of convergence. 509 00:28:37,020 --> 00:28:40,090 In the problem that we had, it's this number 1 here. 510 00:28:40,090 --> 00:28:43,030 This thing works for x less than 1. 511 00:28:43,030 --> 00:28:46,190 In our case, it's maybe x less than R. So 512 00:28:46,190 --> 00:28:47,960 that's some symmetric interval, right? 513 00:28:47,960 --> 00:28:53,630 That's the same as minus R less than x less than R, 514 00:28:53,630 --> 00:28:58,310 and so where there's convergence. 515 00:28:58,310 --> 00:29:00,310 OK, where the series converges. 516 00:29:00,310 --> 00:29:00,810 Converges. 517 00:29:07,210 --> 00:29:12,150 And then there's the region where 518 00:29:12,150 --> 00:29:16,500 every computation that you give will give you nonsense. 519 00:29:16,500 --> 00:29:24,740 So x greater than R is the sum a_n x^n diverges. 520 00:29:28,710 --> 00:29:42,030 And x equals R is very delicate, borderline, 521 00:29:42,030 --> 00:29:44,990 and will not be used by us. 522 00:29:50,060 --> 00:29:54,750 OK, we're going to stick inside the radius of convergence. 523 00:29:54,750 --> 00:29:57,620 Now the way you'll be able to recognize this, 524 00:29:57,620 --> 00:29:58,920 is the following. 525 00:29:58,920 --> 00:30:03,110 What always happens is that these numbers 526 00:30:03,110 --> 00:30:15,370 tend to 0 exponentially fast, fast for x in R, 527 00:30:15,370 --> 00:30:26,600 and doesn't even tend to 0 at all for x greater than R. 528 00:30:26,600 --> 00:30:30,200 All right so it'll be totally obvious. 529 00:30:30,200 --> 00:30:32,210 When you look at this series here, 530 00:30:32,210 --> 00:30:34,040 what's happening when x less than R 531 00:30:34,040 --> 00:30:37,400 is that the numbers are getting smaller and smaller, less 532 00:30:37,400 --> 00:30:37,990 than 1. 533 00:30:37,990 --> 00:30:39,290 When x is bigger than 1, the numbers 534 00:30:39,290 --> 00:30:40,539 are getting bigger and bigger. 535 00:30:40,539 --> 00:30:42,850 There's no chance that the series converges. 536 00:30:42,850 --> 00:30:45,870 So that's going to be the case with all power series. 537 00:30:45,870 --> 00:30:47,680 There's going to be a cutoff. 538 00:30:47,680 --> 00:30:49,369 And it'll be one particular number. 539 00:30:49,369 --> 00:30:51,785 And below that it'll be obvious that you have convergence, 540 00:30:51,785 --> 00:30:53,368 and you'll be able to do computations. 541 00:30:53,368 --> 00:30:55,690 And above that every formula will be wrong 542 00:30:55,690 --> 00:30:57,220 and won't make sense. 543 00:30:57,220 --> 00:30:59,370 So it's a very clean thing. 544 00:30:59,370 --> 00:31:01,660 There is this very subtle borderline, 545 00:31:01,660 --> 00:31:04,350 but we're not going to discuss that in this class. 546 00:31:04,350 --> 00:31:10,383 And it's actually not used in direct studies of power series. 547 00:31:10,383 --> 00:31:13,820 AUDIENCE: How can you tell when the numbers are declining 548 00:31:13,820 --> 00:31:16,780 exponentially fast, whereas just-- In other words 549 00:31:16,780 --> 00:31:18,030 1/x [INAUDIBLE]? 550 00:31:18,030 --> 00:31:20,470 PROFESSOR: OK so, the question is 551 00:31:20,470 --> 00:31:23,070 why was I able to tell you this word here? 552 00:31:23,070 --> 00:31:25,510 Why was I able to tell you not only is it going to 0, 553 00:31:25,510 --> 00:31:27,550 but it's going exponentially fast? 554 00:31:27,550 --> 00:31:29,000 I'm telling you extra information. 555 00:31:29,000 --> 00:31:32,380 I'm telling you it always goes exponentially fast. 556 00:31:32,380 --> 00:31:33,556 You can identify it. 557 00:31:36,150 --> 00:31:37,620 In other words, you'll see it. 558 00:31:37,620 --> 00:31:39,490 And it will happen every single time. 559 00:31:39,490 --> 00:31:41,600 I'm just promising you that it works that way. 560 00:31:41,600 --> 00:31:43,900 And it's really for the same reason 561 00:31:43,900 --> 00:31:46,840 that it works that way here, that these are powers. 562 00:31:46,840 --> 00:31:48,760 And what's going on over here is there are, 563 00:31:48,760 --> 00:31:52,181 it's close to powers, with these a_n's. 564 00:31:52,181 --> 00:31:54,640 All right? 565 00:31:54,640 --> 00:31:56,960 There's a long discussion of radius of convergence 566 00:31:56,960 --> 00:31:58,350 in many textbooks. 567 00:31:58,350 --> 00:32:04,670 But really it's not necessary, all right, for this purpose? 568 00:32:04,670 --> 00:32:05,418 Yeah? 569 00:32:05,418 --> 00:32:07,090 AUDIENCE: How do you find R? 570 00:32:07,090 --> 00:32:08,400 PROFESSOR: The question was how do you find R? 571 00:32:08,400 --> 00:32:10,316 Yes, so I just said, there's a long discussion 572 00:32:10,316 --> 00:32:12,660 for how you find the radius of convergence in textbooks. 573 00:32:12,660 --> 00:32:15,800 But we will not be discussing that here. 574 00:32:15,800 --> 00:32:17,680 And it won't be necessary for you. 575 00:32:17,680 --> 00:32:20,890 Because it will be obvious in any given series what the R is. 576 00:32:20,890 --> 00:32:23,650 It will always either 1 or infinity. 577 00:32:23,650 --> 00:32:25,850 It will always work for all x, or maybe it'll 578 00:32:25,850 --> 00:32:26,820 stop at some point. 579 00:32:26,820 --> 00:32:29,280 But it'll be very clear where it stops, 580 00:32:29,280 --> 00:32:33,201 as it is for the geometric series. 581 00:32:33,201 --> 00:32:36,070 All right? 582 00:32:36,070 --> 00:32:39,639 OK, so now I need to give you the basic facts, 583 00:32:39,639 --> 00:32:40,805 and give you a few examples. 584 00:32:44,400 --> 00:32:46,420 So why are we looking at these series? 585 00:32:51,240 --> 00:32:55,010 Well the answer is we're looking at these series 586 00:32:55,010 --> 00:32:59,270 because the role that they play is exactly 587 00:32:59,270 --> 00:33:03,420 the reverse of this equation here. 588 00:33:03,420 --> 00:33:06,540 That is -- and this is a theme which I have tried to emphasize 589 00:33:06,540 --> 00:33:09,730 throughout this course -- you can read equalities in two 590 00:33:09,730 --> 00:33:11,870 directions. 591 00:33:11,870 --> 00:33:15,610 Both are interesting, typically. 592 00:33:15,610 --> 00:33:18,116 You can think, I don't know what the value of this is. 593 00:33:18,116 --> 00:33:19,240 Here's a way of evaluating. 594 00:33:19,240 --> 00:33:21,540 And in other words, the right side 595 00:33:21,540 --> 00:33:23,090 is a formula for the left side. 596 00:33:23,090 --> 00:33:25,400 Or you can think of the left side 597 00:33:25,400 --> 00:33:27,140 as being a formula for the right side. 598 00:33:30,720 --> 00:33:34,230 And the idea of series is that they're flexible enough 599 00:33:34,230 --> 00:33:35,780 to represent all of the functions 600 00:33:35,780 --> 00:33:39,300 that we've encountered in this course. 601 00:33:39,300 --> 00:33:42,069 This is the tool which is very much like the decimal expansion 602 00:33:42,069 --> 00:33:43,610 which allows you to represent numbers 603 00:33:43,610 --> 00:33:45,130 like the square root of 2. 604 00:33:45,130 --> 00:33:47,520 Now we're going to be representing all the numbers, 605 00:33:47,520 --> 00:33:52,550 all the functions that we know: e^x, arctangent, sine, cosine. 606 00:33:52,550 --> 00:33:55,280 All of those functions become completely flexible, 607 00:33:55,280 --> 00:33:57,970 and completely available to us, and computationally available 608 00:33:57,970 --> 00:33:59,900 to us directly. 609 00:33:59,900 --> 00:34:01,440 So that's what this is a tool for. 610 00:34:01,440 --> 00:34:03,580 And it's just like decimal expansions 611 00:34:03,580 --> 00:34:05,240 giving you handle on all real numbers. 612 00:34:08,920 --> 00:34:12,110 So here's how it works. 613 00:34:12,110 --> 00:34:28,170 The rules for convergent power series 614 00:34:28,170 --> 00:34:33,920 are just like polynomials. 615 00:34:41,280 --> 00:34:43,660 All of the manipulations that you do for power series 616 00:34:43,660 --> 00:34:46,120 are essentially the same as for polynomials. 617 00:34:46,120 --> 00:34:49,190 So what kinds of things do we do with polynomials? 618 00:34:49,190 --> 00:34:50,130 We add them. 619 00:34:53,280 --> 00:34:54,445 We multiply them together. 620 00:34:57,630 --> 00:35:00,790 We do substitutions. 621 00:35:00,790 --> 00:35:01,290 Right? 622 00:35:01,290 --> 00:35:04,970 We take one function of another function. 623 00:35:04,970 --> 00:35:06,030 We divide them. 624 00:35:10,110 --> 00:35:11,590 OK. 625 00:35:11,590 --> 00:35:15,040 And these are all really not very surprising operations. 626 00:35:15,040 --> 00:35:17,710 And we will be able to do them with power series too. 627 00:35:17,710 --> 00:35:20,370 The ones that are interesting, really interesting 628 00:35:20,370 --> 00:35:24,360 for calculus, are the last two. 629 00:35:24,360 --> 00:35:29,790 We differentiate them, and we integrate them. 630 00:35:34,070 --> 00:35:36,060 And all of these operations we'll 631 00:35:36,060 --> 00:35:38,110 be able to do for power series as well. 632 00:35:42,950 --> 00:35:49,120 So now let's explain the high points of this. 633 00:35:49,120 --> 00:35:51,530 Which is mainly just the differentiation 634 00:35:51,530 --> 00:35:53,110 and the integration part. 635 00:35:53,110 --> 00:36:03,200 So if I take a series like this and so forth, 636 00:36:03,200 --> 00:36:08,054 the formula for its derivative is just like polynomials. 637 00:36:08,054 --> 00:36:10,220 That's what I just said, it's just like polynomials. 638 00:36:10,220 --> 00:36:12,570 So the derivative of the constant is 0. 639 00:36:12,570 --> 00:36:15,850 The derivative of this term is a_1. 640 00:36:15,850 --> 00:36:19,340 This one is plus 2 a_2 x 2 x. 641 00:36:19,340 --> 00:36:23,920 This one is 3 a_3 x^2, et cetera. 642 00:36:23,920 --> 00:36:27,070 That's the formula. 643 00:36:27,070 --> 00:36:40,672 Similarly if I integrate, well there's 644 00:36:40,672 --> 00:36:42,130 an unknown constant which I'm going 645 00:36:42,130 --> 00:36:44,930 to put first rather than last. 646 00:36:44,930 --> 00:36:46,927 Which corresponds sort of to the a_0 term which 647 00:36:46,927 --> 00:36:48,010 is going to get wiped out. 648 00:36:48,010 --> 00:36:50,770 That a_0 term suddenly becomes a_0 x. 649 00:36:50,770 --> 00:36:56,980 And the anti-derivative of this next term is a_1 x^2 / 2. 650 00:36:56,980 --> 00:37:02,680 And the next term is a_2 x^3 / 3, and so forth. 651 00:37:05,200 --> 00:37:06,456 Yeah, question? 652 00:37:06,456 --> 00:37:08,357 AUDIENCE: Is that a series or a polynomial? 653 00:37:08,357 --> 00:37:10,190 PROFESSOR: Is this a series or a polynomial? 654 00:37:10,190 --> 00:37:11,180 Good question. 655 00:37:11,180 --> 00:37:14,700 It's a polynomial if it ends. 656 00:37:14,700 --> 00:37:19,490 If it goes on infinitely far, then it's a series. 657 00:37:19,490 --> 00:37:22,420 They look practically the same, polynomials and series. 658 00:37:22,420 --> 00:37:26,286 There's this little dot, dot, dot here. 659 00:37:26,286 --> 00:37:27,660 Is this a series or a polynomial? 660 00:37:27,660 --> 00:37:28,780 It's the same rule. 661 00:37:28,780 --> 00:37:30,460 If it stops at a finite stage, this one 662 00:37:30,460 --> 00:37:32,210 stops at a finite stage. 663 00:37:32,210 --> 00:37:35,074 If it goes on forever, it goes on forever. 664 00:37:35,074 --> 00:37:37,980 AUDIENCE: So I thought that the series add up finite numbers. 665 00:37:37,980 --> 00:37:41,735 You can add up terms of x in series? 666 00:37:45,520 --> 00:37:47,380 PROFESSOR: So an interesting question. 667 00:37:47,380 --> 00:37:49,435 So the question that was just asked 668 00:37:49,435 --> 00:37:53,060 is I thought that a series added up finite numbers. 669 00:37:53,060 --> 00:37:55,140 You could add up x? 670 00:37:55,140 --> 00:37:56,900 That was what you said, right? 671 00:37:56,900 --> 00:38:01,140 OK now notice that I pulled that off 672 00:38:01,140 --> 00:38:03,720 on you by changing the letter a to the letter 673 00:38:03,720 --> 00:38:10,170 x at the very beginning of this commentary here. 674 00:38:10,170 --> 00:38:11,760 This is a series. 675 00:38:11,760 --> 00:38:15,630 For each individual value of x, it's a number. 676 00:38:15,630 --> 00:38:17,790 So in other words, if I plug in here x = 1/2, 677 00:38:17,790 --> 00:38:20,650 I'm going to add 1 + 1/2 + 1/4 + 1/8, 678 00:38:20,650 --> 00:38:22,470 and I'll get a number which is 2. 679 00:38:22,470 --> 00:38:25,030 And I'll plug in a number over here, and I'll get a number. 680 00:38:25,030 --> 00:38:27,820 On the other hand, I can do this for each value of x. 681 00:38:27,820 --> 00:38:31,770 So the interpretation of this is that it's a function of x. 682 00:38:31,770 --> 00:38:34,380 And similarly this is a function of x. 683 00:38:34,380 --> 00:38:37,400 It works when you plug in the possible values 684 00:38:37,400 --> 00:38:42,330 x between -1 and 1. 685 00:38:42,330 --> 00:38:44,760 So there's really no distinction there, it's 686 00:38:44,760 --> 00:38:47,010 just I slipped it passed you. 687 00:38:47,010 --> 00:38:48,130 These are functions of x. 688 00:38:50,790 --> 00:38:53,090 And the notion of a power series is this idea 689 00:38:53,090 --> 00:38:54,860 that you put coefficients on a series, 690 00:38:54,860 --> 00:38:57,040 but then you allow yourself the flexibility 691 00:38:57,040 --> 00:38:58,990 to stick powers here. 692 00:38:58,990 --> 00:39:01,884 And that's exactly what we're doing. 693 00:39:01,884 --> 00:39:03,300 OK there are other kinds of series 694 00:39:03,300 --> 00:39:05,210 where you stick other interesting functions in here 695 00:39:05,210 --> 00:39:06,168 like sines and cosines. 696 00:39:06,168 --> 00:39:08,620 There are lots of other series that people study. 697 00:39:08,620 --> 00:39:10,465 And these are the simplest ones. 698 00:39:10,465 --> 00:39:12,360 And all those examples are extremely 699 00:39:12,360 --> 00:39:14,515 helpful for representing functions. 700 00:39:14,515 --> 00:39:18,920 But we're only going to do this example here. 701 00:39:18,920 --> 00:39:23,610 All right, so here are the two rules. 702 00:39:23,610 --> 00:39:29,880 And now there's only one other complication here 703 00:39:29,880 --> 00:39:35,350 which I have to explain to you before giving you 704 00:39:35,350 --> 00:39:41,270 a bunch of examples to show you that this works extremely well. 705 00:39:41,270 --> 00:39:43,960 And the last thing that I have to do 706 00:39:43,960 --> 00:39:45,460 for you is explain to you something 707 00:39:45,460 --> 00:39:46,580 called Taylor's formula. 708 00:39:55,280 --> 00:40:00,730 Taylor's formula is the way you get from the representations 709 00:40:00,730 --> 00:40:04,430 that we're used to of functions, to a representation in the form 710 00:40:04,430 --> 00:40:06,180 of these coefficients. 711 00:40:06,180 --> 00:40:08,240 When I gave you the function e^x, 712 00:40:08,240 --> 00:40:11,280 it didn't look like a polynomial. 713 00:40:11,280 --> 00:40:14,010 And we have to figure out which of these guys 714 00:40:14,010 --> 00:40:19,090 it is, if it's going to fall into our category here. 715 00:40:19,090 --> 00:40:20,370 And here's the formula. 716 00:40:20,370 --> 00:40:23,430 I'll explain to you how it works in a second. 717 00:40:23,430 --> 00:40:25,650 So the formula is f(x) turns out-- 718 00:40:25,650 --> 00:40:28,560 There's a formula in terms of the derivatives of f. 719 00:40:28,560 --> 00:40:31,260 Namely, you differentiate n times, 720 00:40:31,260 --> 00:40:34,260 and you evaluate it at 0, and you divide by n factorial, 721 00:40:34,260 --> 00:40:37,640 and multiply by x^n. 722 00:40:37,640 --> 00:40:40,510 So here's Taylor's formula. 723 00:40:40,510 --> 00:40:44,270 This tells you what the Taylor series is. 724 00:40:44,270 --> 00:40:48,330 Now about half of our job for the next few minutes 725 00:40:48,330 --> 00:40:51,630 is going to be to give examples of this. 726 00:40:51,630 --> 00:40:56,810 But let me just explain to you why this has to be. 727 00:40:56,810 --> 00:41:00,830 If you pick out this number here, this is the a_n, 728 00:41:00,830 --> 00:41:03,290 the magic number a_n here. 729 00:41:03,290 --> 00:41:05,450 So let's just illustrate it. 730 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, 731 00:41:15,090 --> 00:41:15,940 dot, dot. 732 00:41:15,940 --> 00:41:19,810 And now I differentiate it, right? 733 00:41:19,810 --> 00:41:24,416 I get a 1 + 2 a_2 x + 3 a_3 x. 734 00:41:26,930 --> 00:41:33,120 If I differentiate it another time, I get 2 a_2 plus 3-- 735 00:41:33,120 --> 00:41:38,560 sorry, 3*2*a_3 x plus dot, dot, dot. 736 00:41:38,560 --> 00:41:47,360 And now a third time, I get 3*2 a_3 plus et cetera. 737 00:41:47,360 --> 00:41:54,000 So this next term is really in disguise, 4*3*2 x a-- sorry, 738 00:41:54,000 --> 00:41:56,410 a_4 x. 739 00:41:56,410 --> 00:41:59,410 That's what really comes down if I kept track of the fourth term 740 00:41:59,410 --> 00:42:00,980 there. 741 00:42:00,980 --> 00:42:04,660 So now here is my function. 742 00:42:04,660 --> 00:42:07,980 But now you see if I plug in x = 0, 743 00:42:07,980 --> 00:42:13,750 I can pick off the third term. 744 00:42:13,750 --> 00:42:21,079 f triple prime of 0 is equal to 3*2 a_3. 745 00:42:21,079 --> 00:42:23,620 Right, because all the rest of those terms, when I plug in 0, 746 00:42:23,620 --> 00:42:25,080 are just 0. 747 00:42:25,080 --> 00:42:26,730 Here's the formula. 748 00:42:26,730 --> 00:42:30,610 And so the pattern here is this. 749 00:42:30,610 --> 00:42:33,860 And what's really going on here is this is really 3*2*1 a_3. 750 00:42:37,110 --> 00:42:48,480 And in general a_n is equal to f, nth derivative, 751 00:42:48,480 --> 00:42:50,410 divided by n!. 752 00:42:50,410 --> 00:42:56,224 And of course, n!, I remind you, is n times n-1 times 753 00:42:56,224 --> 00:42:57,570 n-2, all the way down to 1. 754 00:43:02,600 --> 00:43:05,980 Now there's one more crazy convention 755 00:43:05,980 --> 00:43:08,220 which is always used. 756 00:43:08,220 --> 00:43:12,300 Which is that there's something very strange here down at 0, 757 00:43:12,300 --> 00:43:16,990 which is that 0 factorial turns out, has to be set equal to 1. 758 00:43:16,990 --> 00:43:19,270 All right, so that's what you do in order 759 00:43:19,270 --> 00:43:20,540 to make this formula work out. 760 00:43:20,540 --> 00:43:22,623 And that's one of the reasons for this convention. 761 00:43:29,470 --> 00:43:30,600 All right. 762 00:43:30,600 --> 00:43:36,680 So my next goal is to give you some examples. 763 00:43:36,680 --> 00:43:45,280 And let's do a couple. 764 00:43:48,130 --> 00:43:51,880 So here's, well you know, I'm going 765 00:43:51,880 --> 00:43:56,120 to have to let you see a few of them next time. 766 00:43:56,120 --> 00:43:58,960 But let me just tell you this one, which 767 00:43:58,960 --> 00:44:00,820 is by far the most impressive. 768 00:44:06,370 --> 00:44:11,840 So what happens with e^x -- if the function is f(x) = e^x -- 769 00:44:11,840 --> 00:44:18,190 is that its derivative is also e^x. 770 00:44:18,190 --> 00:44:21,390 And its second derivative is also e^x. 771 00:44:21,390 --> 00:44:23,010 And it just keeps on going that way. 772 00:44:23,010 --> 00:44:25,180 They're all the same. 773 00:44:25,180 --> 00:44:30,010 So that means that these numbers in Taylor's formula, 774 00:44:30,010 --> 00:44:33,020 in the numerator-- The nth derivative 775 00:44:33,020 --> 00:44:37,090 is very easy to evaluate. 776 00:44:37,090 --> 00:44:39,280 It's just e^x. 777 00:44:39,280 --> 00:44:43,890 And if I evaluate it at x = 0, I just get 1. 778 00:44:43,890 --> 00:44:46,990 So all of those numerators are 1. 779 00:44:46,990 --> 00:44:54,721 So the formula here, is the sum n equals 0 to infinity, of 1/n! 780 00:44:54,721 --> 00:44:55,220 x^n. 781 00:45:00,960 --> 00:45:04,730 In particular, we now have an honest formula for e 782 00:45:04,730 --> 00:45:06,800 to the first power. 783 00:45:06,800 --> 00:45:08,330 Which is just e. 784 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. 785 00:45:13,790 --> 00:45:16,590 Plus 1, This is the n = 1 term. 786 00:45:16,590 --> 00:45:18,870 Plus 1/2! 787 00:45:18,870 --> 00:45:20,550 plus 1/3! 788 00:45:20,550 --> 00:45:21,150 plus 1/4! 789 00:45:27,040 --> 00:45:27,540 Right? 790 00:45:27,540 --> 00:45:31,360 So this is our first honest formula for e. 791 00:45:31,360 --> 00:45:33,960 And also, this is how you compute 792 00:45:33,960 --> 00:45:35,340 the exponential function. 793 00:45:41,750 --> 00:45:49,490 Finally if you take a function like sin x, 794 00:45:49,490 --> 00:45:52,130 what you'll discover is that we can complete the sort 795 00:45:52,130 --> 00:45:55,957 of strange business that we did at the beginning of the course 796 00:45:55,957 --> 00:46:03,480 -- or cos x -- where we took the linear and quadratic 797 00:46:03,480 --> 00:46:04,950 approximations. 798 00:46:04,950 --> 00:46:10,060 Now we're going to get complete formulas for these functions. 799 00:46:10,060 --> 00:46:16,297 sin x turns out to be equal to x - x^3 / 3! 800 00:46:16,297 --> 00:46:19,233 + x^5 / 5! 801 00:46:19,233 --> 00:46:24,860 - x^7 / 7!, et cetera. 802 00:46:24,860 --> 00:46:30,680 And cos x = 1 - x^2 / 2! 803 00:46:30,680 --> 00:46:36,950 -- that's the same as this 2 here -- plus x^4 / 4! 804 00:46:36,950 --> 00:46:42,590 minus x^6 / 6!, plus et cetera. 805 00:46:42,590 --> 00:46:48,030 Now these may feel like they're hard to memorize because I've 806 00:46:48,030 --> 00:46:49,425 just pulled them out of a hat. 807 00:46:51,990 --> 00:46:55,630 I do expect you to know them. 808 00:46:55,630 --> 00:46:58,990 They're actually extremely similar formulas. 809 00:46:58,990 --> 00:47:02,230 The exponential here just has this collection of factorials. 810 00:47:02,230 --> 00:47:06,860 The sine is all the odd powers with alternating signs. 811 00:47:06,860 --> 00:47:10,260 And the cosine is all the even powers with alternating signs. 812 00:47:10,260 --> 00:47:14,130 So all three of them form part of the same family. 813 00:47:14,130 --> 00:47:16,140 So this will actually make it easier 814 00:47:16,140 --> 00:47:18,150 for you to remember, rather than harder. 815 00:47:21,490 --> 00:47:24,100 And so with that, I'll leave the practice 816 00:47:24,100 --> 00:47:26,210 on differentiation for next time. 817 00:47:26,210 --> 00:47:27,570 And good luck, everybody. 818 00:47:27,570 --> 00:47:29,940 I'll talk to you individually.