1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:02,33 The following content is provided under a Creative 3 00:00:02,33 --> 00:00:03,217 Commons license. 4 00:00:03,217 --> 00:00:06,74 Your support will help MIT OpenCourseWare continue to 5 00:00:06,74 --> 00:00:09,25 offer high quality educational resources for free. 6 00:00:10,05 --> 00:00:12,55 To make a donation or to view additional materials for 7 00:00:12,55 --> 00:00:15,485 hundreds of MIT courses, visit mitopencourseware@ocw.mit.edu. 8 00:00:21,51 --> 00:00:24,84 PROFESSOR: Last time we left off with a question having to 9 00:00:24,84 --> 00:00:26,185 do with playing with blocks. 10 00:00:26,185 --> 00:00:30,45 And this is supposed to give us a visceral feel for something 11 00:00:30,45 --> 00:00:31,705 anyway, having to do with series. 12 00:00:32,82 --> 00:00:36,06 And the question was whether I could stack these blocks, build 13 00:00:36,06 --> 00:00:39,46 up a stack so that I'm going to try here up. 14 00:00:39,46 --> 00:00:42,56 I'm already off balance here, see. 15 00:00:42,92 --> 00:00:48,75 The question is can I build this so that the-- let's draw 16 00:00:48,75 --> 00:00:54,46 a picture of it so that the first block is like this. 17 00:00:54,87 --> 00:00:56,335 The next block is like this. 18 00:00:56,8 --> 00:00:58,15 And maybe the next block is like this. 19 00:00:58,15 --> 00:01:00,34 And notice there is no visible means of 20 00:01:00,34 --> 00:01:02,41 support for this block. 21 00:01:02,41 --> 00:01:06,54 It's completely to the left of the first block. 22 00:01:08,31 --> 00:01:11,055 And the question is, will this fall down? 23 00:01:11,055 --> 00:01:13,73 24 00:01:13,73 --> 00:01:21,69 Or at least, or more precisely, eventually we'll ask you 25 00:01:21,69 --> 00:01:23,58 know, how far can we go? 26 00:01:24,65 --> 00:01:30,05 Now before you answer this question, the claim is it that 27 00:01:30,05 --> 00:01:33,48 this is a kind of a natural, physical question, which 28 00:01:33,48 --> 00:01:37,04 involves some important answer. 29 00:01:37,33 --> 00:01:40,29 No matter whether the answer is, you can do it or you can't. 30 00:01:40,29 --> 00:01:43,08 So this is a good kind of math question where no matter what 31 00:01:43,08 --> 00:01:45,39 the answer is, when you figure out the answer, you're 32 00:01:45,39 --> 00:01:47,35 going to get something interesting out of it. 33 00:01:47,35 --> 00:01:48,235 Because they're two possibilities. 34 00:01:48,57 --> 00:01:53,49 Either there is a limit to how far to the left we can go-- in 35 00:01:53,49 --> 00:01:56,7 which case that's a very interesting number-- or 36 00:01:56,7 --> 00:01:58,81 else there is no limit. 37 00:01:58,81 --> 00:02:00,37 You can go arbitrarily far. 38 00:02:00,37 --> 00:02:02,83 And that's also interesting and curious. 39 00:02:03,24 --> 00:02:05,54 And that's the difference between convergence and 40 00:02:05,54 --> 00:02:09,28 divergence, the thing that we were talking about up 41 00:02:09,28 --> 00:02:10,994 to now concerning series. 42 00:02:11,38 --> 00:02:18,12 So my first question is, do you think that I can get it so that 43 00:02:18,12 --> 00:02:22,41 this thing doesn't fall down with, well you see I have about 44 00:02:22,41 --> 00:02:23,71 eight blocks here or so. 45 00:02:24,02 --> 00:02:25,57 So you can vote now. 46 00:02:25,57 --> 00:02:28,55 How many in favor that I can succeed in doing this sort of 47 00:02:28,55 --> 00:02:30,45 thing with maybe more than three blocks. 48 00:02:30,45 --> 00:02:31,36 How many in favor? 49 00:02:32,04 --> 00:02:33,595 All right somebody is voting twice. 50 00:02:33,86 --> 00:02:34,56 That's good. 51 00:02:34,56 --> 00:02:35,15 I like that. 52 00:02:35,98 --> 00:02:37,005 How about opposed? 53 00:02:37,005 --> 00:02:39,93 54 00:02:39,93 --> 00:02:43,035 So that was really close to a tie. 55 00:02:43,45 --> 00:02:43,836 All right. 56 00:02:44,78 --> 00:02:46,965 But I think the there was slightly more opposed. 57 00:02:49,22 --> 00:02:49,57 I don't know. 58 00:02:49,57 --> 00:02:51,53 You guys who are in the back maybe could tell. 59 00:02:52 --> 00:02:53,475 Anyway it was pretty close. 60 00:02:53,81 --> 00:02:54,25 All right. 61 00:02:54,25 --> 00:02:57,43 So now I'm going-- because this is a real life thing-- I'm 62 00:02:57,43 --> 00:02:58,2 going to try to do it. 63 00:02:58,2 --> 00:02:58,475 All right? 64 00:02:59,476 --> 00:03:00,195 All right. 65 00:03:00,55 --> 00:03:02,77 So now I'm going to tell you what the trick is. 66 00:03:04,24 --> 00:03:08,565 The trick is to do it backwards. 67 00:03:08,89 --> 00:03:12,12 When most people are playing with blocks, they decide to 68 00:03:12,12 --> 00:03:14,125 build it from the bottom up. 69 00:03:14,42 --> 00:03:14,71 Right? 70 00:03:15,67 --> 00:03:19,28 But we're going to build it from the top down, 71 00:03:19,28 --> 00:03:20,375 from the top down. 72 00:03:20,76 --> 00:03:22,82 And that's going to make it possible for us to do the 73 00:03:22,82 --> 00:03:24,26 optimal thing at each stage. 74 00:03:25,08 --> 00:03:28,4 So when I build it from the top down, the best I can do 75 00:03:28,4 --> 00:03:30,29 is well, it'll fall off. 76 00:03:30,52 --> 00:03:33,745 I need to have it you know, halfway across. 77 00:03:34,04 --> 00:03:35,215 That's the best I can do. 78 00:03:35,47 --> 00:03:37,62 So the top one I'm going to build like that. 79 00:03:38 --> 00:03:40,75 I'm going to take it as far to the left as I can. 80 00:03:41,73 --> 00:03:44,45 And then I'm going to put the next one down as far 81 00:03:44,45 --> 00:03:45,46 to the left as I can. 82 00:03:46,16 --> 00:03:49,57 And then the next one as far to the left as I can. 83 00:03:50,45 --> 00:03:51,4 That was a little too far. 84 00:03:51,74 --> 00:03:54,78 And then I'm going to do the next one as far 85 00:03:54,78 --> 00:03:55,956 to the left as I can. 86 00:03:56,382 --> 00:03:58,99 And then I'm going to do the next one-- well let's line it 87 00:03:58,99 --> 00:04:03,57 up first-- as far to the left as I can. 88 00:04:04,45 --> 00:04:04,8 OK? 89 00:04:05,15 --> 00:04:08,574 And then the next one as far to the left as I can. 90 00:04:09,036 --> 00:04:09,92 All right. 91 00:04:09,92 --> 00:04:15,23 Now those of you who are in this line can see, 92 00:04:15,23 --> 00:04:16,209 all right, I succeeded. 93 00:04:16,65 --> 00:04:18,74 All right, that's over the edge. 94 00:04:18,74 --> 00:04:19,376 All right? 95 00:04:20,05 --> 00:04:21,16 So it can be done. 96 00:04:21,604 --> 00:04:22,39 All right. 97 00:04:22,39 --> 00:04:25,45 98 00:04:25,45 --> 00:04:25,91 All right. 99 00:04:26,59 --> 00:04:32,75 So now we know that we can get farther than you know, 100 00:04:32,75 --> 00:04:34,26 we can make it overflow. 101 00:04:34,5 --> 00:04:36,57 So the question now is, how far can I get? 102 00:04:37,39 --> 00:04:37,815 OK. 103 00:04:38,24 --> 00:04:39,79 Do you think I can get to here? 104 00:04:40,63 --> 00:04:43,26 Can I get to the end over here? 105 00:04:43,26 --> 00:04:49,43 So how many people think I can get this far over to here? 106 00:04:50,17 --> 00:04:51,73 How many people think I can get this far? 107 00:04:51,73 --> 00:04:53,11 Well you know, remember. 108 00:04:53,39 --> 00:04:55,92 I'm going to have to use more than just this one more 109 00:04:55,92 --> 00:04:56,68 block that I've got. 110 00:04:58,03 --> 00:04:58,28 Right? 111 00:04:58,28 --> 00:05:01,19 Obviously I'm thinking, actually I do have some 112 00:05:01,19 --> 00:05:02,005 more blocks at home. 113 00:05:02,33 --> 00:05:03,37 But, OK. 114 00:05:03,37 --> 00:05:04,085 We're not going to. 115 00:05:04,085 --> 00:05:06,07 But anyway, do you think I can get over to here? 116 00:05:06,51 --> 00:05:07,7 How many people say yes? 117 00:05:09,71 --> 00:05:11,36 And how many people say no? 118 00:05:13,06 --> 00:05:15,37 More people said no then yes. 119 00:05:15,37 --> 00:05:15,68 All right. 120 00:05:15,68 --> 00:05:18,16 So maybe the stopping place is some mysterious 121 00:05:18,16 --> 00:05:19,63 number in between here. 122 00:05:19,995 --> 00:05:20,69 All right? 123 00:05:20,69 --> 00:05:21,72 Well OK. 124 00:05:21,72 --> 00:05:24,47 So now we're going to do the arithmetic. 125 00:05:24,84 --> 00:05:28,27 And we're going to figure out what happens with this problem. 126 00:05:28,87 --> 00:05:29,185 OK? 127 00:05:29,5 --> 00:05:30,75 So let's do it. 128 00:05:32,175 --> 00:05:36,78 All right, so now again the idea is, the idea is we're 129 00:05:36,78 --> 00:05:44,21 going to start with the top, the top block. 130 00:05:44,21 --> 00:05:46,71 131 00:05:46,71 --> 00:05:48,45 We'll call that block number one. 132 00:05:48,45 --> 00:05:53,07 133 00:05:53,07 --> 00:05:56,62 And then the farthest if you like to the right, that you 134 00:05:56,62 --> 00:05:59,07 can put a block underneath it, is exactly halfway. 135 00:05:59,07 --> 00:06:01,8 136 00:06:01,8 --> 00:06:04,61 All right well, that's the best job I can do. 137 00:06:04,61 --> 00:06:08,8 Now in order to make my units work out easily, I'm going 138 00:06:08,8 --> 00:06:12,47 to decide to call the length of the block 2. 139 00:06:13,16 --> 00:06:13,625 All right? 140 00:06:14,09 --> 00:06:17,33 And that means if I start at location 0, then the 141 00:06:17,33 --> 00:06:20,88 first place where I am is supposed to be halfway. 142 00:06:22,55 --> 00:06:24,07 And that will be 1. 143 00:06:24,07 --> 00:06:26,65 144 00:06:26,65 --> 00:06:31,66 OK so the first step in the process is 1 more to the right. 145 00:06:31,66 --> 00:06:34,1 Or if you like, if I were building up-- which is what you 146 00:06:34,1 --> 00:06:35,9 would actually have to do in real life-- it would 147 00:06:35,9 --> 00:06:36,91 be 1 to the left. 148 00:06:39,28 --> 00:06:40,695 OK now the next one. 149 00:06:41,61 --> 00:06:44,74 Now here is the way that you start figuring 150 00:06:44,74 --> 00:06:45,46 out the arithmetic. 151 00:06:45,46 --> 00:06:47,51 The next one is based on a physical principle. 152 00:06:48,19 --> 00:06:53,35 Which is that the farthest I can stick this next block 153 00:06:53,35 --> 00:06:57,95 underneath is what's called the center of mass of these two, 154 00:06:57,95 --> 00:07:00,21 which is exactly halfway here. 155 00:07:00,21 --> 00:07:02,715 That is there's 1/4 of this guy, and a 1/4 of that 156 00:07:02,715 --> 00:07:04,225 guy balancing each other. 157 00:07:04,66 --> 00:07:04,9 Right? 158 00:07:04,9 --> 00:07:06,16 So that's as far as I can go. 159 00:07:06,16 --> 00:07:08,18 If I go farther than that, it'll fall over. 160 00:07:08,625 --> 00:07:10,6 So that's the absolute farthest I can do. 161 00:07:10,6 --> 00:07:13,93 So the next block is going to be over here. 162 00:07:14,63 --> 00:07:17,775 And 1/4 of 2 is 1/2. 163 00:07:18,21 --> 00:07:20,88 So this is 3/2 here. 164 00:07:21,552 --> 00:07:23,656 All right so we went to 1. 165 00:07:24,43 --> 00:07:27,63 We went to it 3/2 here. 166 00:07:27,96 --> 00:07:31 And then I'm going to keep on going with this eventually. 167 00:07:31,3 --> 00:07:34,6 All right so we're going to figure out what happens 168 00:07:34,6 --> 00:07:35,18 with this stack. 169 00:07:36,77 --> 00:07:37,045 Question? 170 00:07:37,774 --> 00:07:40,72 AUDIENCE: How do you know that this is the best 171 00:07:40,72 --> 00:07:41,516 way to optimize? 172 00:07:42,67 --> 00:07:44,575 PROFESSOR: The question is how do I know that this is 173 00:07:44,575 --> 00:07:45,32 the best way to optimize? 174 00:07:46,87 --> 00:07:48,775 I can't answer that question. 175 00:07:49,21 --> 00:07:51,43 But I can tell you that it's the best way if I start 176 00:07:51,43 --> 00:07:53,86 with a top like this, and the next one like this. 177 00:07:53,86 --> 00:07:56,59 Right, because I'm doing the farthest possible 178 00:07:56,59 --> 00:07:57,155 at each stage. 179 00:07:57,47 --> 00:07:59,84 That actually has a name in computer science that's 180 00:07:59,84 --> 00:08:01,145 called the greedy algorithm. 181 00:08:01,61 --> 00:08:03,815 I'm trying to do the best possible at each stage. 182 00:08:04,52 --> 00:08:07,31 The greedy algorithm starting from the bottom, is an 183 00:08:07,31 --> 00:08:08,695 extremely bad strategy. 184 00:08:09,42 --> 00:08:12,79 Because when you do that, you stack it this way, 185 00:08:12,79 --> 00:08:14,12 and it almost falls over. 186 00:08:14,12 --> 00:08:15,56 And then the next time you can't do anything. 187 00:08:16,05 --> 00:08:18,205 So the greedy algorithm is terrible from the bottom. 188 00:08:19,03 --> 00:08:21,85 This is the greedy algorithm starting from the top, and it 189 00:08:21,85 --> 00:08:24,72 turns out to do much better then the greedy algorithm 190 00:08:24,72 --> 00:08:25,58 starting from the bottom. 191 00:08:25,58 --> 00:08:27,39 But of course I'm not addressing whether there might 192 00:08:27,39 --> 00:08:30,42 not be some other incredibly clever strategy where I wiggle 193 00:08:30,42 --> 00:08:33 around and make it go up. 194 00:08:34,26 --> 00:08:36,01 I'm not addressing that question. 195 00:08:36,13 --> 00:08:37,65 It turns out this is the best you can do. 196 00:08:37,65 --> 00:08:39,66 But that's not clear. 197 00:08:40,79 --> 00:08:44,65 All right so now, here we have this thing. 198 00:08:44,65 --> 00:08:47,6 And now I have to figure out what the arithmetic pattern is, 199 00:08:47,6 --> 00:08:50,34 so that I can figure out what I was doing with those shapes. 200 00:08:50,34 --> 00:08:53,85 201 00:08:53,85 --> 00:08:58,132 So let's figure out a thought experiment here. 202 00:08:58,132 --> 00:08:59,08 All right? 203 00:08:59,57 --> 00:09:03,04 Now the thought experiment I want to imagine for you is, 204 00:09:03,04 --> 00:09:08,45 you've got a stack of a bunch of blocks, and this is 205 00:09:08,45 --> 00:09:10,07 the first N blocks. 206 00:09:10,07 --> 00:09:14,15 207 00:09:14,15 --> 00:09:14,64 All right? 208 00:09:15,31 --> 00:09:17,55 And now we're going to put one underneath it. 209 00:09:19,01 --> 00:09:25,23 And what we're going to figure out is the center of mass of 210 00:09:25,23 --> 00:09:27,55 those N blocks, which I'm going to call C sub N. 211 00:09:29,06 --> 00:09:29,56 OK. 212 00:09:30,96 --> 00:09:33,09 And that's the place where I'm going to put this 213 00:09:33,09 --> 00:09:34,32 very next block. 214 00:09:34,32 --> 00:09:36,105 I'll put it in a different color here. 215 00:09:36,48 --> 00:09:38,935 Here's the new, next block over. 216 00:09:39,56 --> 00:09:42,36 And the next block over is the N plus 1st block. 217 00:09:42,36 --> 00:09:49,9 218 00:09:49,9 --> 00:09:54,38 And now I want you to think about what's going on here. 219 00:09:54,38 --> 00:09:58,15 If the center of mass of the first N blocks is this number, 220 00:09:58,15 --> 00:10:00,71 this new one, it's of length 2. 221 00:10:01,7 --> 00:10:05,46 And its center of mass is 1 further to the right 222 00:10:05,46 --> 00:10:07,23 than the center of mass that we had before. 223 00:10:07,85 --> 00:10:10,58 So in other words, I've added to this configuration of N 224 00:10:10,58 --> 00:10:12,88 blocks, 1 more block, which is shifted. 225 00:10:13,22 --> 00:10:16,56 Whose center mass is not lined up with the center of mass of 226 00:10:16,56 --> 00:10:19,84 this, but actually over farther to the right. 227 00:10:20,2 --> 00:10:25,378 All right so the new center of mass of this new block-- and 228 00:10:25,378 --> 00:10:27,56 this is the extra piece of information that I want to 229 00:10:27,56 --> 00:10:32,94 observe-- is that this thing has a center of mass 230 00:10:32,94 --> 00:10:36,355 at C sub N plus 1. 231 00:10:36,74 --> 00:10:39,43 It's 1 unit over because this total length is 2. 232 00:10:39,87 --> 00:10:44,28 So right in the middle there is 1 over, according to my units. 233 00:10:45,561 --> 00:10:48,58 All right now this is going to make it possible for me to 234 00:10:48,58 --> 00:10:50,89 figure out what the new center of mass is. 235 00:10:51,97 --> 00:11:02,485 So C sub N plus 1 is the center of mass of N plus 1 blocks. 236 00:11:04,86 --> 00:11:08,53 Now this is really only in the horizontal variable, right? 237 00:11:08,91 --> 00:11:10,535 I'm not keeping track of the center of mass. 238 00:11:10,86 --> 00:11:12,94 Actually this thing is hard to build because the center 239 00:11:12,94 --> 00:11:13,86 of mass is also rising. 240 00:11:14,24 --> 00:11:15,83 It's getting higher and higher. 241 00:11:15,83 --> 00:11:18,5 But I'm only keeping track of its left and right 242 00:11:18,5 --> 00:11:18,836 characteristic. 243 00:11:19,51 --> 00:11:21,59 So this is the x-coordinate of it. 244 00:11:21,59 --> 00:11:27,76 245 00:11:27,76 --> 00:11:29,13 All right so now here's the idea. 246 00:11:31,59 --> 00:11:35,62 I'm combining the white ones, the N blocks, with the 247 00:11:35,62 --> 00:11:37,335 pink one, which is the one on the bottom. 248 00:11:37,63 --> 00:11:41,02 And there are N of the white ones. 249 00:11:41,02 --> 00:11:42,52 And there's 1 of the pink one. 250 00:11:42,52 --> 00:11:44,44 And so in order to get the center of mass of the whole, 251 00:11:44,44 --> 00:11:46,86 I have to take the weighted average of the two. 252 00:11:47,7 --> 00:11:55,48 That's N times C sub N plus 1, times the center of mass of the 253 00:11:55,48 --> 00:11:58,11 pink one, which is C sub N plus 1. 254 00:12:00,27 --> 00:12:02,095 And then I have to divide if it's the weighted average of 255 00:12:02,095 --> 00:12:05,19 the total of N plus 1 blocks, by N plus 1. 256 00:12:06,56 --> 00:12:09,44 This is going to give me the new center of mass of my 257 00:12:09,44 --> 00:12:11,855 configuration at the N plus 1st stage. 258 00:12:11,855 --> 00:12:14,8 259 00:12:14,8 --> 00:12:16,66 And now I can just do the arithmetic and figure 260 00:12:16,66 --> 00:12:17,49 out what this is. 261 00:12:19,34 --> 00:12:21,335 And the two C sub Ns combine. 262 00:12:21,81 --> 00:12:28,26 I get N plus 1, times C sub N plus 1, divided by N plus 1. 263 00:12:30,25 --> 00:12:33,68 And if I combine these two things and do the cancellation, 264 00:12:33,68 --> 00:12:37,045 that gives me this recurrence formula, C sub N plus 1 is 265 00:12:37,045 --> 00:12:40,635 equal to C sub N plus, there's a little extra. 266 00:12:41,04 --> 00:12:41,75 These two cancel. 267 00:12:42,01 --> 00:12:43,125 That gives me the C sub N. 268 00:12:43,38 --> 00:12:46,74 But then I also have 1 over N plus 1. 269 00:12:46,74 --> 00:12:54,23 270 00:12:54,23 --> 00:12:57,27 Well that's how much gain I can get in the center of mass 271 00:12:57,27 --> 00:12:58,62 by adding one more block. 272 00:12:58,62 --> 00:13:02,27 That's how much I can shift things over, depending on how 273 00:13:02,27 --> 00:13:05,51 we're thinking of things to the left or the right, depending on 274 00:13:05,51 --> 00:13:07,137 which direction we're building them. 275 00:13:07,137 --> 00:13:10,96 276 00:13:10,96 --> 00:13:13,84 All right, so now I'm going to work out the formulas. 277 00:13:13,84 --> 00:13:17,345 First of all C sub 1, that was the center of the first block. 278 00:13:17,68 --> 00:13:19,92 I put its left-ended 0, the center of the 279 00:13:19,92 --> 00:13:21,24 first block is at 1. 280 00:13:22,25 --> 00:13:23,7 That mean that C sub 1 is 1. 281 00:13:25,91 --> 00:13:26,225 OK? 282 00:13:27,31 --> 00:13:31,09 C sub 2, according to this formula-- and actually I've 283 00:13:31,09 --> 00:13:32,99 worked it out, we'll check it in this thing-- C sub 2 284 00:13:32,99 --> 00:13:34,71 is C sub 1 plus 1 over 2. 285 00:13:35,53 --> 00:13:37,105 All right, so that's the case, N equals 1. 286 00:13:37,38 --> 00:13:38,84 So this is 1 plus 1/2. 287 00:13:40,13 --> 00:13:41,15 That's what we already did. 288 00:13:43,42 --> 00:13:44,96 That's the 3/2 number. 289 00:13:45,79 --> 00:13:49,955 Now the next one is C sub 2 plus 1/3. 290 00:13:51,32 --> 00:13:52,43 That's the formula again. 291 00:13:53,53 --> 00:13:57,205 And so that comes out to be 1 plus 1/2 plus 1/3. 292 00:13:59,59 --> 00:14:01,94 And now you can see what the pattern is. 293 00:14:02,74 --> 00:14:07,84 C sub N, if you just keep on going here, C sub N is going 294 00:14:07,84 --> 00:14:14,79 to be 1 plus 1/2 plus 1/3 plus 1/4 plus 1/N. 295 00:14:14,79 --> 00:14:21,41 296 00:14:21,41 --> 00:14:24,47 So now I would like you to vote again. 297 00:14:25,6 --> 00:14:27,94 Do you think I can-- now that we have the formula-- do you 298 00:14:27,94 --> 00:14:29 think I can get over to here? 299 00:14:30,83 --> 00:14:33,18 How many people think I can get over to here? 300 00:14:33,18 --> 00:14:36,16 301 00:14:36,16 --> 00:14:38,84 How many people think I can't get over to here? 302 00:14:40,41 --> 00:14:42,7 There's still a lot of people who do. 303 00:14:42,7 --> 00:14:45,32 So it's still almost 50/50. 304 00:14:46,42 --> 00:14:47,14 That's amazing. 305 00:14:47,47 --> 00:14:49,8 Well so we'll address that in a few minutes. 306 00:14:49,8 --> 00:14:51,49 So now let me tell you what's going on. 307 00:14:52,69 --> 00:14:54,8 This C sub N of course, is the same as what we 308 00:14:54,8 --> 00:14:56,39 called last time S sub N. 309 00:14:57,53 --> 00:15:01,23 And remember that we actually estimated the size of this guy. 310 00:15:01,23 --> 00:15:03,34 This is related to what's called the harmonic series. 311 00:15:04,62 --> 00:15:10,3 And what we showed is that log N is less than S sub N, which 312 00:15:10,3 --> 00:15:13,64 is less than S sub N plus 1. 313 00:15:14,69 --> 00:15:15,335 All right? 314 00:15:15,335 --> 00:15:17,91 315 00:15:17,91 --> 00:15:23,15 Now I'm going to call your attention to the red part, 316 00:15:23,15 --> 00:15:26,71 which is the divergence part of this estimate, which is 317 00:15:26,71 --> 00:15:29,23 this one for the time being, all right. 318 00:15:30,29 --> 00:15:31,83 Just saying that this thing is growing. 319 00:15:32,78 --> 00:15:42,57 And what this is saying is that as N goes to infinite, log N 320 00:15:42,57 --> 00:15:52,23 goes to infinite, So that means that S sub N goes to 321 00:15:52,23 --> 00:15:57,6 infinite, because of this inequality here. 322 00:15:57,98 --> 00:15:59,473 It's bigger than log N. 323 00:16:00,3 --> 00:16:03,25 And so if N is big enough, we can get as far as we like. 324 00:16:04,002 --> 00:16:04,85 All right? 325 00:16:06,03 --> 00:16:08,045 So I can get to here. 326 00:16:08,53 --> 00:16:10,87 And at least half of you, at least the ones who voted, 327 00:16:10,87 --> 00:16:12,33 that was I don't know. 328 00:16:12,33 --> 00:16:13,89 We have a quorum here, but I'm not sure. 329 00:16:13,89 --> 00:16:16,045 We certainly didn't have a majority on either side. 330 00:16:16,47 --> 00:16:19,14 Anyway this thing does go to infinite. 331 00:16:19,42 --> 00:16:21,63 So in principle, if I had enough blocks, I could 332 00:16:21,63 --> 00:16:23,61 get it over to here. 333 00:16:25,298 --> 00:16:27,6 All right, and that's the meaning of divergence 334 00:16:27,6 --> 00:16:28,3 in this case. 335 00:16:28,3 --> 00:16:32,43 336 00:16:32,43 --> 00:16:36,85 On the other hand, I want to discuss with you, and the 337 00:16:36,85 --> 00:16:39,84 reason why I use this example, is I want to discuss with you 338 00:16:39,84 --> 00:16:45,7 also what's going on with this other inequality here, and 339 00:16:45,7 --> 00:16:47,04 what its significance is. 340 00:16:49,36 --> 00:16:52,97 Which is that it's going to take us a lot of numbers N, 341 00:16:52,97 --> 00:16:56,565 a lot of blocks, to get up to a certain level. 342 00:16:57,05 --> 00:16:59,35 In other words, I can't do it with just eight 343 00:16:59,35 --> 00:17:00,49 blocks or nine blocks. 344 00:17:00,8 --> 00:17:02,3 In order to get over here, I'd have to use 345 00:17:02,3 --> 00:17:04,65 quite a few of them. 346 00:17:06 --> 00:17:08,62 So let's just see how many it is. 347 00:17:08,62 --> 00:17:11,12 348 00:17:11,12 --> 00:17:12,975 So I worked this out carefully. 349 00:17:14,23 --> 00:17:15,645 And let's see what I got. 350 00:17:15,645 --> 00:17:18,15 351 00:17:18,15 --> 00:17:30,78 So to get across the lab tables, all right. 352 00:17:31,76 --> 00:17:34,406 This distance here, I already did this secretly. 353 00:17:35,78 --> 00:17:39,07 But I don't actually even have enough of these to show you. 354 00:17:39,69 --> 00:17:42,85 Well 1, 2, 3, 4, 5, 6, and 1/2. 355 00:17:43,85 --> 00:17:44,75 I guess that's enough. 356 00:17:44,75 --> 00:17:45,83 So it's 6 and 1/2. 357 00:17:46,24 --> 00:17:49,03 So it's two lab tables is 13 of these blocks. 358 00:17:50 --> 00:17:50,37 All right. 359 00:17:51,19 --> 00:17:58,01 So they're 13 blocks, which is equal to 26 units. 360 00:18:00,31 --> 00:18:03,535 OK, that's how far to get across I need. 361 00:18:04,02 --> 00:18:06,09 And the first one is already 2. 362 00:18:06,09 --> 00:18:08,59 So it's really 26 minus 2, which is 24. 363 00:18:09,47 --> 00:18:10,78 Which that's what I need. 364 00:18:11,45 --> 00:18:11,84 OK. 365 00:18:13,11 --> 00:18:22,91 So I need log N to be equal to 24, roughly speaking, 366 00:18:22,91 --> 00:18:24,23 in order to get that far. 367 00:18:25,09 --> 00:18:27,82 So let's just see how big that is. 368 00:18:28,195 --> 00:18:28,825 All right. 369 00:18:30,3 --> 00:18:31,76 I think I worked this out. 370 00:18:31,76 --> 00:18:42,01 371 00:18:42,01 --> 00:18:43,17 So let's see. 372 00:18:43,17 --> 00:18:49,27 That means that N is equal to either the 24th-- and if you 373 00:18:49,27 --> 00:18:59,14 realize that these blocks are 3 centimeters high-- OK let's see 374 00:18:59,14 --> 00:19:00,45 how many that we would need here. 375 00:19:00,45 --> 00:19:01,39 That's kind of a lot. 376 00:19:02,21 --> 00:19:09,71 Let's see, it's 3 centimeters times either the 24th, which 377 00:19:09,71 --> 00:19:14,012 is about 8 times 10 to the 8th meters. 378 00:19:15,66 --> 00:19:15,925 OK. 379 00:19:17,6 --> 00:19:20,81 And that is twice the distance to the moon. 380 00:19:20,81 --> 00:19:32,31 381 00:19:32,31 --> 00:19:33,72 So OK, I could do it maybe. 382 00:19:35,19 --> 00:19:37,53 But I would need a lot of blocks. 383 00:19:37,98 --> 00:19:38,435 Right? 384 00:19:38,435 --> 00:19:41,15 So that's not very plausible here, all right. 385 00:19:42,19 --> 00:19:45,5 So those of you who voted against this were actually 386 00:19:45,5 --> 00:19:46,64 sort of half right. 387 00:19:47,08 --> 00:19:50,34 And in fact, if you wanted to get it to the wall over there, 388 00:19:50,34 --> 00:19:53,9 which is over 30 feet, the height would be about 389 00:19:53,9 --> 00:19:56,385 the diameter of the observable universe. 390 00:19:57,36 --> 00:19:59,13 That's kind of a long way. 391 00:20:01,58 --> 00:20:05,92 There's one other thing that I wanted to point out to 392 00:20:05,92 --> 00:20:08,385 you about this shape here. 393 00:20:09,34 --> 00:20:13,61 Which is that if you lean to the left, right? 394 00:20:13,61 --> 00:20:15,58 If you put your head like this-- of course you have to be 395 00:20:15,58 --> 00:20:23,92 on your side to look at it-- this curve is the shape 396 00:20:23,92 --> 00:20:25,41 of a logarithmic curve. 397 00:20:25,81 --> 00:20:29,24 So in other words, if you think of the vertical as the x-axis, 398 00:20:29,24 --> 00:20:32,68 and the horizontal that way, is the vertical, is the up 399 00:20:32,68 --> 00:20:36,64 direction, then this thing is growing very, very, 400 00:20:36,64 --> 00:20:37,61 very, very slowly. 401 00:20:38,51 --> 00:20:43,46 If you send the x-axis all the way up to the moon, the graph 402 00:20:43,46 --> 00:20:46,75 still hasn't gotten across the lab tables here. 403 00:20:47,54 --> 00:20:48,905 It's only partway there. 404 00:20:48,905 --> 00:20:52,29 If you go twice the distance to the moon up that way, it's 405 00:20:52,29 --> 00:20:54,34 gotten finally to that end. 406 00:20:54,34 --> 00:20:56,645 All right so that's how slowly the logarithm grows. 407 00:20:56,99 --> 00:20:58,275 It grows very, very slowly. 408 00:20:58,65 --> 00:21:00,46 And if you look at it another way, if you stand on your 409 00:21:00,46 --> 00:21:03,626 head, you can see an exponential curve. 410 00:21:05,42 --> 00:21:08,47 So you get some sense as to the growth properties 411 00:21:08,47 --> 00:21:09,255 of these functions. 412 00:21:10,42 --> 00:21:17,23 And fortunately these are protecting us from all kinds of 413 00:21:17,23 --> 00:21:20,2 stuff that would happen if there weren't exponentially 414 00:21:20,2 --> 00:21:21,43 small tails in the world. 415 00:21:21,75 --> 00:21:23,91 Like you know, I could walk through this wall which 416 00:21:23,91 --> 00:21:24,84 I wouldn't like doing. 417 00:21:27 --> 00:21:32,15 OK, now so this is our last example. 418 00:21:32,41 --> 00:21:35,1 And the important number, unfortunately we didn't 419 00:21:35,1 --> 00:21:36,515 discover another important number. 420 00:21:36,79 --> 00:21:39,085 There wasn't an amazing number place where this stopped. 421 00:21:40,25 --> 00:21:43,405 All we discovered again is some property of infinite. 422 00:21:44,16 --> 00:21:45,72 So infinite is still a nice number. 423 00:21:46,15 --> 00:21:50,94 And the theme here is just that infinite isn't just one thing, 424 00:21:50,94 --> 00:21:54,135 it has a character which is a rate of growth. 425 00:21:54,46 --> 00:21:55,84 And you shouldn't just think of there being 426 00:21:55,84 --> 00:21:56,93 one order of infinite. 427 00:21:57,33 --> 00:21:58,53 There are lots of different orders. 428 00:21:58,53 --> 00:22:00,19 And some of them have different meaning from others. 429 00:22:01,28 --> 00:22:05,69 All right so that's the theme I wanted to do, and just have a 430 00:22:05,69 --> 00:22:07,126 visceral example of infinite. 431 00:22:07,9 --> 00:22:13,23 Now, we're going to move on now to some other 432 00:22:13,23 --> 00:22:15,476 kinds of techniques. 433 00:22:16,21 --> 00:22:18,965 And this is going to be our last subject. 434 00:22:20,79 --> 00:22:24,02 What we're going to talk about is what are 435 00:22:24,02 --> 00:22:25,4 known as power series. 436 00:22:27,32 --> 00:22:29,92 And we've already seen our first power series. 437 00:22:29,92 --> 00:22:32,71 438 00:22:32,71 --> 00:22:35,24 And I'm going to remind you of that. 439 00:22:35,24 --> 00:22:43,98 440 00:22:43,98 --> 00:22:45,49 Here we are with power series. 441 00:22:45,49 --> 00:22:51,49 442 00:22:51,49 --> 00:22:53,98 Our first series was this one. 443 00:22:53,98 --> 00:22:58,87 444 00:22:58,87 --> 00:23:02,89 And we mentioned last time that it was equal to 1 over 1 445 00:23:02,89 --> 00:23:06,43 minus x, for x less than 1. 446 00:23:06,43 --> 00:23:09,77 447 00:23:09,77 --> 00:23:11,67 Well this one is known as the geometric series. 448 00:23:12,04 --> 00:23:14,68 You didn't use the letter x last time, I used the letter a. 449 00:23:15,05 --> 00:23:16,91 But this is known as the geometric series. 450 00:23:16,91 --> 00:23:24,95 451 00:23:24,95 --> 00:23:31,55 Now I'm going to show you one reason why this is true, 452 00:23:31,55 --> 00:23:32,635 why the formula holds. 453 00:23:33,9 --> 00:23:37,68 And it's just the kind of manipulation that was done 454 00:23:37,68 --> 00:23:39,273 when these things were first introduced. 455 00:23:41,27 --> 00:23:43,965 And here's the idea of a proof. 456 00:23:46,25 --> 00:23:53,38 So suppose that this sum is equal to some number x, 457 00:23:53,38 --> 00:23:57,36 which is the sum of all of these numbers here. 458 00:23:57,36 --> 00:24:00,54 459 00:24:00,54 --> 00:24:02,45 The first thing that I'm going to do is I'm 460 00:24:02,45 --> 00:24:04,02 going to multiply by x. 461 00:24:05,5 --> 00:24:09,6 OK, so if I multiply by x-- let's think about that-- I 462 00:24:09,6 --> 00:24:13,81 multiply by x on both the left and the right-hand side. 463 00:24:13,81 --> 00:24:19,595 Then on the left side, I get x plus x-squared plus x-cubed 464 00:24:19,595 --> 00:24:20,665 plus, and so forth. 465 00:24:20,99 --> 00:24:23,18 And on the right side, I get S times x. 466 00:24:23,18 --> 00:24:27,38 467 00:24:27,38 --> 00:24:31,15 And now I'm going to subtract the two equations, 468 00:24:31,15 --> 00:24:32,08 one from the other. 469 00:24:32,91 --> 00:24:35,85 And there's a very, very substantial cancellation. 470 00:24:36,23 --> 00:24:39,18 This whole tail here gets canceled off. 471 00:24:39,18 --> 00:24:40,69 And the only thing that's left is the 1. 472 00:24:41,03 --> 00:24:45,63 So when I subtract, I get 1 on the left-hand side. 473 00:24:45,95 --> 00:24:51,52 And on the right-hand side, I get S minus S times x. 474 00:24:53,09 --> 00:24:53,5 All right? 475 00:24:53,5 --> 00:24:58,9 476 00:24:58,9 --> 00:25:02,735 And now that can be rewritten as S times 1 minus x. 477 00:25:03,85 --> 00:25:06,18 And so I've got my formula here. 478 00:25:06,18 --> 00:25:10,214 This is 1 over 1 minus x is equal to S. 479 00:25:12,599 --> 00:25:13,553 All right. 480 00:25:13,553 --> 00:25:17,85 481 00:25:17,85 --> 00:25:24,455 Now this reasoning has one flaw. 482 00:25:26,34 --> 00:25:27,273 It's not complete. 483 00:25:28,12 --> 00:25:31,6 And this reasoning is basically correct. 484 00:25:33,08 --> 00:25:45,68 But it's incomplete because it requires that S exists. 485 00:25:45,68 --> 00:25:50,27 486 00:25:50,27 --> 00:25:52,97 For example, it doesn't make any sense in 487 00:25:52,97 --> 00:25:54,33 the case x equals 1. 488 00:25:55,1 --> 00:25:59,05 So for example in the case x equals 1, we have 1 plus 489 00:25:59,05 --> 00:26:03,25 1 plus 1 plus et cetera, equals whatever we call S. 490 00:26:03,76 --> 00:26:06,37 And then when we multiply through by 1, we get 1 plus 1 491 00:26:06,37 --> 00:26:10,28 plus 1 plus, equals S times 1. 492 00:26:10,64 --> 00:26:13,06 And now you see that the subtraction gives us infinite 493 00:26:13,06 --> 00:26:15,85 minus infinite is equal to infinite minus infinite. 494 00:26:16,33 --> 00:26:18,5 That's what's really going on in the argument 495 00:26:18,5 --> 00:26:19,79 in this context. 496 00:26:19,79 --> 00:26:21,005 So it's just nonsense. 497 00:26:21,34 --> 00:26:23,145 I mean it doesn't give us anything meaningful. 498 00:26:24,38 --> 00:26:27,09 So this argument, it's great. 499 00:26:27,09 --> 00:26:29,266 And it gives us the right answer, but not always. 500 00:26:30 --> 00:26:33,13 And the times when it gives us the answer, the correct answer, 501 00:26:33,13 --> 00:26:35,325 is when the series is convergent. 502 00:26:36,8 --> 00:26:38,51 And that's why we care about convergence. 503 00:26:38,8 --> 00:26:42,06 Because we want manipulations like this to be allowed. 504 00:26:42,06 --> 00:26:47,78 505 00:26:47,78 --> 00:26:50,62 So the good case, this is the red case that we were 506 00:26:50,62 --> 00:26:53,876 describing last time, that's the bad case. 507 00:26:54,79 --> 00:26:58,18 What we want is the good case, the convergent case. 508 00:26:58,82 --> 00:27:01,42 And that is the case when x is less than 1. 509 00:27:01,77 --> 00:27:03,182 So this is the convergent case. 510 00:27:03,182 --> 00:27:11,29 511 00:27:11,29 --> 00:27:11,625 Yep. 512 00:27:11,96 --> 00:27:14,64 OK, so they're much more detailed things to check 513 00:27:14,64 --> 00:27:15,71 exactly what's going on. 514 00:27:15,71 --> 00:27:18,24 But I'm going to just say general words about how you 515 00:27:18,24 --> 00:27:19,315 recognize convergence. 516 00:27:20,16 --> 00:27:22,37 And then we're not going to worry about so much about 517 00:27:22,37 --> 00:27:24,45 convergence, because it works very, very well. 518 00:27:24,74 --> 00:27:27,7 And it's always easy to diagnose when there's 519 00:27:27,7 --> 00:27:29,21 convergence with a power series. 520 00:27:31,59 --> 00:27:33,45 All right so here's the general setup. 521 00:27:33,45 --> 00:27:44,7 522 00:27:44,7 --> 00:27:50,13 The general setup is that we have not just the coefficients 523 00:27:50,13 --> 00:27:56,385 1 all the time, but any numbers here, dot, dot, dot. 524 00:27:56,66 --> 00:27:59,04 And we abbreviate that with the summation notation. 525 00:27:59,31 --> 00:28:04,15 This is the sum a n x to the n, n equals 0 to infinite. 526 00:28:05,36 --> 00:28:07,26 And that's what's known as a power series. 527 00:28:07,26 --> 00:28:12,37 528 00:28:12,37 --> 00:28:17,66 Fortunately there is a very simple rule about how 529 00:28:17,66 --> 00:28:19,573 power series converge. 530 00:28:20,26 --> 00:28:21,21 And it's the following. 531 00:28:23,2 --> 00:28:28,48 There's a magic number R which depends on these numbers here 532 00:28:28,48 --> 00:28:30,98 such that-- and this thing is known as a radius of 533 00:28:30,98 --> 00:28:38,12 convergence-- and the problem that we had, 534 00:28:38,12 --> 00:28:39,745 it's this number 1 here. 535 00:28:40,09 --> 00:28:42,04 This thing works for x less than 1. 536 00:28:43,03 --> 00:28:45,37 In our case, it's maybe x less R. 537 00:28:46,08 --> 00:28:47,96 So that's some symmetric interval, right? 538 00:28:47,96 --> 00:28:55,39 That's the same as minus R, less then x, less than R, and 539 00:28:55,39 --> 00:28:56,64 so where there's convergence. 540 00:28:58,31 --> 00:29:00,81 OK, where the series converges. 541 00:29:00,81 --> 00:29:07,21 542 00:29:07,21 --> 00:29:13,27 And then there's the region where every computation that 543 00:29:13,27 --> 00:29:14,485 you give will give you nonsense. 544 00:29:16,5 --> 00:29:24,74 So x greater than R is the sum a sub n x to the n, diverges. 545 00:29:24,74 --> 00:29:28,71 546 00:29:28,71 --> 00:29:43,37 And x equals R is very delicate, borderline, and 547 00:29:43,37 --> 00:29:44,99 will not be used by us. 548 00:29:44,99 --> 00:29:50,06 549 00:29:50,06 --> 00:29:52,915 OK we're going to stick inside the radius of convergence. 550 00:29:54,75 --> 00:29:57,62 Now the way you'll be able to recognize this, 551 00:29:57,62 --> 00:29:58,465 is the following. 552 00:29:58,92 --> 00:30:05,37 What always happens is that these numbers tend to 0 553 00:30:05,37 --> 00:30:19,98 exponentially fast for x in R, and doesn't even tend to 0 at 554 00:30:19,98 --> 00:30:24,97 all for x greater than R. 555 00:30:26,6 --> 00:30:29,605 Right so it'll be totally obvious. 556 00:30:30,2 --> 00:30:33,495 When you look at this series here, what's happening when x 557 00:30:33,495 --> 00:30:35,65 less than R is that the numbers are getting smaller and 558 00:30:35,65 --> 00:30:37,99 smaller, less than 1. 559 00:30:37,99 --> 00:30:39,56 When x is bigger than 1, the numbers are getting 560 00:30:39,56 --> 00:30:40,1 bigger and bigger. 561 00:30:40,1 --> 00:30:41,85 There's no chance that the series converges. 562 00:30:42,85 --> 00:30:45,456 So that's going to be the case with all power series. 563 00:30:45,87 --> 00:30:47,26 There's going to be a cut off. 564 00:30:47,68 --> 00:30:49,44 And it'll be one particular number. 565 00:30:49,81 --> 00:30:52,04 And below that it'll be obvious that you have convergence, and 566 00:30:52,04 --> 00:30:52,895 you'll be able to do computations. 567 00:30:53,26 --> 00:30:55,69 And above that every formula will be wrong 568 00:30:55,69 --> 00:30:56,45 and won't make sense. 569 00:30:57,22 --> 00:30:58,89 So it's a very clean thing. 570 00:30:59,37 --> 00:31:02,43 There is this very subtle borderline, but we're not going 571 00:31:02,43 --> 00:31:04,35 to discuss that in this class. 572 00:31:04,35 --> 00:31:08,91 And it's actually not used in direct studies of power series. 573 00:31:10,383 --> 00:31:13,82 AUDIENCE: How can you tell when the numbers are declining 574 00:31:13,82 --> 00:31:16,78 exponentially fast, whereas just, in other words 1 575 00:31:16,78 --> 00:31:17,735 over x [INAUDIBLE]? 576 00:31:18,03 --> 00:31:21,66 PROFESSOR: OK so, the question is why was I able to 577 00:31:21,66 --> 00:31:23,07 tell you this word here? 578 00:31:23,07 --> 00:31:25,81 Why was I able to tell you not only is it going to 0, but it's 579 00:31:25,81 --> 00:31:27,04 going exponentially fast? 580 00:31:27,55 --> 00:31:29 I'm telling you extra information. 581 00:31:29 --> 00:31:31,79 I'm telling you it always goes exponentially fast. 582 00:31:32,38 --> 00:31:33,556 You can identify it. 583 00:31:33,556 --> 00:31:36,15 584 00:31:36,15 --> 00:31:37,62 In other words, you'll see it. 585 00:31:37,62 --> 00:31:39,155 And it will happen every single time. 586 00:31:39,49 --> 00:31:41,34 I'm just promising you that it works that way. 587 00:31:41,6 --> 00:31:44,83 And it's really for the same reason that it works that way 588 00:31:44,83 --> 00:31:45,785 here, that these are powers. 589 00:31:46,84 --> 00:31:49,43 And what's going on over here is there are, it's close to 590 00:31:49,43 --> 00:31:51,02 powers with this a sub ns. 591 00:31:52,181 --> 00:31:53,04 All right? 592 00:31:54,64 --> 00:31:56,96 There's a long discussion of radius of convergence 593 00:31:56,96 --> 00:31:57,71 in many textbooks. 594 00:31:58,35 --> 00:32:03,23 But really it's not necessary, all right, for this purpose? 595 00:32:04,67 --> 00:32:05 Yeah? 596 00:32:05,418 --> 00:32:07,09 AUDIENCE: How do you find R? 597 00:32:07,09 --> 00:32:08,4 PROFESSOR: The question was how do you find R? 598 00:32:08,4 --> 00:32:10,43 Yes, so I just said, there's a long discussion for how 599 00:32:10,43 --> 00:32:12,13 you find the radius of convergence in textbooks. 600 00:32:12,66 --> 00:32:15,05 But we will not be discussing that here. 601 00:32:15,8 --> 00:32:17,68 And it won't be necessary for you. 602 00:32:17,68 --> 00:32:20,89 Because it will be obvious in any given series what the R is. 603 00:32:20,89 --> 00:32:22,755 It will always either 1 or infinite. 604 00:32:23,65 --> 00:32:25,85 It will always work for all x, or maybe it'll 605 00:32:25,85 --> 00:32:26,82 stop at some point. 606 00:32:26,82 --> 00:32:30,12 But it'll be very clear where it stops, as it is for 607 00:32:30,12 --> 00:32:31,92 the geometric series. 608 00:32:33,201 --> 00:32:34,1 All right? 609 00:32:36,07 --> 00:32:39,96 OK so now I need to give you the basic facts, and 610 00:32:39,96 --> 00:32:40,805 give you a few examples. 611 00:32:40,805 --> 00:32:44,4 612 00:32:44,4 --> 00:32:46,42 So why are we looking at these series? 613 00:32:46,42 --> 00:32:51,24 614 00:32:51,24 --> 00:32:56,09 Well the answer is we're looking at these series because 615 00:32:56,09 --> 00:33:00,06 the role that they play is exactly the reverse of 616 00:33:00,06 --> 00:33:02,03 this equation here. 617 00:33:03,42 --> 00:33:06,54 That is, and this is a theme which I have tried to emphasize 618 00:33:06,54 --> 00:33:09,555 throughout this course, you can read equalities in 619 00:33:09,555 --> 00:33:10 two directions. 620 00:33:11,87 --> 00:33:14,49 Both are interesting, typically. 621 00:33:15,61 --> 00:33:18,22 You can think, I don't know what the value of this is. 622 00:33:18,22 --> 00:33:19,24 Here's a way of evaluating. 623 00:33:19,24 --> 00:33:22,06 And in other words, the right side is a formula 624 00:33:22,06 --> 00:33:22,825 for the left side. 625 00:33:23,09 --> 00:33:26,18 Or you can think of the left side as being a formula 626 00:33:26,18 --> 00:33:27,14 for the right side. 627 00:33:27,14 --> 00:33:30,72 628 00:33:30,72 --> 00:33:34,35 And the idea of series is that they're flexible enough to 629 00:33:34,35 --> 00:33:36,65 represent all of the functions that we've encountered 630 00:33:36,65 --> 00:33:37,43 in this course. 631 00:33:39,3 --> 00:33:41,78 This is the tool which is very much like the decimal 632 00:33:41,78 --> 00:33:43,88 expansion which allows you to represent numbers like 633 00:33:43,88 --> 00:33:44,76 the square root of 2. 634 00:33:45,13 --> 00:33:47,52 Now we're going to be representing all the numbers, 635 00:33:47,52 --> 00:33:49,95 all the functions that we know, e to the x, 636 00:33:49,95 --> 00:33:52,24 arctangent, sine, cosine. 637 00:33:52,55 --> 00:33:55,44 All of those functions become completely flexible, and 638 00:33:55,44 --> 00:33:57,56 completely available to us, and computationally 639 00:33:57,56 --> 00:33:59,226 available to us directly. 640 00:33:59,9 --> 00:34:01,44 So that's what this is a tool for. 641 00:34:01,44 --> 00:34:04,25 And it's just like decimal expansions giving you handle 642 00:34:04,25 --> 00:34:05,24 on all real numbers. 643 00:34:05,24 --> 00:34:08,92 644 00:34:08,92 --> 00:34:09,975 So here's how it works. 645 00:34:12,11 --> 00:34:28,67 The rules for convergent power series are just 646 00:34:28,67 --> 00:34:33,92 like polynomials. 647 00:34:33,92 --> 00:34:41,28 648 00:34:41,28 --> 00:34:43,96 All of the manipulations that you do for power series are 649 00:34:43,96 --> 00:34:45,49 essentially the same as for polynomials. 650 00:34:46,12 --> 00:34:48,14 So what kinds of things do we do with polynomials? 651 00:34:49,19 --> 00:34:50,13 We add them. 652 00:34:50,13 --> 00:34:53,28 653 00:34:53,28 --> 00:34:54,445 We multiply them together. 654 00:34:54,445 --> 00:34:57,63 655 00:34:57,63 --> 00:34:58,695 We do substitutions. 656 00:35:01,06 --> 00:35:01,29 Right? 657 00:35:01,29 --> 00:35:03,04 We take one function of another function. 658 00:35:04,97 --> 00:35:06,03 We divide them. 659 00:35:06,03 --> 00:35:10,11 660 00:35:10,11 --> 00:35:10,41 OK. 661 00:35:11,59 --> 00:35:14,665 And these are all really not very surprising operations. 662 00:35:15,04 --> 00:35:17,29 And we will be able to do them with power series too. 663 00:35:17,71 --> 00:35:20,5 The ones that are interesting, really interesting for 664 00:35:20,5 --> 00:35:23,58 calculus, are the last two. 665 00:35:24,36 --> 00:35:29,79 We differentiate them, and we integrate them. 666 00:35:29,79 --> 00:35:34,07 667 00:35:34,07 --> 00:35:36,74 And all of these operations we'll be able to do for 668 00:35:36,74 --> 00:35:38,11 power series as well. 669 00:35:38,11 --> 00:35:42,95 670 00:35:42,95 --> 00:35:48,805 So now let's explain the high points of this. 671 00:35:49,12 --> 00:35:51,66 Which is mainly just the differentiation and 672 00:35:51,66 --> 00:35:52,656 the integration part. 673 00:35:53,11 --> 00:36:04,02 So if I take a series like this and so forth, the formula for 674 00:36:04,02 --> 00:36:07,6 it's derivative is just like polynomials. 675 00:36:08,29 --> 00:36:10,04 That's what I just said, it's just like polynomials. 676 00:36:10,22 --> 00:36:12,24 So the derivative of the constant is 0. 677 00:36:12,57 --> 00:36:15,375 The derivative of this term is a sub 1. 678 00:36:15,85 --> 00:36:18,845 This one is plus 2 a sub 2 x. 679 00:36:19,34 --> 00:36:23,555 This one is 3a sub 3 x-squared, et cetera. 680 00:36:23,92 --> 00:36:24,69 That's the formula. 681 00:36:27,07 --> 00:36:41,82 Similarly if I integrate, well there's an unknown constant 682 00:36:41,82 --> 00:36:44,03 which I'm going to put first rather than last. 683 00:36:44,93 --> 00:36:47,08 Which corresponds sort of to the a sub 0, term which is 684 00:36:47,08 --> 00:36:48,01 going to get wiped out. 685 00:36:48,01 --> 00:36:50,51 That a sub 0 term suddenly becomes a sub 0 x. 686 00:36:50,77 --> 00:36:54,73 And the anti-derivative of this next term is a sub 687 00:36:54,73 --> 00:36:56,18 1 x-squared over 2. 688 00:36:56,98 --> 00:37:02,68 And the next term is a sub 2 x-cubed over 3, and so forth. 689 00:37:02,68 --> 00:37:05,2 690 00:37:05,2 --> 00:37:06,01 Yeah, question? 691 00:37:06,456 --> 00:37:08,69 AUDIENCE: Is that a series or a polynomial? 692 00:37:08,69 --> 00:37:09,9 PROFESSOR: Is this a series or a polynomial? 693 00:37:10,19 --> 00:37:10,775 Good question. 694 00:37:11,18 --> 00:37:14,275 It's a polynomial if it ends. 695 00:37:14,7 --> 00:37:18,56 If it goes on infinitely far, then it's a series. 696 00:37:19,49 --> 00:37:22,01 They look practically the same, polynomials and series. 697 00:37:22,42 --> 00:37:24,532 There's this little dot, dot, dot here. 698 00:37:26,4 --> 00:37:27,66 Is this a series or a polynomial? 699 00:37:27,66 --> 00:37:28,51 It's the same rule. 700 00:37:28,78 --> 00:37:30,82 If it stops at a finite stage, this one stops 701 00:37:30,82 --> 00:37:31,23 at a finite stage. 702 00:37:32,21 --> 00:37:34,1 If it goes on forever, it goes on forever. 703 00:37:35,074 --> 00:37:37,98 AUDIENCE: So I thought that the series add up finite numbers. 704 00:37:37,98 --> 00:37:41,735 You can add up terms of x in series? 705 00:37:41,735 --> 00:37:45,52 706 00:37:45,52 --> 00:37:47,38 PROFESSOR: So an interesting question. 707 00:37:47,38 --> 00:37:51,35 So the question that was just asked is I thought that a 708 00:37:51,35 --> 00:37:53,06 series added up finite numbers. 709 00:37:53,06 --> 00:37:54,215 You could add up x? 710 00:37:55,14 --> 00:37:56,405 That was what you said, right? 711 00:37:56,9 --> 00:38:02,68 OK now notice that I pulled that off on you by changing the 712 00:38:02,68 --> 00:38:07,35 letter a to the letter x at the very beginning of this 713 00:38:07,35 --> 00:38:08,42 commentary here. 714 00:38:10,17 --> 00:38:11,36 This is a series. 715 00:38:11,76 --> 00:38:15,17 For each individual value of x, it's a number. 716 00:38:15,63 --> 00:38:17,964 So in other words, it I plug in here x equals 1/2, I'm going to 717 00:38:17,964 --> 00:38:21,09 add 1 plus 1/2 plus 1/4 plus 1/8; and I'll get a 718 00:38:21,09 --> 00:38:22,07 number which is 2. 719 00:38:22,47 --> 00:38:24,62 And I'll plug in a number over here, and I'll get a number. 720 00:38:25,03 --> 00:38:27,26 On the other hand, I can do this for each value of x. 721 00:38:27,82 --> 00:38:30,71 So the interpretation of this is that it's a function of x. 722 00:38:31,77 --> 00:38:33,57 And similarly this is a function of x. 723 00:38:34,38 --> 00:38:37,77 It works when you plug in the possible values x 724 00:38:37,77 --> 00:38:39,86 between minus 1 and 1. 725 00:38:42,33 --> 00:38:45,06 So there's really no distinction there, just I 726 00:38:45,06 --> 00:38:46,22 slipped it passed you. 727 00:38:47,01 --> 00:38:48,13 These are functions of x. 728 00:38:48,13 --> 00:38:50,79 729 00:38:50,79 --> 00:38:53,29 And the notion of a power series is this idea that you 730 00:38:53,29 --> 00:38:55,86 put coefficients on a series, but then you allow yourself 731 00:38:55,86 --> 00:38:58,02 the flexibility to stick powers here. 732 00:38:58,99 --> 00:39:00,76 And that's exactly what we're doing. 733 00:39:02,12 --> 00:39:04,12 OK there are other kinds if series where you stick other 734 00:39:04,12 --> 00:39:06,12 interesting functions in here like sines and cosines. 735 00:39:06,12 --> 00:39:08,62 There are lots of other series that people study. 736 00:39:08,62 --> 00:39:10,465 And these are the simplest ones. 737 00:39:10,465 --> 00:39:13,37 And all those examples are extremely helpful for 738 00:39:13,37 --> 00:39:14,515 representing functions. 739 00:39:14,515 --> 00:39:16,52 But we're only going to do this example here. 740 00:39:18,92 --> 00:39:22,876 All right, so here are the two rules. 741 00:39:23,61 --> 00:39:30,52 And now there's only one other complication here which I have 742 00:39:30,52 --> 00:39:36,87 to explain to you before giving you a bunch of examples to show 743 00:39:36,87 --> 00:39:39,02 you that this works extremely well. 744 00:39:41,27 --> 00:39:45,09 And the last thing that I have to do for you is explain to 745 00:39:45,09 --> 00:39:46,58 you something called Taylor's formula. 746 00:39:46,58 --> 00:39:55,28 747 00:39:55,28 --> 00:40:00,73 Taylor's formula is the way you get from the representations 748 00:40:00,73 --> 00:40:04,13 that we're used to a functions, to a representation in the 749 00:40:04,13 --> 00:40:05,26 form of these coefficients. 750 00:40:06,18 --> 00:40:08,62 When I gave you the function e to the x, it didn't 751 00:40:08,62 --> 00:40:09,53 look like a polynomial. 752 00:40:11,28 --> 00:40:15,08 And we have to figure out which of these guys it is, if it's 753 00:40:15,08 --> 00:40:17,74 going to fall into our category here. 754 00:40:19,09 --> 00:40:20,045 And here's the formula. 755 00:40:20,37 --> 00:40:22,18 I'll explain to you how it works in a second. 756 00:40:23,43 --> 00:40:26,67 So the formula is f of x, turns out there's a formula in terms 757 00:40:26,67 --> 00:40:27,84 of the derivatives of f. 758 00:40:28,56 --> 00:40:31,99 Namely, you differentiate n times, and you evaluate it 759 00:40:31,99 --> 00:40:35,18 at 0, and you divide by n factorial, and multiply 760 00:40:35,18 --> 00:40:36,237 by x to the n. 761 00:40:37,64 --> 00:40:39,49 So here's Taylor's formula. 762 00:40:40,51 --> 00:40:43,46 This tells you what the Taylor series is. 763 00:40:44,27 --> 00:40:48,72 Now about half of our job for the next few minutes is going 764 00:40:48,72 --> 00:40:51,23 to be to give examples of this. 765 00:40:51,63 --> 00:40:55,92 But let me just explain to you why this has to be. 766 00:40:56,81 --> 00:41:00,87 If you pick out this number here, this is the a sub n, the 767 00:41:00,87 --> 00:41:02,75 magic number a sub n here. 768 00:41:03,29 --> 00:41:04,982 So let's just illustrate it. 769 00:41:05,45 --> 00:41:11,053 If f of x happens to be a sub zero plus a sub 1 x plus a sub 770 00:41:11,053 --> 00:41:15,94 2 x-squared plus a sub 2 x-cubed plus dot, dot, dot. 771 00:41:15,94 --> 00:41:18,875 And now I differentiate it, right? 772 00:41:19,81 --> 00:41:27,15 I get a sub 1 plus 2 sub a 2x plus 3 a sub 3 x If I 773 00:41:27,15 --> 00:41:34,75 differentiate it another time, I get 2 a sub 2, plus 3 times 2 774 00:41:34,75 --> 00:41:38,56 a sub 3 x plus dot, dot, dot. 775 00:41:38,56 --> 00:41:45,93 And now a third time, I get 3 times 2 a sub 3 plus et cetera. 776 00:41:47,36 --> 00:41:50,39 So this next term is really in disguise, 4 times 777 00:41:50,39 --> 00:41:54,482 3 times 2 a sub 4x. 778 00:41:56,41 --> 00:41:58,66 That's what really comes down if I kept track of 779 00:41:58,66 --> 00:41:59,62 the fourth term there. 780 00:42:00,98 --> 00:42:02,67 So now here is my function. 781 00:42:04,66 --> 00:42:08,5 But now you see if I plug in x equals 0, I can 782 00:42:08,5 --> 00:42:11,765 pick off the third term. 783 00:42:13,75 --> 00:42:19,87 f triple prime of 0 is equal to 3 times 2 times a sub 3. 784 00:42:21,59 --> 00:42:23,08 Right, because all the rest of those terms when I 785 00:42:23,08 --> 00:42:24,18 plug in 0 are just 0. 786 00:42:25,08 --> 00:42:26,265 Here's the formula. 787 00:42:26,73 --> 00:42:30,61 And so the pattern here is this. 788 00:42:30,61 --> 00:42:32,86 And what's really going on here is this is really 3 times 789 00:42:32,86 --> 00:42:34,788 2 times 1 equals a sub 3. 790 00:42:37,11 --> 00:42:48,48 And in general a sub n is equal to f to the nth derivative 791 00:42:48,48 --> 00:42:49,966 divided by n factorial. 792 00:42:50,41 --> 00:42:53,99 And of course, n factorial, I remind you, is n times n minus 793 00:42:53,99 --> 00:42:57,57 1 times n minus 2, all the way down. 794 00:42:57,57 --> 00:43:02,6 795 00:43:02,6 --> 00:43:05,98 Now there's one more crazy convention 796 00:43:05,98 --> 00:43:07,125 which is always used. 797 00:43:08,22 --> 00:43:12,3 Which is that there's something very strange here down at 0, 798 00:43:12,3 --> 00:43:15,21 which is that 0 factorials turns out, have to 799 00:43:15,21 --> 00:43:16,57 be set equal to 1. 800 00:43:16,99 --> 00:43:19,58 All right, so that's what you do in order to make 801 00:43:19,58 --> 00:43:20,54 this formula work out. 802 00:43:20,54 --> 00:43:22,425 And that's one of the reasons for this convention. 803 00:43:22,425 --> 00:43:29,47 804 00:43:29,47 --> 00:43:29,846 All right. 805 00:43:30,6 --> 00:43:35,29 So my next goal is to give you some examples. 806 00:43:36,68 --> 00:43:45,28 And let's do a couple. 807 00:43:45,28 --> 00:43:48,13 808 00:43:48,13 --> 00:43:53,99 So here's, well you know, I'm going to have to let you see 809 00:43:53,99 --> 00:43:55,37 a few of them next time. 810 00:43:56,12 --> 00:43:59,23 But let me just tell you this one, which is by 811 00:43:59,23 --> 00:44:00,82 far the most impressive. 812 00:44:00,82 --> 00:44:06,37 813 00:44:06,37 --> 00:44:10,87 So what happens with e to the x, if a function of f of x is 814 00:44:10,87 --> 00:44:18,19 e to the x, is that it's derivative is also x to the x. 815 00:44:18,19 --> 00:44:21,39 And its second derivative is also e to the x. 816 00:44:21,39 --> 00:44:23,01 And it just keeps on going that way. 817 00:44:23,01 --> 00:44:23,94 They're all the same. 818 00:44:25,18 --> 00:44:30,01 So that means that these numbers in Taylor's formula, 819 00:44:30,01 --> 00:44:33,5 in the numerator, the nth derivative is very 820 00:44:33,5 --> 00:44:35,266 easy to evaluate. 821 00:44:37,09 --> 00:44:38,16 It's just e to the x. 822 00:44:39,28 --> 00:44:42,81 And if I evaluated at x equals 0, I just get 1. 823 00:44:43,89 --> 00:44:45,55 So all of those numerators are 1. 824 00:44:46,99 --> 00:44:52,62 So the formula here, is the sum n equals 0 to infinite, of 1 825 00:44:52,62 --> 00:44:55,987 divided by n factorial x to the n. 826 00:44:55,987 --> 00:45:00,96 827 00:45:00,96 --> 00:45:04,56 In particular, we now have an honest formula for 828 00:45:04,56 --> 00:45:05,77 e to the first power. 829 00:45:06,8 --> 00:45:07,89 Which is just e. 830 00:45:08,33 --> 00:45:11,705 Which if I plug it in, x equals 1, I get 1. 831 00:45:11,97 --> 00:45:14,415 This is the n equals 0 term plus 1. 832 00:45:14,75 --> 00:45:19,2 This is the n equals 1 term plus 1 over 2 factorial plus 833 00:45:19,2 --> 00:45:21,991 1 over 3 factorial plus 1 over 4 factorial. 834 00:45:21,991 --> 00:45:27,07 835 00:45:27,07 --> 00:45:27,54 Right? 836 00:45:27,54 --> 00:45:31,36 So this is our first honest formula for e. 837 00:45:31,36 --> 00:45:34,21 And also, this is how you compute the 838 00:45:34,21 --> 00:45:35,34 exponential function. 839 00:45:35,34 --> 00:45:41,75 840 00:45:41,75 --> 00:45:49,98 Finally if you take a function like sine x, what you'll 841 00:45:49,98 --> 00:45:52,78 discover is that we can complete the sort of strange 842 00:45:52,78 --> 00:45:56,57 business that we did at the beginning of the course-- or 843 00:45:56,57 --> 00:46:03,48 cosine x-- where we took the linear and quadratic 844 00:46:03,48 --> 00:46:03,97 approximations. 845 00:46:04,95 --> 00:46:08,62 Now we're going to get complete formulas for these functions. 846 00:46:10,06 --> 00:46:15,18 Sine x turns out to be equal to x minus x-cubed over 3 847 00:46:15,18 --> 00:46:20,62 factorial plus x to the 5th over 5 factorial minus x to the 848 00:46:20,62 --> 00:46:23,5 7th over 7 factorial, et cetera. 849 00:46:24,86 --> 00:46:29,51 And cosine x is equal to 1, minus x-squared over 2 850 00:46:29,51 --> 00:46:35,52 factorial-- that's the same as this 2 here-- plus x to the 4th 851 00:46:35,52 --> 00:46:41,07 over 4 factorial minus x to the 6th over 6 factorial, 852 00:46:41,07 --> 00:46:41,783 plus et cetera. 853 00:46:42,59 --> 00:46:48,03 Now these may feel like they're hard to memorize because I've 854 00:46:48,03 --> 00:46:49,425 just pulled them out of a hat. 855 00:46:49,425 --> 00:46:51,99 856 00:46:51,99 --> 00:46:54,67 I do expect you to know them. 857 00:46:55,63 --> 00:46:58,283 They're actually extremely similar formulas. 858 00:46:58,99 --> 00:47:01,815 The exponential here just has this collection of factorials. 859 00:47:02,23 --> 00:47:06,86 The sine is all the odd powers with alternating sines. 860 00:47:06,86 --> 00:47:09,95 And the cosine is all the even powers with alternating sines. 861 00:47:10,26 --> 00:47:13,815 So all three of them form part of the same family. 862 00:47:14,13 --> 00:47:16,9 So this will actually make it easier for you to remember, 863 00:47:16,9 --> 00:47:18,15 rather than harder. 864 00:47:18,15 --> 00:47:21,49 865 00:47:21,49 --> 00:47:25,38 And so with that, I'll leave the practice on differentiation 866 00:47:25,38 --> 00:47:26,21 for next time. 867 00:47:26,21 --> 00:47:27,57 And good luck, everybody. 868 00:47:27,57 --> 00:47:29,99 I'll talk to individually. 869 00:47:29,99 --> 00:47:30,585