1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:01,87 NARRATOR: The following content is provided under a 3 00:00:01,87 --> 00:00:03,165 Creative Commons license. 4 00:00:03,61 --> 00:00:06,74 Your support will help MIT OpenCourseWare continue to 5 00:00:06,74 --> 00:00:09,23 offer high quality educational resources for free. 6 00:00:09,62 --> 00:00:12,78 To make a donation, or to view additional materials from 7 00:00:12,78 --> 00:00:16,16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,16 --> 00:00:16,9 at ocw.mit.edu. 9 00:00:21,58 --> 00:00:27,7 PROFESSOR: So, Professor Jerison is relaxing in sunny 10 00:00:27,7 --> 00:00:31,8 London, Ontario today and sent me in as his substitute again. 11 00:00:32,13 --> 00:00:33,76 I'm glad to the here and see you all again. 12 00:00:33,76 --> 00:00:39,19 13 00:00:39,19 --> 00:00:43,31 He said that he'd already talked about power series and 14 00:00:43,31 --> 00:00:49,605 Taylor's formula oh on last week right, on Friday? 15 00:00:50,85 --> 00:00:53,99 So I'm going to go a little further with that and show you 16 00:00:53,99 --> 00:00:58,25 some examples, show you some applications, and then I have 17 00:00:58,25 --> 00:01:02,58 this course evaluation survey that I'll hand out in the last 18 00:01:02,58 --> 00:01:03,805 10 minutes or so of the class. 19 00:01:03,805 --> 00:01:06,89 20 00:01:06,89 --> 00:01:10,5 I also have this handout that he made that says 18.01 21 00:01:10,5 --> 00:01:11,44 end of term, 2007. 22 00:01:12,52 --> 00:01:15,69 If you didn't pick this up coming in, grab it going out. 23 00:01:15,69 --> 00:01:17,99 People tend not to pick it up when they walk in, I see. 24 00:01:18,85 --> 00:01:20,28 So grab this when you're going out. 25 00:01:22,01 --> 00:01:23,39 There's some things missing from it. 26 00:01:23,39 --> 00:01:27,37 He has not decided when his office hours will 27 00:01:27,37 --> 00:01:28,52 be at the end of term. 28 00:01:28,52 --> 00:01:30,975 He will have them, just hasn't decided when. 29 00:01:31,28 --> 00:01:34,575 So, check the website for that information. 30 00:01:34,575 --> 00:01:38 31 00:01:38 --> 00:01:40,91 And we're looking forward to the final exam, which 32 00:01:40,91 --> 00:01:47,03 is [UNINTELLIGIBLE] 33 00:01:47,03 --> 00:01:49,59 Any questions about this technical stuff? 34 00:01:49,59 --> 00:01:52,9 35 00:01:52,9 --> 00:01:55,21 All right, let's talk about power series for a little bit. 36 00:01:56,86 --> 00:02:00,56 So I thought I should review for you what the story 37 00:02:00,56 --> 00:02:01,98 with power series is. 38 00:02:01,98 --> 00:02:21,79 39 00:02:21,79 --> 00:02:23,22 OK, could I have your attention please? 40 00:02:23,22 --> 00:02:26,92 41 00:02:26,92 --> 00:02:31,64 So, power series is a way of writing a function as a sum 42 00:02:31,64 --> 00:02:33,165 of interval powers of x. 43 00:02:33,56 --> 00:02:36,413 These a, 0, a1, and so on, are numbers. 44 00:02:38,09 --> 00:02:43,06 An example of a power series is a polynomial. 45 00:02:43,06 --> 00:02:48,05 46 00:02:48,05 --> 00:02:55,35 Not to be forgotten, one type of power series is one which 47 00:02:55,35 --> 00:02:59,51 goes on for a finite number of terms and then ends, so that 48 00:02:59,51 --> 00:03:02,715 all of the all the higher a sub i's are all 0. 49 00:03:03,65 --> 00:03:06,15 This is a perfectly good example of a power series 50 00:03:06,15 --> 00:03:08,075 it's a very special kind of power series. 51 00:03:09,04 --> 00:03:11,94 Part of what I want to tell you today is that power series 52 00:03:11,94 --> 00:03:14,55 behave, if almost exactly like, polynomials. 53 00:03:14,97 --> 00:03:17,66 There's just one thing that you have to be careful about when 54 00:03:17,66 --> 00:03:21,54 you're using power series that isn't a concern for 55 00:03:21,54 --> 00:03:23,93 polynomials, and I'll show you what that is in a minute. 56 00:03:24,54 --> 00:03:26,89 So, you should think of them as generalized polynomials. 57 00:03:29,32 --> 00:03:45,85 The one thing that you have to be careful about is that 58 00:03:45,85 --> 00:03:54,73 there's a number, which I'll call r, where r can be between 59 00:03:54,73 --> 00:03:57,07 0 and it can also be ifninity. 60 00:03:57,35 --> 00:04:03,41 It's a number between 0 and infinity, inclusive, so that 61 00:04:03,41 --> 00:04:06,83 when the absolute value of x is less than r. 62 00:04:06,83 --> 00:04:11,26 So when x is smaller than r in size, the sum converges. 63 00:04:11,26 --> 00:04:17,22 64 00:04:17,22 --> 00:04:20,13 that's sum converges to a finite value. 65 00:04:21,17 --> 00:04:25,12 And when x is bigger than r in absolute value, 66 00:04:25,12 --> 00:04:26,05 the sum diverges. 67 00:04:26,05 --> 00:04:30,26 68 00:04:30,26 --> 00:04:32,165 This r is called the radius of convergence. 69 00:04:32,165 --> 00:04:42,96 70 00:04:42,96 --> 00:04:45,75 So we'll see some examples of what the radius of convergence 71 00:04:45,75 --> 00:04:49,625 is in various powers series as well, and how you find it also. 72 00:04:49,625 --> 00:04:55,89 73 00:04:55,89 --> 00:05:00,55 Let me go on and give you a few more of the properties about 74 00:05:00,55 --> 00:05:02,98 power series which I think that professor Jerison 75 00:05:02,98 --> 00:05:04,99 talked about earlier. 76 00:05:05,41 --> 00:05:07,27 So one of them is there's a radius of convergence. 77 00:05:08,74 --> 00:05:10,2 Here's another one. 78 00:05:10,2 --> 00:05:15,76 79 00:05:15,76 --> 00:05:21,09 If you're inside of the radius convergence, then the function 80 00:05:21,09 --> 00:05:22,01 has all it's derivatives. 81 00:05:23,066 --> 00:05:34,265 It has all it's derivatives, just like a polynomial does. 82 00:05:34,53 --> 00:05:36,545 You can differentiate it over and over again. 83 00:05:37,21 --> 00:05:46,1 In terms of those derivatives, the number a sub n in the power 84 00:05:46,1 --> 00:05:48,85 series can be expressed in terms of the value of 85 00:05:48,85 --> 00:05:50,38 the derivative at 0. 86 00:05:52 --> 00:05:53,7 And this is called Taylor's formula. 87 00:05:53,7 --> 00:05:58,54 88 00:05:58,54 --> 00:06:02,5 So I'm saying that inside of this radius of convergence, the 89 00:06:02,5 --> 00:06:06,57 function that we're looking at, this f of x, can be written as 90 00:06:06,57 --> 00:06:12,63 the value of the function at 0, that's a sub 0, plus the 91 00:06:12,63 --> 00:06:13,505 value of the derivative. 92 00:06:13,83 --> 00:06:17,25 This bracket n means you take the derivative n times. 93 00:06:17,63 --> 00:06:21,4 So, when n is one, you take the derivative once at 0 divided 94 00:06:21,4 --> 00:06:24,82 by one factorial, which is one, and multiplied by x. 95 00:06:25,1 --> 00:06:26,876 That's the linear term in the power series. 96 00:06:27,63 --> 00:06:30,005 Then, the qaudratic term is you take the second derivative. 97 00:06:30,26 --> 00:06:33,13 Remember to divide by 2 factorial which is 2. 98 00:06:33,74 --> 00:06:37,45 Multiply that by x squared and so on out. 99 00:06:39,73 --> 00:06:44,28 So, the coefficients in the power series just record the 100 00:06:44,28 --> 00:06:47,9 values of derivatives of the function at x equals 0. 101 00:06:48,08 --> 00:06:50,045 They Can be computed that way also. 102 00:06:52,27 --> 00:06:52,83 Let's see. 103 00:06:53,16 --> 00:06:55,76 I think that's a end of my summary of things 104 00:06:55,76 --> 00:06:57,07 that he talked about. 105 00:06:57,07 --> 00:06:59,28 I think he did one example, and I'll repeat that 106 00:06:59,28 --> 00:07:03,415 example of a power series. 107 00:07:04,68 --> 00:07:06,83 This example wasn't due to David Jerison; it was 108 00:07:06,83 --> 00:07:07,86 due to Leonard Euler. 109 00:07:07,86 --> 00:07:11,59 110 00:07:11,59 --> 00:07:15,49 It's the example of where the function is the exponential 111 00:07:15,49 --> 00:07:16,53 function e to the x. 112 00:07:16,53 --> 00:07:19,28 113 00:07:19,28 --> 00:07:22,02 So, let's see. 114 00:07:22,98 --> 00:07:26,04 I will just repeat for you the computations of the power 115 00:07:26,04 --> 00:07:28,13 series for e to the x, just because it's such an 116 00:07:28,13 --> 00:07:29,24 important thing to do. 117 00:07:30,46 --> 00:07:32,71 So, in order to do that, I have to know what the derivative of 118 00:07:32,71 --> 00:07:35,411 e to the x is, and what the second derivative of e to the x 119 00:07:35,411 --> 00:07:41,25 is, and so on, because that comes into the Taylor formula 120 00:07:41,25 --> 00:07:41,67 for the coefficients. 121 00:07:42,69 --> 00:07:44,94 But we know what the derivaative of e to the x is, 122 00:07:44,94 --> 00:07:49 it's just e to the x again, and it's that way all the way down. 123 00:07:49,31 --> 00:07:52,45 All the derivatives are e to the x over and over again. 124 00:07:53,31 --> 00:07:57,31 So when I evaluate this at x equal 0, well, the value of e 125 00:07:57,31 --> 00:08:01,415 to the x is one, the value of e to the x is one and x equals 0. 126 00:08:01,91 --> 00:08:04,88 You get a value of one all the way down. 127 00:08:05,18 --> 00:08:08,765 So all these derivatives at 0 have the value one. 128 00:08:10,09 --> 00:08:16,49 And now, when I plug into this formula, I find e to the x is 129 00:08:16,49 --> 00:08:24,62 1, plus 1 times x, plus 1 over 2 factorial times x squared 130 00:08:24,62 --> 00:08:30,95 plus 1 over 3 factorial times x cubed and so on. 131 00:08:32,25 --> 00:08:35,22 So all of these numbers are one, and all you wind up 132 00:08:35,22 --> 00:08:36,769 with is the factorials and the denominators. 133 00:08:37,13 --> 00:08:38,87 That's the power series for e to the x. 134 00:08:38,87 --> 00:08:40,68 This was a discovery of Leonhard Euler in 135 00:08:40,68 --> 00:08:42,05 1740 or something. 136 00:08:42,05 --> 00:08:42,842 Yes Ma'am. 137 00:08:42,842 --> 00:08:45,554 AUDIENCE: When your writing out the power series, how far do 138 00:08:45,554 --> 00:08:46,458 you have to write it out? 139 00:08:46,91 --> 00:08:48,7 PROFESSOR: How far do you have to write the power series 140 00:08:48,7 --> 00:08:49,96 before it becomes well defined? 141 00:08:51,1 --> 00:08:54,34 Before its a satisfactory solution to an exam problem, 142 00:08:54,34 --> 00:08:56,32 I suppose, is another way to phrase the question. 143 00:08:57,71 --> 00:09:00,06 Write until you can see what the pattern is. 144 00:09:00,92 --> 00:09:02,23 I can see what the pattern is. 145 00:09:02,23 --> 00:09:03,94 Is there anyone who's in doubt about what the 146 00:09:03,94 --> 00:09:05,8 next term might be? 147 00:09:07,89 --> 00:09:09,9 Some people would tell you that you have to write the 148 00:09:09,9 --> 00:09:11,44 summation convention thing. 149 00:09:11,9 --> 00:09:12,49 Don't believe them. 150 00:09:13,91 --> 00:09:15,82 If you right out enough terms to make it clear, 151 00:09:15,82 --> 00:09:16,605 that's good enough. 152 00:09:17,02 --> 00:09:17,33 OK? 153 00:09:19,24 --> 00:09:20,07 Is that an answer for you? 154 00:09:20,07 --> 00:09:21,042 AUDIENCE: Yes, Thank you. 155 00:09:22,99 --> 00:09:25,98 PROFESSOR: OK, so that's a basic example. 156 00:09:25,98 --> 00:09:28,96 Let's do another basic example of a powers series. 157 00:09:28,96 --> 00:09:31,56 Oh yes, and by the way, whenever you write out a power 158 00:09:31,56 --> 00:09:34,745 series, you should say what the readius of convergence is. 159 00:09:35,24 --> 00:09:37,83 For now, I will just to tell you that the radius of 160 00:09:37,83 --> 00:09:41,14 convergence of this power series is infiinity; that is, 161 00:09:41,14 --> 00:09:44,95 this sum always convergence for any value of x. 162 00:09:46,33 --> 00:09:48,12 I'll say a little more about that in a few minutes. 163 00:09:49,78 --> 00:09:51,933 AUDIENCE: Which functions can be written as power series? 164 00:09:52,88 --> 00:09:54,95 PROFESSOR: Which functions can be written as power series? 165 00:09:57,06 --> 00:09:58,335 That's an excellent question. 166 00:10:00,03 --> 00:10:06,38 Any function that has a reasonable expression can be 167 00:10:06,38 --> 00:10:07,575 written as a power series. 168 00:10:08,84 --> 00:10:11,22 I'm not giving you a very good answer because the true answer 169 00:10:11,22 --> 00:10:12,26 is a little bit complicated. 170 00:10:12,73 --> 00:10:16,05 But any of the functions that occur in Calculus like sines 171 00:10:16,05 --> 00:10:19,66 cosines, tangents, they all have power series 172 00:10:19,66 --> 00:10:21,22 expansions, OK? 173 00:10:21,74 --> 00:10:27,07 We'll see more examples Let's do another example. 174 00:10:27,07 --> 00:10:28,005 Here's another example. 175 00:10:30,13 --> 00:10:31,52 I guess this was example one. 176 00:10:31,52 --> 00:10:35,52 177 00:10:35,52 --> 00:10:39,805 So, this example, I think, was due to Newton, not Euler. 178 00:10:42,14 --> 00:10:46,33 Let's find the power series expansion of this function, 179 00:10:46,33 --> 00:10:47,78 1 over 1 plus x. 180 00:10:48,2 --> 00:10:52,43 Well, I think that somewhere along the line, you learned 181 00:10:52,43 --> 00:10:57,23 about the geometric series which tells you what the 182 00:10:57,23 --> 00:10:59,37 answer to this is, and I'll just write it out. 183 00:11:00,19 --> 00:11:12,49 The geometric series tells you that this function can be 184 00:11:12,49 --> 00:11:16,01 written as an alternating sum of powers of x. 185 00:11:16,46 --> 00:11:18,81 You may wonder where these minuses came from. 186 00:11:18,81 --> 00:11:21,3 Well, if you really think about the geometric series, as you 187 00:11:21,3 --> 00:11:24,63 probably remembered, there was a minus sign here, and 188 00:11:24,63 --> 00:11:26,54 that gets replaced by these minus signs. 189 00:11:28,42 --> 00:11:30,27 I think maybe Jerison talked about this also. 190 00:11:31,81 --> 00:11:33,96 Anyway, here's another basic example. 191 00:11:34,64 --> 00:11:37,07 Remember what the graph of this function looks like when 192 00:11:37,07 --> 00:11:39,77 x is equal to minus one. 193 00:11:41,62 --> 00:11:43,86 There's a little problem here because the denominator 194 00:11:43,86 --> 00:11:47,6 become 0, so the graph has a poll there. 195 00:11:47,6 --> 00:11:53,84 It goes up to infinity at x equals minus one, and that's an 196 00:11:53,84 --> 00:11:58,52 indication that the radius of convergence is not infinity. 197 00:11:59,15 --> 00:12:01,74 If you try to converge to this infinite number by putting 198 00:12:01,74 --> 00:12:04,6 in x equals minus one here, you'll have a big problem. 199 00:12:04,93 --> 00:12:07,49 In fact, you see when you put in x equals minus one, you keep 200 00:12:07,49 --> 00:12:09,8 getting one in every term, and it gets bigger and bigger 201 00:12:09,8 --> 00:12:10,705 and does not converge. 202 00:12:11,39 --> 00:12:14,94 In this example, the radius of convergence is one. 203 00:12:14,94 --> 00:12:18,57 204 00:12:18,57 --> 00:12:21,795 OK, so, let's do a new example now. 205 00:12:22,21 --> 00:12:25,41 Oh, and by the way, I should say you can calculate these 206 00:12:25,41 --> 00:12:27,47 numbers using Taylor's formula. 207 00:12:27,77 --> 00:12:29,94 If you haven't seen it, check it out. 208 00:12:29,94 --> 00:12:36,58 Calculate The iterated derivatives of this function 209 00:12:36,58 --> 00:12:40,35 and plug in x equals 0 and see that you get plus one, minus 210 00:12:40,35 --> 00:12:41,93 one, plus one, minus one, and so on. 211 00:12:41,93 --> 00:12:42,23 Yes sir. 212 00:12:42,706 --> 00:12:46,29 AUDIENCE: Per the radius of convergence as stated, if you 213 00:12:46,29 --> 00:12:48,09 do minus one it'll fall out. 214 00:12:48,09 --> 00:12:50,37 If you put in one though, seems like it would be fine. 215 00:12:50,74 --> 00:12:53,52 PROFESSOR: The questions is I can see that there's a problem 216 00:12:53,52 --> 00:12:56,71 at x equals minus one, why is there also a problem at x 217 00:12:56,71 --> 00:12:59,24 equals one where the graph is perfectly smooth and 218 00:12:59,24 --> 00:13:00,38 innocuous and finite. 219 00:13:00,76 --> 00:13:03,04 That's another excellent question. 220 00:13:04,49 --> 00:13:08,28 The problem is, that if you go off to a radius of one in any 221 00:13:08,28 --> 00:13:10,97 direction and there's a problem, that's it. 222 00:13:11,53 --> 00:13:13,19 That's what the radius of convergence is. 223 00:13:13,53 --> 00:13:17,68 Here, what does happen if I put an x equals plus one? 224 00:13:18,07 --> 00:13:19,54 So, let' look at the partial sums. 225 00:13:20,49 --> 00:13:22,705 Do x equals plus one in your mind here. 226 00:13:23,06 --> 00:13:27,85 So I'll get a partial sum one, then 0, then one, 227 00:13:27,85 --> 00:13:29,675 then 0, then one. 228 00:13:29,99 --> 00:13:32,09 So even though it doesn't go up to infinity, it still 229 00:13:32,09 --> 00:13:32,9 does not converge. 230 00:13:32,9 --> 00:13:35,046 AUDIENCE: And anything in between? 231 00:13:35,5 --> 00:13:38,47 PROFESSOR: Any of these other things will also fail to 232 00:13:38,47 --> 00:13:39,47 converge in this example. 233 00:13:41,33 --> 00:13:43,93 Well, that's the only two real numbers at the edge. 234 00:13:43,93 --> 00:13:46,94 235 00:13:46,94 --> 00:13:49,05 OK, let's do a different example now. 236 00:13:49,05 --> 00:13:50,793 How about a trig function? 237 00:13:50,793 --> 00:13:55,422 238 00:13:55,422 --> 00:14:00,38 I'm going to compute the power series expansion for the sine 239 00:14:00,38 --> 00:14:03,7 of x, and I'm going to do it using Taylor's formula. 240 00:14:04,4 --> 00:14:07,17 So Taylor's formula says that I have to start computing 241 00:14:07,17 --> 00:14:09,585 derivatives of the sine of x. 242 00:14:09,585 --> 00:14:22,1 243 00:14:22,1 --> 00:14:23,83 Sounds like it's going to be a lot of work. 244 00:14:25,28 --> 00:14:28,005 Let's see, the derivative of the sine is the cosine. 245 00:14:28,005 --> 00:14:30,87 246 00:14:30,87 --> 00:14:33,59 and the derivative of the cosine, the second derivative 247 00:14:33,59 --> 00:14:35,51 of the sine, is what? 248 00:14:36,53 --> 00:14:39,88 Remember the minus, it's minus sine of x. 249 00:14:40,27 --> 00:14:42,94 OK, now I want to take the third derivative of the sine, 250 00:14:42,94 --> 00:14:46,02 which is the derivative of sine, prime, prime, so it's 251 00:14:46,02 --> 00:14:47,12 the derivative of this. 252 00:14:47,76 --> 00:14:50,72 And we just decided to derivative of sine is cosine, 253 00:14:50,72 --> 00:14:53,73 so I get cosine, but I have this minus sign in front. 254 00:14:53,73 --> 00:14:56,66 255 00:14:56,66 --> 00:15:00,32 And now I want to differentiate again, so the cosine becomes a 256 00:15:00,32 --> 00:15:04,52 minus sine, and that minus sine cancels with this minus 257 00:15:04,52 --> 00:15:06,795 sine to give me sine. 258 00:15:08,02 --> 00:15:08,52 of x. 259 00:15:08,52 --> 00:15:09,11 You follow that? 260 00:15:10,102 --> 00:15:12,31 It's a lot of minus one's canceling out there. 261 00:15:13,66 --> 00:15:18,22 So, all of a sudden, I'm right back where I started; these two 262 00:15:18,22 --> 00:15:21,16 are the same and the pattern will now repeat 263 00:15:21,16 --> 00:15:22 forever and ever. 264 00:15:22,78 --> 00:15:25,15 Higher and higher derivatives of sines are just plus or 265 00:15:25,15 --> 00:15:26,87 minus sines and cosines. 266 00:15:28,83 --> 00:15:33,79 Now Taylor's formula says I should now substitute x equals 267 00:15:33,79 --> 00:15:36,67 0 into this and see what happens, so let's do that. 268 00:15:37,58 --> 00:15:42,9 When x is equals to 0, the sine is 0 and the cosine is one. 269 00:15:43,24 --> 00:15:46,365 The sine is 0, so minus 0 is also 0. 270 00:15:47,41 --> 00:15:51,46 The cosine is one, but now there's a minus one, and now 271 00:15:51,46 --> 00:15:56,68 I'm back where I started, and so the pattern will repeat. 272 00:15:58,76 --> 00:16:02,69 OK, so the values of the derivatives are all 0es and 273 00:16:02,69 --> 00:16:06,64 plus and minus ones and they go through that pattern, four-fold 274 00:16:06,64 --> 00:16:08,48 periodicity, over and over again. 275 00:16:09,67 --> 00:16:14,53 So we can write out what the sine of x is using 276 00:16:14,53 --> 00:16:15,41 Taylor's formula. 277 00:16:15,41 --> 00:16:18 278 00:16:18 --> 00:16:22,01 So I put in the value at 0 which is 0, then I put 279 00:16:22,01 --> 00:16:26,8 in the derivative which is 1 multiplied by x. 280 00:16:27,62 --> 00:16:31,52 Then, I have the second derivative divided by 2 281 00:16:31,52 --> 00:16:34,79 factorial, but the second derivatve at 0 is 0. 282 00:16:35,15 --> 00:16:36,83 So I'm going to drop that term out. 283 00:16:38,28 --> 00:16:41,365 Now I have the third derivative which is minus one. 284 00:16:41,365 --> 00:16:43,93 285 00:16:43,93 --> 00:16:46,29 Remember the 3 factorial in the denominator. 286 00:16:46,79 --> 00:16:48,125 That's the coefficient of x cubed. 287 00:16:50,05 --> 00:16:50,91 What's the fourth derivative? 288 00:16:51,68 --> 00:16:53,4 Well, here we are, it's on the board, it's 0. 289 00:16:54,4 --> 00:16:58,15 So I drop that term out go up to the fifth term, 290 00:16:58,15 --> 00:16:59,39 the fifth power of x. 291 00:16:59,83 --> 00:17:01,995 It's derivative is now one. 292 00:17:02,26 --> 00:17:06,29 We've gone through the pattern and we're back at plus one as 293 00:17:06,29 --> 00:17:09,86 the value of the iterated derivative, so now I get 1/5 294 00:17:09,86 --> 00:17:12,28 factorial times x to the fifth. 295 00:17:13,18 --> 00:17:15,72 Now, you tell me, have we done enough terms to see 296 00:17:15,72 --> 00:17:16,66 what the pattern is? 297 00:17:17,9 --> 00:17:22,17 I guess an x term will be a minus 1/7 factorial 298 00:17:22,17 --> 00:17:23,03 x to the seventh. 299 00:17:23,75 --> 00:17:25,924 Let me write this out again just so we have it. 300 00:17:28,16 --> 00:17:31,68 So it's x minus x cubed over 3 factorial plus x to the 301 00:17:31,68 --> 00:17:33,06 fifth over 5 factorial. 302 00:17:34,58 --> 00:17:37,65 You guessed it, and so on. 303 00:17:38,74 --> 00:17:40,51 That's the power series expansion for the 304 00:17:40,51 --> 00:17:43,935 sine of x, OK? 305 00:17:43,935 --> 00:17:46,95 306 00:17:46,95 --> 00:17:49,7 So, the sines alternate, and these denominators 307 00:17:49,7 --> 00:17:51,7 get very big, don't they? 308 00:17:52,25 --> 00:17:54,02 Factorials grow very fast. 309 00:17:54,41 --> 00:17:55,375 Let me make a remark. 310 00:17:55,83 --> 00:17:57,55 r is infinity here. 311 00:17:58,8 --> 00:18:02,07 The radius of convergence of this power series again is 312 00:18:02,07 --> 00:18:03,723 infinity, and let me just say why. 313 00:18:03,723 --> 00:18:11,01 314 00:18:11,01 --> 00:18:14,81 The general term is going to be like x to the 2n plus 1 divided 315 00:18:14,81 --> 00:18:19,29 by 2n plus one factorial, an odd number I can 316 00:18:19,29 --> 00:18:20,64 write as 2n plus 1. 317 00:18:21,9 --> 00:18:26,04 And what I want to say is about what happens to the size of 318 00:18:26,04 --> 00:18:31,96 this as n goes to infinity. 319 00:18:34 --> 00:18:35,28 So let's just think about this. 320 00:18:35,28 --> 00:18:37,775 For a fixed x, let's fix the number of x. 321 00:18:38,27 --> 00:18:41,57 Look at powers of x and think about the size of this 322 00:18:41,57 --> 00:18:43,445 expression when n gets to be large. 323 00:18:45,93 --> 00:18:47,14 Let's just do that for a second. 324 00:18:47,14 --> 00:18:53,27 So, for x to the 2n plus 1 over 2n plus 1 factorial, 325 00:18:53,27 --> 00:18:54,42 I can write out like this. 326 00:18:54,42 --> 00:19:09,25 It's x over one times x over x over 3, times x over 2n plus 1. 327 00:19:09,25 --> 00:19:13,22 I've multiplied x by itself 2n plus 1 times in the numerator, 328 00:19:13,22 --> 00:19:16,85 and I've multiplied the numbers 1, 2, 3, 4, and so on, by each 329 00:19:16,85 --> 00:19:19,81 other in the denominator, and that gives me the factorial. 330 00:19:19,81 --> 00:19:21,57 So I've just written this out like this. 331 00:19:22,33 --> 00:19:26,82 Now x is fixed, so maybe it's a million, OK? 332 00:19:26,82 --> 00:19:28,275 It's big, but fixed. 333 00:19:28,7 --> 00:19:30,195 What happens to these numbers? 334 00:19:30,65 --> 00:19:32,05 Well at first, they're pretty big. 335 00:19:32,05 --> 00:19:34,255 This is a million over 2, this is a million over 3. 336 00:19:34,56 --> 00:19:39,14 But when n gets to be maybe a million, then this 337 00:19:39,14 --> 00:19:39,98 is about one half. 338 00:19:41,18 --> 00:19:47,16 If n is a billion, then this is about 1/2,000, right? 339 00:19:48,32 --> 00:19:50,53 The denominator just keep getting bigger and bigger, 340 00:19:50,53 --> 00:19:53,48 but the numerators stay the same; they're always x. 341 00:19:54,47 --> 00:19:57,875 So when I take the product, if I go far enough out, I'm going 342 00:19:57,875 --> 00:20:00,34 to be multiplying, by very, very small numbers and 343 00:20:00,34 --> 00:20:01,415 more and more of them. 344 00:20:01,73 --> 00:20:05,98 And so no matter what x is, these numbers 345 00:20:05,98 --> 00:20:07,22 will converge to 0. 346 00:20:07,22 --> 00:20:10,04 They'll get smaller and smaller as x gets to be bigger. 347 00:20:11,55 --> 00:20:15,06 That's the sign that x is inside of the 348 00:20:15,06 --> 00:20:16,1 radius of convergence. 349 00:20:16,59 --> 00:20:22,13 This is the sign for you that this series converges 350 00:20:22,13 --> 00:20:23,145 for that value of x. 351 00:20:23,78 --> 00:20:31,61 And because I could do this for any x, this works. 352 00:20:33,6 --> 00:20:39,78 This convergence to 0 for any fixed s. 353 00:20:40,26 --> 00:20:44,51 That's what tells you that the raidus of convergence 354 00:20:44,51 --> 00:20:53,07 is infinity because of the formula that the radius of 355 00:20:53,07 --> 00:20:54,295 convergence talks about. 356 00:20:54,57 --> 00:20:57,805 If r is equal infinity, this is no condition on x. 357 00:20:58,16 --> 00:21:02,48 Every number is less than infinity in absolute value, So 358 00:21:02,48 --> 00:21:06,59 if this covergence to 0 of the general term works for 359 00:21:06,59 --> 00:21:08,895 every x, then radius of convergence is infinity. 360 00:21:10,32 --> 00:21:12,33 Well that was kind of fast, but I think that you've heard 361 00:21:12,33 --> 00:21:14,56 something about that earlier as well. 362 00:21:16,02 --> 00:21:19,14 Anyway, so we've got the sine function, a new function 363 00:21:19,14 --> 00:21:20,465 with its own power series. 364 00:21:20,88 --> 00:21:22,96 It's a way of computing sine of x. 365 00:21:23,73 --> 00:21:27,99 If you take enough terms you'll a good evaluation of the 366 00:21:27,99 --> 00:21:29,34 sine of x for any x. 367 00:21:30 --> 00:21:32,43 This tells you a lot about the function sine of x but 368 00:21:32,43 --> 00:21:33,75 not everything at all. 369 00:21:33,75 --> 00:21:37,85 For example, from this formula, it's very hard to see that 370 00:21:37,85 --> 00:21:38,955 the sine of x is periodic. 371 00:21:39,745 --> 00:21:41,485 I'ts not obvious at all. 372 00:21:41,93 --> 00:21:44,83 Somewhere hidden away in this expression is that number 373 00:21:44,83 --> 00:21:46,726 pi, the half of the period. 374 00:21:47,4 --> 00:21:49,92 But that's not clear from the power series at all. 375 00:21:51,1 --> 00:21:53,32 So the power series are very good for some things, 376 00:21:53,32 --> 00:21:55,51 but they hide other properties of functions. 377 00:21:55,51 --> 00:21:58,15 378 00:21:58,15 --> 00:22:02,07 So I want to spend a few minutes telling you about what 379 00:22:02,07 --> 00:22:05,9 you can do with a power series once you have one to get 380 00:22:05,9 --> 00:22:08,53 new power series, and new power series from old. 381 00:22:08,53 --> 00:22:18,3 382 00:22:18,3 --> 00:22:24,56 This is also called operations on power series. 383 00:22:25,49 --> 00:22:28,27 So what are the things that we can do to a power series? 384 00:22:28,27 --> 00:22:29,905 Well one of the things you can do is multiply. 385 00:22:29,905 --> 00:22:33,99 386 00:22:33,99 --> 00:22:37,88 For example, what if I want to compute a power series for 387 00:22:37,88 --> 00:22:39,42 x times the sine of x. 388 00:22:40,97 --> 00:22:43,905 Well I a power series for the sine of x, I just did it. 389 00:22:44,16 --> 00:22:45,92 How about a power series for x? 390 00:22:45,92 --> 00:22:48,91 391 00:22:48,91 --> 00:22:50,545 Actually, I did that here too. 392 00:22:51,48 --> 00:22:54,65 The function x is a very simple polynomial. 393 00:22:55,12 --> 00:22:58,93 It's a polynomial where, about 0, a1 is 1, and all the 394 00:22:58,93 --> 00:23:00,22 other coefficients are 0. 395 00:23:00,93 --> 00:23:04,09 So x itself is a power series, a very simple one. 396 00:23:04,09 --> 00:23:06,67 The sine of x is a powers series. 397 00:23:08,44 --> 00:23:11,06 What I want to encourage you to do is treat power series 398 00:23:11,06 --> 00:23:13,905 just like polynomials and multiply them together. 399 00:23:14,33 --> 00:23:16,64 We'll see other operations too. 400 00:23:16,96 --> 00:23:20,39 So, to compute the power series for x times the sine of x, I 401 00:23:20,39 --> 00:23:23,5 just take this one and multiply it by x. 402 00:23:23,5 --> 00:23:26,65 403 00:23:26,65 --> 00:23:31,04 It distributes through x squared minus x to the fourth 404 00:23:31,04 --> 00:23:35,18 over 3 factorial plus x to the sixth over 5 405 00:23:35,18 --> 00:23:40,39 factorial, and so on. 406 00:23:42,19 --> 00:23:44,68 Again, the radius of convergence is going to be 407 00:23:44,68 --> 00:23:47,4 the smaller of the two radii of convergence here. 408 00:23:48,22 --> 00:23:50,48 So it's r equals infinity in this case. 409 00:23:51,84 --> 00:23:53,72 OK, you can multiply power series together. 410 00:23:54,01 --> 00:23:57,8 It can be a pain if the power series are very long, but 411 00:23:57,8 --> 00:23:59,66 if one of them is x, it's pretty simple. 412 00:24:01,8 --> 00:24:03,72 OK, it's one thing I can do. 413 00:24:06,04 --> 00:24:07,22 Notice something by the way. 414 00:24:08,91 --> 00:24:10,175 You know that even an odd functions? 415 00:24:10,175 --> 00:24:13,18 416 00:24:13,18 --> 00:24:17,49 So, sine is an odd function, x is an odd function, and the 417 00:24:17,49 --> 00:24:19,93 product of two large functions is an even function. 418 00:24:20,57 --> 00:24:24,07 That's reflected in the fact that all the powers that occur 419 00:24:24,07 --> 00:24:25,475 in the power series are even. 420 00:24:26,79 --> 00:24:30,37 For an odd function, like the sine, all the powers that 421 00:24:30,37 --> 00:24:32,135 occur are odd powers of x. 422 00:24:32,6 --> 00:24:33,38 That's always true. 423 00:24:33,38 --> 00:24:37,51 424 00:24:37,51 --> 00:24:40,01 OK, we can multiply, and I can also differentiate. 425 00:24:40,01 --> 00:24:48,66 426 00:24:48,66 --> 00:24:57,66 Let's just do a case of that and use the process of 427 00:24:57,66 --> 00:25:00,41 differentiation to find out what the power series for the 428 00:25:00,41 --> 00:25:05,23 cosine of x is by writing the cosine of x as the 429 00:25:05,23 --> 00:25:08,685 derivative of the sine and differentiating term by term. 430 00:25:09,01 --> 00:25:11,97 So, I'll take this expression for the power series of the 431 00:25:11,97 --> 00:25:14,75 sine and differentiate it term by term, and I'll get the 432 00:25:14,75 --> 00:25:16,61 power series for cosine. 433 00:25:18,21 --> 00:25:19,03 So, let's see. 434 00:25:19,03 --> 00:25:21,056 The derivative of x is one. 435 00:25:22,51 --> 00:25:26,37 The derivative of x cubed is 3x squared, and then there's a 3 436 00:25:26,37 --> 00:25:27,66 factorial in the denominator. 437 00:25:28,91 --> 00:25:33,03 The derivative of x to the fifth is 5x to the fourth, and 438 00:25:33,03 --> 00:25:35,83 there's a 5 factorial in the denominator, and 439 00:25:35,83 --> 00:25:36,6 so on and so on. 440 00:25:38,68 --> 00:25:40,46 And know some cancellation happens. 441 00:25:40,95 --> 00:25:45,96 So this is one minus, well, the 3 cancels with the last factor 442 00:25:45,96 --> 00:25:48,995 in this 3 factorial and leaves you with 2 factorial. 443 00:25:48,995 --> 00:25:52,46 444 00:25:52,46 --> 00:25:56,27 The 5 cancels with the last factor in the 5 factorial and 445 00:25:56,27 --> 00:25:58,92 leaves you with a 4 factorial in the denominator. 446 00:25:58,92 --> 00:26:01,57 447 00:26:01,57 --> 00:26:03,81 So, there you go, there's the power series 448 00:26:03,81 --> 00:26:05,635 expansion for the cosine. 449 00:26:05,98 --> 00:26:09,76 It's got all even powers of x they alternate, and you have 450 00:26:09,76 --> 00:26:11,34 factorials in the denominator. 451 00:26:12,72 --> 00:26:16,06 And of course, you could derive that expression by using 452 00:26:16,06 --> 00:26:19,38 Taylor's formula by the same kind of calculation you did 453 00:26:19,38 --> 00:26:22,71 here, taking higher and higher derivatives of the cosine. 454 00:26:22,97 --> 00:26:26,52 You get the same periodic pattern of derivatives and 455 00:26:26,52 --> 00:26:28,75 values of derivatives at x equals 0. 456 00:26:30,08 --> 00:26:33,53 But here's a cleaner, simpler way to do it because 457 00:26:33,53 --> 00:26:35,62 we already knew the derivative of the sine. 458 00:26:36,83 --> 00:26:38,68 When you differentiate, you keep the same 459 00:26:38,68 --> 00:26:39,63 radius of convergence. 460 00:26:39,63 --> 00:26:44,42 461 00:26:44,42 --> 00:26:50,06 OK, so we can multiply, and I can add to and multiply that 462 00:26:50,06 --> 00:26:51,25 constant, things like that. 463 00:26:52,4 --> 00:26:53,28 How about integrating? 464 00:26:54,28 --> 00:26:56,58 That's what half of this course was about isn't it? 465 00:26:56,58 --> 00:26:58,55 So, let's integrate something. 466 00:26:58,55 --> 00:27:07,21 467 00:27:07,21 --> 00:27:16,1 So, the integration I'm going to do is this one: the 468 00:27:16,1 --> 00:27:19,31 integral from 0 to x of dt over one plus x. 469 00:27:20,16 --> 00:27:21,98 What is that integral as a function? 470 00:27:21,98 --> 00:27:28,36 471 00:27:28,36 --> 00:27:32,35 So, when I find the anti-derivative of this, I get 472 00:27:32,35 --> 00:27:37,89 the natural log of 1 plus t, and then when I evaluate that 473 00:27:37,89 --> 00:27:41,805 at t equals x, I get the natural log of 1 plus x. 474 00:27:42,6 --> 00:27:47,41 And when I evaluate the natural log at 0, I get the natural 475 00:27:47,41 --> 00:27:53,24 log of one, which is 0, so this is what you get, OK? 476 00:27:55,07 --> 00:28:03,1 This is really valid, by the way, for x bigger 477 00:28:03,1 --> 00:28:04,14 than minus one. 478 00:28:05,69 --> 00:28:09,06 But you don't want to think about this quite like this 479 00:28:09,06 --> 00:28:10,24 when x is smaller than that. 480 00:28:10,24 --> 00:28:13,77 481 00:28:13,77 --> 00:28:19,94 Now, I'm going to try to apply power series methods here and 482 00:28:19,94 --> 00:28:23,31 use this integral to find a power series for the natural 483 00:28:23,31 --> 00:28:29,9 log, and I'll do it by plugging in to this expression what the 484 00:28:29,9 --> 00:28:33,06 power series for 1/1 plus t was. 485 00:28:34,76 --> 00:28:36,5 I know what that is because I wrote it down on 486 00:28:36,5 --> 00:28:37,33 the board up here. 487 00:28:38,33 --> 00:28:43,23 Change the variable from x to t there, and so 1/1 plus t is 488 00:28:43,23 --> 00:28:49,41 1 minus t, plus t squared, minust t cubed, and so on. 489 00:28:49,41 --> 00:28:52,62 490 00:28:52,62 --> 00:28:57,92 So that's the thing in the in inside of the integral, and now 491 00:28:57,92 --> 00:29:02,62 it's legal to integrate that term by term, so let's do that. 492 00:29:03,82 --> 00:29:05,35 I'm going to get something which I will then 493 00:29:05,35 --> 00:29:07,74 evaluate at x and at 0. 494 00:29:09,23 --> 00:29:12,61 So, when I integrate one I get x, and when they 495 00:29:12,61 --> 00:29:14,9 integrate t, I get t. 496 00:29:14,9 --> 00:29:19,15 497 00:29:19,15 --> 00:29:24,56 Intergrating t, I get t squared over 2, and t squared gives me 498 00:29:24,56 --> 00:29:29,97 t cubed over 3, and so on and so on. 499 00:29:29,97 --> 00:29:32,52 500 00:29:32,52 --> 00:29:37,36 Then, when I put in t equals x, while I just replace all the 501 00:29:37,36 --> 00:29:41,1 t's by x's, and when I put in t equals 0, I get 0. 502 00:29:41,78 --> 00:29:43,575 So this equals x. 503 00:29:43,95 --> 00:29:50,43 So, I've discovered that the natural log of 1 plus x is x 504 00:29:50,43 --> 00:29:57,01 minus x squared over 2, plus x cubed over 3 minus x to the 505 00:29:57,01 --> 00:30:00,188 fourth over 4, and so on and so on. 506 00:30:02,02 --> 00:30:04,11 There's the power series expansion for the natural 507 00:30:04,11 --> 00:30:05,18 log of 1 plus x. 508 00:30:05,18 --> 00:30:07,8 509 00:30:07,8 --> 00:30:11,06 Because I began with a power series who's radius of 510 00:30:11,06 --> 00:30:15,57 convergence was just one, I began with this power series, 511 00:30:15,57 --> 00:30:18,15 the radius of convergence of this is also going to be one. 512 00:30:18,15 --> 00:30:22,2 513 00:30:22,2 --> 00:30:25,83 Also, because this function as I just pointed out, this 514 00:30:25,83 --> 00:30:29,87 function goes bad when x becomes less than minus one, so 515 00:30:29,87 --> 00:30:32,32 some problem happens, and that's reflected in the 516 00:30:32,32 --> 00:30:33,17 radius of convergence. 517 00:30:35,59 --> 00:30:40,37 Cool, so, you can integrate. 518 00:30:41,11 --> 00:30:44,25 That is the correct power series expansion for the 519 00:30:44,25 --> 00:30:48,06 natural log of 1 plus x, and another victory of Euler's was 520 00:30:48,06 --> 00:30:50,85 to use this kind of power series expansion to calculate 521 00:30:50,85 --> 00:30:53,28 natural algorithms in a much more efficient way than 522 00:30:53,28 --> 00:30:54,35 people had done before. 523 00:30:54,35 --> 00:30:57,85 524 00:30:57,85 --> 00:31:17,482 OK, one more property, what are we at here, 3 or 4? 525 00:31:18,69 --> 00:31:28,91 Substitute, very appropriate for me as a substitute teacher 526 00:31:28,91 --> 00:31:29,96 to tell you about substitution. 527 00:31:29,96 --> 00:31:32,81 528 00:31:32,81 --> 00:31:35,48 So I'm going to try to find the power series expansion 529 00:31:35,48 --> 00:31:36,885 of e to the minus t squared. 530 00:31:36,885 --> 00:31:41,74 531 00:31:41,74 --> 00:31:45,19 And the way I'll do that is by taking the power series 532 00:31:45,19 --> 00:31:50,74 expansion for e to the x, which we have up there, and make the 533 00:31:50,74 --> 00:32:00,85 substitution x equals minus t squared in the expansion 534 00:32:00,85 --> 00:32:01,63 for x to the x. 535 00:32:01,63 --> 00:32:02,32 Did you have a question? 536 00:32:03,272 --> 00:32:05,66 AUDIENCE: Well, it's just concerning the radius 537 00:32:05,66 --> 00:32:06,25 of convergence. 538 00:32:07,624 --> 00:32:11,74 You can't define x so that is always positive, and if so, it 539 00:32:11,74 --> 00:32:14,06 wouldn't have a radius of convergence, right? 540 00:32:14,36 --> 00:32:19,37 PROFESSOR: Like I say, again the worry is this natural log 541 00:32:19,37 --> 00:32:21,51 of 1 plus x function is perfectly well 542 00:32:21,51 --> 00:32:22,76 behaved for large x. 543 00:32:24,12 --> 00:32:27,32 Why does the power series fail to converge for large x? 544 00:32:27,72 --> 00:32:31,11 Well suppose that x is bigger than one, then here you get 545 00:32:31,11 --> 00:32:35,16 bigger and bigger powers of x, which will grow to infinity, 546 00:32:35,16 --> 00:32:40,23 and they grow large faster than the numbers 2, 3, 4, 5, 6. 547 00:32:40,33 --> 00:32:45,23 They grow exponentially, and these just grow linearly. 548 00:32:45,85 --> 00:32:50,32 So, again, when x is bigger than one, the general term will 549 00:32:50,32 --> 00:32:53,12 go off to infinity, even though the function that you're 550 00:32:53,12 --> 00:32:56,39 talking about, log of net of 1 plus x is perfectly good. 551 00:32:56,75 --> 00:33:01,52 So the power series is not good outside of the 552 00:33:01,52 --> 00:33:02,24 radius of convergence. 553 00:33:02,24 --> 00:33:03,55 It's just a fact of life. 554 00:33:04,72 --> 00:33:05,15 Yes? 555 00:33:05,15 --> 00:33:18,49 AUDIENCE: [INAUDIBLE] 556 00:33:18,49 --> 00:33:19,61 PROFESSOR: Talk to me after class. 557 00:33:20,29 --> 00:33:22,62 The question is why is it the smaller of the two 558 00:33:22,62 --> 00:33:23,465 radii of convergence? 559 00:33:24,11 --> 00:33:31,13 The basic answer is, well, you can't expect it to be bigger 560 00:33:31,13 --> 00:33:34,22 than that smaller one, because the power series only gives you 561 00:33:34,22 --> 00:33:37,45 information inside of that range about the function, so. 562 00:33:37,45 --> 00:33:41,01 AUDIENCE: [INAUDIBLE] 563 00:33:41,01 --> 00:33:43,58 PROFESSOR: Well, in this case, both of the radii of 564 00:33:43,58 --> 00:33:44,635 convergence are infinity. 565 00:33:44,89 --> 00:33:48,33 x has radius of convergence infinity for sure, and 566 00:33:48,33 --> 00:33:49,26 sine of x does too. 567 00:33:49,26 --> 00:33:52,14 So you get insanity in that case, OK? 568 00:33:52,14 --> 00:33:54,85 569 00:33:54,85 --> 00:33:59,42 Let's just do this, and then I'm going to integrate this and 570 00:33:59,42 --> 00:34:01,785 that'll be the end of what I have time for today. 571 00:34:03,83 --> 00:34:05,88 What's the power series expansion for this? 572 00:34:05,88 --> 00:34:08,77 The power series expansion of this is going to be a function 573 00:34:08,77 --> 00:34:12,634 of t, right, because the variable here is t. 574 00:34:13,53 --> 00:34:17,37 I get it by taking my expansion for x to the x and putting 575 00:34:17,37 --> 00:34:20,46 in what x is in terms of t. 576 00:34:20,46 --> 00:34:36,7 577 00:34:36,7 --> 00:34:40,95 I just put in minus t squared in place of x there in the 578 00:34:40,95 --> 00:34:43,02 series expansion for e to the x. 579 00:34:44,12 --> 00:34:46,583 I can work this out a little bit better. 580 00:34:47,37 --> 00:34:49,03 minus t squared is what it is. 581 00:34:49,03 --> 00:34:52,95 This is going to give me a t to the fourth and the minus 582 00:34:52,95 --> 00:34:55,47 squared is going to give me a plus, so I get t to the 583 00:34:55,47 --> 00:34:56,879 fourth over 2 factorial. 584 00:34:58,73 --> 00:35:03,52 Then I get minus t quantity cubed, so there'll be a minus 585 00:35:03,52 --> 00:35:07,383 sign and a t to the sixth and the denominator 3 factorial. 586 00:35:08,19 --> 00:35:11,36 So the signs are going to alternate, the powers are all 587 00:35:11,36 --> 00:35:15,38 even, and the denominators are these factorials. 588 00:35:15,38 --> 00:35:20,16 589 00:35:20,16 --> 00:35:25,65 Several times as this course has gone on, the error function 590 00:35:25,65 --> 00:35:26,813 has made an appearance. 591 00:35:27,216 --> 00:35:35,72 The error function gets normalized by putting a 2 over 592 00:35:35,72 --> 00:35:43,09 the square root of pi in front, and it's the integral of e to 593 00:35:43,09 --> 00:35:46,27 the minus t squared d t from 0 to x. 594 00:35:46,83 --> 00:35:55,24 This normalization is here because as x gets to the 595 00:35:55,24 --> 00:36:00,84 large the value becomes one. 596 00:36:01,3 --> 00:36:04,17 So this error function is very important in the 597 00:36:04,17 --> 00:36:04,86 theory of probability. 598 00:36:06,08 --> 00:36:08,54 I think you calculated this fact at some 599 00:36:08,54 --> 00:36:09,76 point in the course. 600 00:36:12,236 --> 00:36:14,59 This is the standard definition of the error function, you 601 00:36:14,59 --> 00:36:16,46 put a 2 over the square root of pi in front. 602 00:36:16,46 --> 00:36:18,425 Let's calculate it's power series expansion. 603 00:36:18,425 --> 00:36:21,32 604 00:36:21,32 --> 00:36:23,56 So there's a 2 over the score root of pi that hurts 605 00:36:23,56 --> 00:36:26,56 nobody here in the front. 606 00:36:27,28 --> 00:36:30,74 And now I want to integrate either the minus t squared, and 607 00:36:30,74 --> 00:36:34,01 I'm going to use this power series expansion for that 608 00:36:34,01 --> 00:36:35,19 to see what you get. 609 00:36:36,35 --> 00:36:38,04 I'm just going to write this out I think. 610 00:36:38,8 --> 00:36:40,98 I did it out carefully in another example over 611 00:36:40,98 --> 00:36:42,775 there so I'll do it a little quicker now. 612 00:36:43,1 --> 00:36:46,69 intergrate this terms by term, you're just integrating powers 613 00:36:46,69 --> 00:36:50,73 of t so it's pretty simple, and then I'm evaluating 614 00:36:50,73 --> 00:36:51,83 at and then 0. 615 00:36:51,83 --> 00:37:01,24 So I get x minus x cubed over 3, plus x to the fifth over 5 616 00:37:01,24 --> 00:37:05,82 times 2 factorial, 5 from integrating the t's of the 617 00:37:05,82 --> 00:37:08,82 fourth, and the 2 factorial from this denominator 618 00:37:08,82 --> 00:37:09,75 that we already had. 619 00:37:11,39 --> 00:37:15,84 And then there's an a minus x to the seventh over 7 times 3 620 00:37:15,84 --> 00:37:20,07 factorial, and plus, and so on, and you can imagine 621 00:37:20,07 --> 00:37:21,62 how they go on from there. 622 00:37:21,62 --> 00:37:24,49 623 00:37:24,49 --> 00:37:29,05 I guess to get this exactly in the form that we began 624 00:37:29,05 --> 00:37:32,51 talking about, I should multiply through. 625 00:37:32,51 --> 00:37:35,93 So the coeifficient of x is 2 over the square root of pi, and 626 00:37:35,93 --> 00:37:39,36 the coefficient of x cubed is minus 2/3 times the square 627 00:37:39,36 --> 00:37:40,47 root of pi, and so on. 628 00:37:41,09 --> 00:37:43,14 But this is a perfectly good way to write this power 629 00:37:43,14 --> 00:37:44,51 series expansion as well. 630 00:37:45,63 --> 00:37:48,42 And, this is a very good way to compute the value 631 00:37:48,42 --> 00:37:49,295 of the error function. 632 00:37:49,57 --> 00:37:52,82 It's a new function in our experience. 633 00:37:53,21 --> 00:37:56,05 Your calculator probably calculates it, and your 634 00:37:56,05 --> 00:37:58,71 calculator probably does it by this method. 635 00:37:58,71 --> 00:38:01,27 636 00:38:01,27 --> 00:38:07,68 OK, so that's my sermon on examples of things you can 637 00:38:07,68 --> 00:38:08,8 do with power series. 638 00:38:10,26 --> 00:38:13,43 So, we're going to do the CEG thing in just a minute. 639 00:38:13,87 --> 00:38:17,16 Professor Jerison wanted me to make an ad for 18.02. 640 00:38:17,74 --> 00:38:20,62 Just in case you were thinking of not taking it next term, 641 00:38:20,62 --> 00:38:21,98 you really should take it. 642 00:38:21,98 --> 00:38:25,07 It will put a lot of things in this course into 643 00:38:25,07 --> 00:38:26,46 context, for one thing. 644 00:38:26,72 --> 00:38:29,03 It's about vector calculus and so on. 645 00:38:29,03 --> 00:38:31,21 So you'll learn about vectors and things like that. 646 00:38:32,19 --> 00:38:35,59 But it comes back and explains some things in this course that 647 00:38:35,59 --> 00:38:38,15 might have been a little bit strange, like these strange 648 00:38:38,15 --> 00:38:46,08 formulas for the product rule and the quotient rule and these 649 00:38:46,08 --> 00:38:47,255 sort of random formulas. 650 00:38:48,71 --> 00:38:51,1 Well, one of the things you learn in 18.02 is that 651 00:38:51,1 --> 00:38:54,185 they're all special cases of the chain rule. 652 00:38:54,56 --> 00:38:59,39 And just to drive that point home, he wanted me to show you 653 00:38:59,39 --> 00:39:04,47 this poem of his that really drives the points home 654 00:39:04,47 --> 00:39:04,75 forcefully, I think. 655 00:39:04,75 --> 00:39:13,22