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 --> 00:00:03,144 Commons license. 4 00:00:03,144 --> 00:00:06,255 Your support will help MIT OpenCourseWare continue to 5 00:00:06,255 --> 00:00:09,243 offer high quality educational resources for free. 6 00:00:09,646 --> 00:00:12,54 To make a donation or to view additional materials from 7 00:00:12,54 --> 00:00:16,15 hundreds of MIT courses visit MIT OpenCourseWare 8 00:00:16,15 --> 00:00:17,09 at ocw.mit.edu. 9 00:00:21,57 --> 00:00:23,04 PROFESSOR: Today we are continuing with 10 00:00:23,04 --> 00:00:23,71 improper integrals. 11 00:00:24,73 --> 00:00:27,371 I still have a little bit more to tell you about them. 12 00:00:27,371 --> 00:00:30,08 13 00:00:30,08 --> 00:00:33,51 What we were discussing at the very end of last time 14 00:00:33,51 --> 00:00:36,195 was improper integrals. 15 00:00:36,195 --> 00:00:41,56 16 00:00:41,56 --> 00:00:44,57 Now and these are going to be improper integrals 17 00:00:44,57 --> 00:00:45,95 of the second kind. 18 00:00:48,22 --> 00:00:50,99 By second kind I mean if they have a singularity 19 00:00:50,99 --> 00:00:52,095 at a finite place. 20 00:00:52,095 --> 00:00:54,8 21 00:00:54,8 --> 00:00:56,24 That would be something like this. 22 00:00:57,26 --> 00:00:59,286 So here's the definition if you like. 23 00:01:00,14 --> 00:01:03,38 Same sort of thing as we did when the singularity 24 00:01:03,38 --> 00:01:04,12 was an infinity. 25 00:01:04,12 --> 00:01:07,29 So if you have the integral from 0 to 1 of f of x. 26 00:01:09,44 --> 00:01:11,4 This is going to be the same thing as the limit. 27 00:01:12,6 --> 00:01:14,856 As a goes to 0 from above. 28 00:01:15,79 --> 00:01:19,16 The integral from 8 to 1 of f of x d x. 29 00:01:21,15 --> 00:01:25,63 The idea here is the same one that we had at infinity. 30 00:01:25,92 --> 00:01:27,32 Let me draw a picture of it. 31 00:01:27,32 --> 00:01:29,82 You have imagine a function which is coming down like 32 00:01:29,82 --> 00:01:31,26 this and here's the point 1. 33 00:01:32,02 --> 00:01:35,63 And we don't know whether the area enclosed is going to be 34 00:01:35,63 --> 00:01:40,255 infinite or finite until we cut it off at someplace a. 35 00:01:40,68 --> 00:01:43,68 And we let a go to 0 from above. 36 00:01:44,22 --> 00:01:46 So really it's 0 plus. 37 00:01:46,8 --> 00:01:49,133 So we're coming in from the right here. 38 00:01:49,82 --> 00:01:52,08 And we're counting up the area in this chunk. 39 00:01:52,81 --> 00:01:56,79 And we're seeing as it expands whether it goes to infinity or 40 00:01:56,79 --> 00:01:58,67 whether it tends to some finite limit. 41 00:01:59,29 --> 00:02:02,25 Right so this is the example and this is the definition. 42 00:02:02,99 --> 00:02:06,42 And just as we did for the other kind of improper 43 00:02:06,42 --> 00:02:11,86 integral, we say that this converges-- so that's the key 44 00:02:11,86 --> 00:02:21,36 word here --if the limit is finite, exists maybe I should 45 00:02:21,36 --> 00:02:26,39 just say and diverges if not. 46 00:02:26,39 --> 00:02:35,65 47 00:02:35,65 --> 00:02:38,945 Let's just take care of the basic examples. 48 00:02:38,945 --> 00:02:42,11 49 00:02:42,11 --> 00:02:44,74 First of all I wrote this one down last time. 50 00:02:44,74 --> 00:02:46,96 We're going to evaluate this one. 51 00:02:47,39 --> 00:02:51,565 The integral from 0 to 1 of 1 over the square root of x. 52 00:02:51,565 --> 00:02:55,74 53 00:02:55,74 --> 00:02:58,65 You almost don't notice the fact that it goes to infinity. 54 00:02:59,92 --> 00:03:01,995 This goes to infinity as x goes to 0. 55 00:03:02,57 --> 00:03:05,45 But if you evaluate it -- first of all we always 56 00:03:05,45 --> 00:03:06,445 write this as a power. 57 00:03:06,86 --> 00:03:07,19 Right? 58 00:03:08,01 --> 00:03:09,03 To get evaluation. 59 00:03:09,87 --> 00:03:13,06 And then I'm not even going to replace the 0 by an a. 60 00:03:13,06 --> 00:03:14,49 I'm just going to leave it as 0. 61 00:03:14,49 --> 00:03:18,66 The antiderivative here is x to the 1/2 times 2. 62 00:03:18,66 --> 00:03:21,54 63 00:03:21,54 --> 00:03:23,75 And then I evaluate that at 0 and 1. 64 00:03:23,75 --> 00:03:25,08 And I get 2. 65 00:03:25,08 --> 00:03:27,13 2 minus 0, which is 2. 66 00:03:29,09 --> 00:03:30,443 All right so this one is convergent. 67 00:03:31,33 --> 00:03:34,18 And not only is it convergent but we can evaluate it. 68 00:03:34,18 --> 00:03:38,31 69 00:03:38,31 --> 00:03:42,94 The second example being not systematic but really giving 70 00:03:42,94 --> 00:03:45,27 you the principal examples that we'll be thinking about. 71 00:03:46,99 --> 00:03:53,92 Is this one here d x over x And this one gives you the 72 00:03:53,92 --> 00:03:55,03 antiderivative as the logarithm. 73 00:03:56,07 --> 00:03:57,56 Valuated at 0 and 1. 74 00:03:58,18 --> 00:04:00,91 And now again you have to have this thought process in your 75 00:04:00,91 --> 00:04:02,385 mind that your really taking the limit. 76 00:04:02,77 --> 00:04:06,44 But this is going to be the log of 1 minus the log of 0. 77 00:04:06,44 --> 00:04:07,83 Really the log of 0 from above. 78 00:04:07,83 --> 00:04:10,195 There is no such thing as the log 0 from below. 79 00:04:10,97 --> 00:04:12,52 And this is minus infinity. 80 00:04:12,82 --> 00:04:16,969 So it's 0 minus minus infinity, which is plus infinity. 81 00:04:19,02 --> 00:04:20,213 And so this one diverges. 82 00:04:20,213 --> 00:04:29,71 83 00:04:29,71 --> 00:04:31,685 All right so what's the general? 84 00:04:32,75 --> 00:04:40,41 So more or less in general for powers anyway, if you work out 85 00:04:40,41 --> 00:04:44,925 this thing for d x over x to the p from 0 to 1. 86 00:04:45,22 --> 00:04:49,39 What you're going to find is that it's 1 over 1 minus p. 87 00:04:50,15 --> 00:04:52,24 When p is less than one. 88 00:04:53,02 --> 00:05:00,24 And it diverges for p bigger than or equal to 1. 89 00:05:00,24 --> 00:05:02,75 90 00:05:02,75 --> 00:05:04,64 Now that's the final result. 91 00:05:05,31 --> 00:05:09,75 If you carry out this integration it's not difficult. 92 00:05:10,92 --> 00:05:15,17 All right so now I just want to try to help 93 00:05:15,17 --> 00:05:16,55 you to remember this. 94 00:05:17,27 --> 00:05:20,5 And to think about how you should think about it. 95 00:05:20,5 --> 00:05:22,84 So I'm going to say it in a few more ways. 96 00:05:23,89 --> 00:05:28,22 All right just repeat what I've said already but try to get it 97 00:05:28,22 --> 00:05:31,535 to percolate and absorb itself. 98 00:05:32,02 --> 00:05:36,19 And in order to do that I have to make the contrast between 99 00:05:36,19 --> 00:05:38,99 the kind of improper integral that I was dealing with before. 100 00:05:39,41 --> 00:05:43,23 Which was not as x goes to 0 here but as x goes to 101 00:05:43,23 --> 00:05:44,425 infinity, the other side. 102 00:05:45,92 --> 00:05:47,023 Let's make this contrast. 103 00:05:47,023 --> 00:05:52,47 104 00:05:52,47 --> 00:05:56,41 First of all, if I look at the angle that we have been paying 105 00:05:56,41 --> 00:05:57,61 attention to right now. 106 00:05:57,61 --> 00:06:00,58 We've just considered things like this. 107 00:06:00,58 --> 00:06:01,83 1 over x to the 1 half. 108 00:06:02,52 --> 00:06:06,335 Which is a lot smaller than one over x. 109 00:06:06,6 --> 00:06:09,245 Which is a lot smaller than say 1 over x squared. 110 00:06:10,74 --> 00:06:11,725 Which would be another example. 111 00:06:12,05 --> 00:06:14,56 This is as x goes to 0. 112 00:06:14,56 --> 00:06:19,99 113 00:06:19,99 --> 00:06:22,28 So this once the smallest one. 114 00:06:22,28 --> 00:06:23,31 This one next smallest one. 115 00:06:23,31 --> 00:06:24,44 And this one is very large. 116 00:06:26,82 --> 00:06:30,985 On the other hand it goes the other way at infinity. 117 00:06:30,985 --> 00:06:36,67 118 00:06:36,67 --> 00:06:38,12 As x tends to infinity. 119 00:06:39,936 --> 00:06:42,875 All right so try to keep that in mind. 120 00:06:43,14 --> 00:06:47,91 And now I'm going to put a little box around 121 00:06:47,91 --> 00:06:49,21 the bad guys here. 122 00:06:50,1 --> 00:06:53,5 This one is divergent. 123 00:06:54,71 --> 00:06:56,555 And this one is divergent. 124 00:06:57,87 --> 00:06:59,62 And this one is divergent. 125 00:06:59,95 --> 00:07:01,02 And this one is divergent. 126 00:07:01,02 --> 00:07:03,51 When the cross over point is 1 over x. 127 00:07:03,51 --> 00:07:06,19 When we get smaller than that, we get to things 128 00:07:06,19 --> 00:07:06,9 which are convergent. 129 00:07:07,39 --> 00:07:10,59 When we get smaller than it on this other scale, 130 00:07:10,59 --> 00:07:11,185 it's convergent. 131 00:07:12,68 --> 00:07:13,995 All right so these guys are divergent. 132 00:07:13,995 --> 00:07:20,3 133 00:07:20,3 --> 00:07:22,81 So they are associated with divergent integrals. 134 00:07:23,43 --> 00:07:26,39 The functions themselves are just tending towards-- well 135 00:07:26,39 --> 00:07:28,96 these tend to infinity, and these tend toward 0. 136 00:07:29,31 --> 00:07:33,83 So I'm not talking about the functions themselves 137 00:07:33,83 --> 00:07:34,49 but the integrals. 138 00:07:35,19 --> 00:07:40,04 Now I want to draw this again here, not small enough. 139 00:07:40,04 --> 00:07:43,81 140 00:07:43,81 --> 00:07:44,84 I want to try this again. 141 00:07:44,84 --> 00:07:48,98 142 00:07:48,98 --> 00:07:50,8 So I'm just going to draw a picture of what it 143 00:07:50,8 --> 00:07:51,76 is that I have here. 144 00:07:51,76 --> 00:07:53,39 But I'm going to combine these two pictures. 145 00:07:54,85 --> 00:08:01,63 So here's the picture for example of y equals 1 over x. 146 00:08:01,63 --> 00:08:04,714 147 00:08:04,714 --> 00:08:05,65 All right. 148 00:08:06,7 --> 00:08:08,03 That's y equals 1 over x. 149 00:08:08,77 --> 00:08:10,215 And that picture is very balanced. 150 00:08:10,47 --> 00:08:12,13 It's symmetric on the two ends. 151 00:08:12,6 --> 00:08:17,325 If I cut it in half then what I get here is 2 halves. 152 00:08:17,66 --> 00:08:21,7 And this one has an infinite area. 153 00:08:23,7 --> 00:08:26,75 That corresponds to the integral from 1 to infinity, 154 00:08:26,75 --> 00:08:29,43 d x over x being infinite. 155 00:08:30,53 --> 00:08:33,82 And the other piece-- which this one we 156 00:08:33,82 --> 00:08:35,31 calculated last time. 157 00:08:35,31 --> 00:08:37,85 This is the one that we just calculated over here at example 158 00:08:37,85 --> 00:08:42,719 2 has the same property. 159 00:08:42,719 --> 00:08:43,586 It's infinite. 160 00:08:45,72 --> 00:08:47,72 And that's the fact that the integral from 0 161 00:08:47,72 --> 00:08:50,79 to 1 is infinite. 162 00:08:52,1 --> 00:08:54,93 Right we lose on both ends. 163 00:08:56,57 --> 00:09:03,04 On the other hand if I take something like-- I'm drawing it 164 00:09:03,04 --> 00:09:06,57 the same way but it's really not the same --y equals 1 over 165 00:09:06,57 --> 00:09:10,705 the square root of x. y equals one over x to the 1 half. 166 00:09:11,01 --> 00:09:17,62 And if I cut that in half here then the x to the 1 half is 167 00:09:17,62 --> 00:09:19,4 actually bigger than this guy. 168 00:09:19,89 --> 00:09:21,74 So this piece is infinite. 169 00:09:21,74 --> 00:09:26,83 170 00:09:26,83 --> 00:09:29,81 And this part over here actually is going to give 171 00:09:29,81 --> 00:09:30,88 us an honest number. 172 00:09:31,31 --> 00:09:33,42 In fact this one is finite. 173 00:09:34,81 --> 00:09:36,55 And we just checked what the number is. 174 00:09:36,55 --> 00:09:38,34 It actually happens to have area 2. 175 00:09:38,34 --> 00:09:46,25 176 00:09:46,25 --> 00:09:49,57 And what's happening here is if you would superimpose this 177 00:09:49,57 --> 00:09:52,34 graph on the other graph what you would see is 178 00:09:52,34 --> 00:09:53,035 that they cross. 179 00:09:54,58 --> 00:09:58,585 And this one sits on top. 180 00:09:58,97 --> 00:10:04,66 So if I drew this one in let's have another color here 181 00:10:04,66 --> 00:10:05,84 --orange let's say. 182 00:10:05,84 --> 00:10:08,79 If this were orange if I set it on top here it 183 00:10:08,79 --> 00:10:10,39 would go this way. 184 00:10:11,13 --> 00:10:13,715 OK and underneath the orange is still infinite. 185 00:10:14,79 --> 00:10:16,07 So both of these are infinite. 186 00:10:16,07 --> 00:10:18,25 On here on the other hand underneath the orange is 187 00:10:18,25 --> 00:10:21 infinite but underneath where the green is finite. 188 00:10:21,7 --> 00:10:22,805 That's a smaller quantity. 189 00:10:23,77 --> 00:10:25,29 Infinity is a lot bigger than 2. 190 00:10:25,29 --> 00:10:26,68 2 is a lot less than infinity. 191 00:10:27,826 --> 00:10:30,835 All right so that's reflected in these comparisons here. 192 00:10:30,835 --> 00:10:33,12 Now if you like if I want to do these in green. 193 00:10:33,61 --> 00:10:37,475 This guy good and this guy is good. 194 00:10:39,9 --> 00:10:42,96 Well let me just repeat that idea over here in this sort 195 00:10:42,96 --> 00:10:47,176 of reversed picture with y equals 1 over x squared. 196 00:10:47,93 --> 00:10:52,94 If I chop that in half then the good part is this end here. 197 00:10:53,64 --> 00:10:54,44 It's finite. 198 00:10:54,44 --> 00:10:56,99 199 00:10:56,99 --> 00:10:59,83 And the bad part is this part of here which is 200 00:10:59,83 --> 00:11:00,746 way more singular. 201 00:11:01,48 --> 00:11:02,06 And it's infinite. 202 00:11:02,06 --> 00:11:07,07 203 00:11:07,07 --> 00:11:10,75 All right so again what I've just tried to do is to give 204 00:11:10,75 --> 00:11:18,225 you some geometric sense and also some visceral sense. 205 00:11:18,58 --> 00:11:22,68 This guy it's tail as it goes out to infinity is much lower. 206 00:11:23,04 --> 00:11:24,96 It's much smaller than 1 over x. 207 00:11:25,47 --> 00:11:28,08 And these guys trapped an infinite amount of area. 208 00:11:28,08 --> 00:11:30,11 This one traps only a finite amount of area. 209 00:11:30,11 --> 00:11:36,676 210 00:11:36,676 --> 00:11:40,88 All right so now I'm just going to give one last example which 211 00:11:40,88 --> 00:11:43,09 combines these two types of pictures. 212 00:11:43,09 --> 00:11:45,81 It's really practically the same as what I said 213 00:11:45,81 --> 00:11:53,32 before but I-- oh have to erase this one too. 214 00:11:53,32 --> 00:12:01,34 215 00:12:01,34 --> 00:12:07,16 So here's another example if you're in so let's take 216 00:12:07,16 --> 00:12:08,16 the following example. 217 00:12:08,16 --> 00:12:09,96 This is somewhat related to the first one that 218 00:12:09,96 --> 00:12:11,15 I gave last time. 219 00:12:11,6 --> 00:12:17,99 If you take a function y equals 1 over x minus 3 squared. 220 00:12:18,77 --> 00:12:20,46 And you think about it's integral. 221 00:12:21,16 --> 00:12:24,61 So let's think about the integral from 0 to infinity, 222 00:12:24,61 --> 00:12:27,09 d x over x minus 3 squared. 223 00:12:27,09 --> 00:12:28,68 And suppose you were faced with this integral. 224 00:12:30,42 --> 00:12:33,21 In order to understand what it's doing you have to pay 225 00:12:33,21 --> 00:12:35,915 attention just two places where it can go wrong. 226 00:12:36,31 --> 00:12:38,055 We're going to split into two pieces. 227 00:12:39,53 --> 00:12:44,54 I'm going say break it up into this one here up to 5, 228 00:12:44,54 --> 00:12:45,39 for the sake of argument. 229 00:12:45,39 --> 00:12:48,09 230 00:12:48,09 --> 00:12:49,715 And say from 5 to infinity. 231 00:12:49,715 --> 00:12:53,672 232 00:12:53,672 --> 00:12:54,545 All right. 233 00:12:54,96 --> 00:12:55,965 So these are the two chunks. 234 00:12:56,26 --> 00:12:58,36 Now why did I break it up into those two pieces? 235 00:12:58,81 --> 00:13:01,89 Because what's happening with this function is that it's 236 00:13:01,89 --> 00:13:04,43 going up like this at 3. 237 00:13:05,25 --> 00:13:08,47 And so if I look at the two halves here. 238 00:13:08,47 --> 00:13:10,64 I'm going to draw them again and I'm going to illustrate 239 00:13:10,64 --> 00:13:15,35 with what colors we've chosen-- which are I guess red and green 240 00:13:15,35 --> 00:13:25,1 --what you'll discover is that this one here which corresponds 241 00:13:25,1 --> 00:13:29,02 to this piece here is infinite. 242 00:13:30,54 --> 00:13:33,18 And it's infinite because there's a square in 243 00:13:33,18 --> 00:13:33,59 the denominator. 244 00:13:34,27 --> 00:13:39,19 And as x goes to 3 this is very much like we 245 00:13:39,19 --> 00:13:40,345 shifted the 3 to 0. 246 00:13:40,67 --> 00:13:42,56 Very much like this 1 over x squared here. 247 00:13:42,56 --> 00:13:43,87 But not in this context. 248 00:13:44,25 --> 00:13:46,33 In the other context where it's going to infinity. 249 00:13:46,33 --> 00:13:49,46 250 00:13:49,46 --> 00:13:54,32 This is the same as at the picture directly above with 251 00:13:54,32 --> 00:13:55,79 the infinite part in red. 252 00:13:57,398 --> 00:13:58,07 All right. 253 00:13:59,65 --> 00:14:04,5 And this part here This part is finite. 254 00:14:04,915 --> 00:14:05,785 All right. 255 00:14:06,24 --> 00:14:08,96 So since we have an infinite part plus a finite part 256 00:14:08,96 --> 00:14:13,03 the conclusion is that this guy converges. 257 00:14:15,07 --> 00:14:18,85 And this one diverges. 258 00:14:18,85 --> 00:14:21,47 259 00:14:21,47 --> 00:14:23,595 But the total unfortunately diverges. 260 00:14:23,93 --> 00:14:25,76 Right because it's got 1 infinity in it. 261 00:14:25,76 --> 00:14:28,37 So this thing diverges. 262 00:14:28,37 --> 00:14:31,99 263 00:14:31,99 --> 00:14:33,58 And that's what happened last time when we 264 00:14:33,58 --> 00:14:34,87 got a crazy number. 265 00:14:34,87 --> 00:14:37,105 If you integrated this you would get some negative number. 266 00:14:37,48 --> 00:14:39,36 If you wrote down the formulas carelessly. 267 00:14:39,85 --> 00:14:41,92 And the reason is that the calculation 268 00:14:41,92 --> 00:14:42,63 actually is nonsense. 269 00:14:44,71 --> 00:14:48,59 So you've gotta be aware if you encounter a singularity in 270 00:14:48,59 --> 00:14:50,43 the middle not to ignore it. 271 00:14:51,84 --> 00:14:52,1 Yeah. 272 00:14:52,1 --> 00:14:52,36 Question. 273 00:14:52,62 --> 00:14:56,43 AUDIENCE: [INAUDIBLE PHRASE] 274 00:14:56,43 --> 00:14:58,25 PROFESSOR: Why do we say that the whole thing diverges? 275 00:14:59,71 --> 00:15:01,79 The reason why we say that is the area under the 276 00:15:01,79 --> 00:15:02,89 whole curve is infinite. 277 00:15:03,69 --> 00:15:05,785 It's the sum of this piece plus this piece. 278 00:15:06,13 --> 00:15:07,62 And so the total is infinite. 279 00:15:08,886 --> 00:15:17,37 AUDIENCE: [INAUDIBLE PHRASE] 280 00:15:17,37 --> 00:15:17,99 PROFESSOR: We're stuck. 281 00:15:17,99 --> 00:15:19,27 This is an ill defined integral. 282 00:15:19,27 --> 00:15:21,72 It's one where you're red flashing warning 283 00:15:21,72 --> 00:15:22,56 sign should be on. 284 00:15:22,56 --> 00:15:24,04 Because you're not going to get the right answer 285 00:15:24,04 --> 00:15:24,51 by computing it. 286 00:15:24,51 --> 00:15:25,65 You'll never get an answer. 287 00:15:26,86 --> 00:15:28,84 Similarly you'll never get an answer with this. 288 00:15:29,23 --> 00:15:30,85 And you will get an answer with that. 289 00:15:32,73 --> 00:15:33,2 OK? 290 00:15:33,2 --> 00:15:37,22 291 00:15:37,22 --> 00:15:38,23 Yeah another question. 292 00:15:39,64 --> 00:15:45,76 AUDIENCE: [INAUDIBLE PHRASE] 293 00:15:45,76 --> 00:15:49,48 PROFESSOR: So the question is if you have a little glance at 294 00:15:49,48 --> 00:15:53,03 an integral, how are you going to decide where you 295 00:15:53,03 --> 00:15:53,77 should be heading? 296 00:15:54,75 --> 00:15:57,395 So I'm going to answer that orally. 297 00:15:58,28 --> 00:16:03,665 Although you know but I'll say one little hint here. 298 00:16:04,01 --> 00:16:08,29 So you always have to check x going to infinity and x going 299 00:16:08,29 --> 00:16:10,485 to minus infinity, if they're in there. 300 00:16:10,74 --> 00:16:14,81 And you also have to check any singularity, like x going 301 00:16:14,81 --> 00:16:16,78 to 3 for sure in this case. 302 00:16:16,78 --> 00:16:18,33 You have to pay attention all the places where 303 00:16:18,33 --> 00:16:19,015 the thing is infinite. 304 00:16:19,35 --> 00:16:21,865 And then you want to focus in on each one separately. 305 00:16:22,33 --> 00:16:25,16 And decide what's going on it at that particular place. 306 00:16:26,92 --> 00:16:29,25 When it's a negative power. 307 00:16:29,59 --> 00:16:35,26 So remember d x over x as x goes to 0 is bad. 308 00:16:36,98 --> 00:16:39,07 And d x over x squared is bad. 309 00:16:39,07 --> 00:16:40,49 D x over x cubed is bad. 310 00:16:40,49 --> 00:16:41,69 All of them are even worse. 311 00:16:42,6 --> 00:16:48,455 So anything of this form is bad and equals 1, 2, 3. 312 00:16:49,65 --> 00:16:51 These are the red box kind. 313 00:16:51,98 --> 00:16:52,725 All right. 314 00:16:55,03 --> 00:16:58,22 That means that any of the integrals that we did in 315 00:16:58,22 --> 00:17:01,89 partial fractions which had a root, which had a factor of 316 00:17:01,89 --> 00:17:04,34 something, the denominator Those are all divergent 317 00:17:04,34 --> 00:17:05,876 integrals if you cross the singularly. 318 00:17:06,67 --> 00:17:09,524 Not a single one of them makes sense across the singularity. 319 00:17:09,524 --> 00:17:12,75 320 00:17:12,75 --> 00:17:14,85 If you have square roots and things like that then you can 321 00:17:14,85 --> 00:17:16,1 repair things like that. 322 00:17:16,1 --> 00:17:18,03 And there's some interesting examples of that. 323 00:17:18,03 --> 00:17:20,825 Such as with the arc sign function and so forth. 324 00:17:21,08 --> 00:17:23,784 Where you have an improper integral which is really OK. 325 00:17:25,94 --> 00:17:26,39 All right. 326 00:17:26,73 --> 00:17:29,21 So that's the best I can do. 327 00:17:29,88 --> 00:17:31,95 It's obviously something you get experience with. 328 00:17:32,26 --> 00:17:32,74 All right. 329 00:17:34,11 --> 00:17:38,55 Now I'm going to move on and this is more or 330 00:17:38,55 --> 00:17:40,385 less our last topic. 331 00:17:42,15 --> 00:17:43,84 Yay but not quite. 332 00:17:43,9 --> 00:17:46,73 Well it's our penultimate topic. 333 00:17:46,73 --> 00:17:48,13 Right because we have one more lecture. 334 00:17:49,43 --> 00:17:49,85 All right. 335 00:17:52,01 --> 00:17:54,335 So that our next topic is series. 336 00:17:54,75 --> 00:17:58,415 Now we'll do it in a sort of a concrete way today. 337 00:17:58,81 --> 00:18:02,775 And then we'll do what are known as power series tomorrow. 338 00:18:02,775 --> 00:18:05,4 339 00:18:05,4 --> 00:18:06,76 So let me tell you about series. 340 00:18:06,76 --> 00:18:20,49 341 00:18:20,49 --> 00:18:22,79 Remember we're talking about infinity and 342 00:18:22,79 --> 00:18:23,687 dealing with infinity. 343 00:18:23,687 --> 00:18:26,82 344 00:18:26,82 --> 00:18:28,73 So we're not just talking about any old series. 345 00:18:28,73 --> 00:18:30,13 We're talking about infinite series. 346 00:18:30,13 --> 00:18:32,65 347 00:18:32,65 --> 00:18:37,69 There is one infinite series which is probably, which is 348 00:18:37,69 --> 00:18:40,96 without question the most important and useful series. 349 00:18:41,98 --> 00:18:46,15 And that's the geometric series but I'm going to introduce it 350 00:18:46,15 --> 00:18:48,405 concretely first in a particular case. 351 00:18:48,405 --> 00:18:51,99 352 00:18:51,99 --> 00:18:54,26 If I draw a picture of this sum. 353 00:18:54,26 --> 00:18:56,13 Which in principle goes on forever. 354 00:18:56,48 --> 00:18:59,92 You can see that it go someplace fairly easily by 355 00:18:59,92 --> 00:19:02,31 marking out what's happening on the number line. 356 00:19:02,73 --> 00:19:05,43 The first step takes us to 1 from 0. 357 00:19:06,63 --> 00:19:10,075 And then if I add this half, I get to 3 halves. 358 00:19:11,79 --> 00:19:15,19 Right, so the first step was 1 and the second step was a half. 359 00:19:16,24 --> 00:19:20,51 Now see if I add this quarter in which is the next piece 360 00:19:20,51 --> 00:19:22,4 then I get some place here. 361 00:19:22,4 --> 00:19:29,12 But what I want to observe is that I can look at it from 362 00:19:29,12 --> 00:19:29,975 the other point of view. 363 00:19:31,13 --> 00:19:34,44 When I move this quarter I got half way to 2 here. 364 00:19:36,24 --> 00:19:40,04 I'm putting 2 in green because I want you to think of it 365 00:19:40,04 --> 00:19:42,64 as being the good kind. 366 00:19:42,95 --> 00:19:43,39 Right. 367 00:19:43,97 --> 00:19:45,075 The kind that has a number. 368 00:19:45,42 --> 00:19:46,66 And not one of the red kind. 369 00:19:47,28 --> 00:19:49,61 We're getting there and we're almost there. 370 00:19:49,92 --> 00:19:52,62 So the next stage we get half way again. 371 00:19:52,62 --> 00:19:54,605 That's the eighth and so forth. 372 00:19:54,605 --> 00:19:56,49 And eventually we get to 2. 373 00:19:56,49 --> 00:19:59,03 So this sum we write equals two. 374 00:20:00,476 --> 00:20:02,84 All right that's kind of a paradox because 375 00:20:02,84 --> 00:20:04,13 we never get to 2. 376 00:20:04,13 --> 00:20:07,74 This is the paradox that Zeno fussed with. 377 00:20:08,08 --> 00:20:12,265 And his conclusion with the rabbit and the hare. 378 00:20:12,96 --> 00:20:14,06 Oh no the rabbit and the tortoise. 379 00:20:14,14 --> 00:20:18,53 Sorry hare chasing-- anyway the rabbit chasing the tortoise. 380 00:20:18,89 --> 00:20:22,47 His conclusion-- I don't know if you're aware of this --but 381 00:20:22,47 --> 00:20:23,77 he understood this paradox. 382 00:20:23,77 --> 00:20:25,92 And he said you know it doesn't look like it ever gets there 383 00:20:25,92 --> 00:20:30,27 because they're infinitely many time between the time-- you 384 00:20:30,27 --> 00:20:32,9 know that the tortoise is always behind, always behind, 385 00:20:32,9 --> 00:20:34,12 always behind, always behind. 386 00:20:34,58 --> 00:20:36,97 So therefore it's impossible that the tortoise 387 00:20:36,97 --> 00:20:38,03 catches up right. 388 00:20:38,71 --> 00:20:40,15 So do you know what his conclusion was? 389 00:20:41,6 --> 00:20:42,95 Time does not exist. 390 00:20:45,28 --> 00:20:46,825 That was actually literally his conclusion. 391 00:20:48,26 --> 00:20:49,95 Because he didn't understand the possibility of a 392 00:20:49,95 --> 00:20:51,03 continuum of time. 393 00:20:51,03 --> 00:20:53,09 Because there were infinitely many things that happened 394 00:20:53,09 --> 00:20:54,91 before the tortoise caught up. 395 00:20:56,61 --> 00:20:57,79 So that was the reasoning. 396 00:20:57,79 --> 00:21:00,61 I mean it's a long time ago but you know he didn't 397 00:21:00,61 --> 00:21:01,27 believe in continuum. 398 00:21:02,43 --> 00:21:02,78 All right. 399 00:21:03,48 --> 00:21:06,14 So anyway that's a small point. 400 00:21:06,94 --> 00:21:19,84 Now the general case here of geometric series is where 401 00:21:19,84 --> 00:21:22,42 I put in a number a instead of a half here. 402 00:21:22,82 --> 00:21:23,87 So what we had before. 403 00:21:23,87 --> 00:21:26,19 So that's 1 plus a plus a squared. 404 00:21:26,89 --> 00:21:28,39 Isn't quite the most general but anyway 405 00:21:28,39 --> 00:21:30,43 I'll write this down. 406 00:21:31,9 --> 00:21:34,72 And your certainly going to want to remember that the 407 00:21:34,72 --> 00:21:39,03 formula for this in the limit is 1 over 1 minus a. 408 00:21:39,51 --> 00:21:43,47 And I remind you that there's only works when the absolute 409 00:21:43,47 --> 00:21:45,4 value is strictly less than 1. 410 00:21:45,81 --> 00:21:47,66 In other words when minus 1 is strictly less 411 00:21:47,66 --> 00:21:48,74 than a is less than 1. 412 00:21:51,13 --> 00:21:53,37 And that's really the issue that we're going to want 413 00:21:53,37 --> 00:21:54,31 to worry about now. 414 00:21:54,31 --> 00:21:57,45 What we're worrying about is this notion of convergence. 415 00:21:58,27 --> 00:22:06,13 And what goes wrong when there isn't convergence, when 416 00:22:06,13 --> 00:22:06,686 there's a divergence. 417 00:22:07,36 --> 00:22:13,22 So let me illustrate divergence before going on. 418 00:22:13,22 --> 00:22:15,995 And this is what we have to avoid if we're going 419 00:22:15,995 --> 00:22:17,035 to understand series. 420 00:22:18,99 --> 00:22:21,475 So here's an example when a is equal to 1. 421 00:22:21,89 --> 00:22:25,833 You get 1 plus 1 plus 1 plus et cetera. 422 00:22:26,62 --> 00:22:29,21 And that's equal to 1 over 1 minus 1. 423 00:22:29,99 --> 00:22:32,37 Which is 1 over 0. 424 00:22:32,37 --> 00:22:33,76 So this is not bad. 425 00:22:33,76 --> 00:22:35,13 It's almost right. 426 00:22:35,36 --> 00:22:36,8 It's sort of infinity equals infinity. 427 00:22:37,75 --> 00:22:40,01 At the edge here we managed to get something which 428 00:22:40,01 --> 00:22:41,39 is sort of almost right. 429 00:22:42,34 --> 00:22:46,1 But you know it's we don't consider this to be logically 430 00:22:46,1 --> 00:22:47,54 to make complete sense. 431 00:22:47,54 --> 00:22:52,19 So it's a little dangerous And so we just say 432 00:22:52,19 --> 00:22:52,79 that it diverges. 433 00:22:52,96 --> 00:22:53,83 And we get rid of this. 434 00:22:54,1 --> 00:22:55,585 So we're still putting it in red. 435 00:22:55,99 --> 00:22:56,5 All right. 436 00:22:58,54 --> 00:23:00,85 The bad guy here so this one diverges. 437 00:23:00,85 --> 00:23:04,67 438 00:23:04,67 --> 00:23:11,01 Similarly if I take a equals minus 1, I get 1 minus 1 plus 1 439 00:23:11,01 --> 00:23:14,78 minus one plus 1 because the odd and even powers in that 440 00:23:14,78 --> 00:23:16,39 formula alternate sign. 441 00:23:17,27 --> 00:23:19,375 And this bounces back and forth. 442 00:23:19,75 --> 00:23:21,09 It never settles down. 443 00:23:21,64 --> 00:23:23,295 It starts at 1. 444 00:23:23,56 --> 00:23:25,42 And then it gets down to 0 and then comes back up to 445 00:23:25,42 --> 00:23:27,52 1, down to 0, back up to 1. 446 00:23:28,4 --> 00:23:29,57 It doesn't settle down. 447 00:23:29,57 --> 00:23:30,6 It bounces back and forth. 448 00:23:30,6 --> 00:23:30,7 It oscillates. 449 00:23:31,58 --> 00:23:33,973 On the other hand if you compare the right hand side. 450 00:23:34,74 --> 00:23:35,98 What's the right hand side? 451 00:23:35,98 --> 00:23:38,23 It's 1 over 1 minus minus 1. 452 00:23:39,73 --> 00:23:40,87 which is a half. 453 00:23:41,515 --> 00:23:42,18 All right. 454 00:23:42,46 --> 00:23:44,71 So if you just paid attention to the formula. 455 00:23:45,12 --> 00:23:47,24 Which is what we were doing when we integrated without 456 00:23:47,24 --> 00:23:48,52 thinking too hard about this. 457 00:23:49,24 --> 00:23:51,32 You get a number here but in fact that's wrong. 458 00:23:51,32 --> 00:23:52,83 Actually it's kind of an interesting number. 459 00:23:52,83 --> 00:23:55,7 It's halfway between the 2 between 0 1. 460 00:23:56,38 --> 00:23:59,57 So again there's some sort of vague sense in which this is 461 00:23:59,57 --> 00:24:01,613 trying to be this answer. 462 00:24:01,966 --> 00:24:02,575 All right. 463 00:24:04,58 --> 00:24:07,64 It's not so bad but we're still going to put this in a red box. 464 00:24:08,744 --> 00:24:09,59 All right. 465 00:24:10,03 --> 00:24:11,91 because this is what we called divergence. 466 00:24:12,71 --> 00:24:15,875 So both of these cases are divergent. 467 00:24:16,13 --> 00:24:19,67 It only really works when a is less than 1. 468 00:24:20,49 --> 00:24:25,92 I'm going to add one more case just to see that mathematicians 469 00:24:25,92 --> 00:24:32,02 are slightly curious about what goes on in other cases. 470 00:24:32,02 --> 00:24:34,89 So this is 1 plus 2 plus 2 squared plus 471 00:24:34,89 --> 00:24:37,255 2 to cube plus etc.. 472 00:24:37,57 --> 00:24:41,44 And that should be equal to-- according to this formula 473 00:24:41,44 --> 00:24:43,5 --1 over 1 minus 2. 474 00:24:44,75 --> 00:24:46,25 Which is negative 1. 475 00:24:48,242 --> 00:24:49,113 All right. 476 00:24:49,86 --> 00:24:52,74 Now this one it clearly wrong, right? 477 00:24:53,46 --> 00:24:59,16 This one it's totally Wrong It certainly diverges. 478 00:24:59,465 --> 00:25:02,37 The left hand side is obviously infinite. 479 00:25:02,37 --> 00:25:03,79 The right hand side is way off. 480 00:25:04,07 --> 00:25:04,97 It's negative 1. 481 00:25:05,96 --> 00:25:11,33 On the other hand it turns out actually that mathematicians 482 00:25:11,33 --> 00:25:12,94 have ways of making sense out of these. 483 00:25:13,36 --> 00:25:15,81 In number theory there's a strange system where 484 00:25:15,81 --> 00:25:16,795 this is actually true. 485 00:25:18,16 --> 00:25:22,6 And what happens in that system is that what you have to throw 486 00:25:22,6 --> 00:25:26,31 out is the idea that 0 is less than you 1. 487 00:25:27,05 --> 00:25:29,15 There is no such thing as negative numbers. 488 00:25:29,98 --> 00:25:31,77 So this number exists. 489 00:25:32,09 --> 00:25:35,7 And it's the additive inverse of 1. 490 00:25:35,7 --> 00:25:41,37 It has this arithmetic property but the statement that 1 is 491 00:25:41,37 --> 00:25:42,975 bigger than 0 does not make sense. 492 00:25:43,33 --> 00:25:45,83 So you have your choice either this diverges or you have to 493 00:25:45,83 --> 00:25:48,08 throw out something like this. 494 00:25:48,63 --> 00:25:51,12 So that's a very curious thing in higher mathematics. 495 00:25:51,51 --> 00:25:56,06 Which if you get to number theory there's fun stuff there. 496 00:25:56,41 --> 00:25:57,21 All right. 497 00:25:58,92 --> 00:26:02,345 OK but for our purposes these things are all out. 498 00:26:02,74 --> 00:26:03,02 All right. 499 00:26:03,3 --> 00:26:03,72 They're gone. 500 00:26:04,01 --> 00:26:05,16 We're not considering them. 501 00:26:05,16 --> 00:26:08,03 Only a between negative 1 and 1. 502 00:26:09,55 --> 00:26:10,07 All right. 503 00:26:10,07 --> 00:26:13,91 504 00:26:13,91 --> 00:26:17,76 Now I want to do something systematic. 505 00:26:18,19 --> 00:26:21,76 And it's more or less on the lines of the powers that 506 00:26:21,76 --> 00:26:23,09 I'm erasing right now. 507 00:26:23,09 --> 00:26:26,63 508 00:26:26,63 --> 00:26:28,36 I want to tell you about series. 509 00:26:28,36 --> 00:26:29,985 Which are kind of borderline convergent. 510 00:26:30,42 --> 00:26:33,93 And then next time when we talk about powers series we'll come 511 00:26:33,93 --> 00:26:36,08 back to this very important series which is the 512 00:26:36,08 --> 00:26:37,08 most important one. 513 00:26:37,08 --> 00:26:40,68 514 00:26:40,68 --> 00:26:46,426 So now let's talk about some series general notations. 515 00:26:47,4 --> 00:26:49,97 And this will help you with the last bit. 516 00:26:49,97 --> 00:26:53,81 517 00:26:53,81 --> 00:26:56,89 This is going to be pretty much the same as what we 518 00:26:56,89 --> 00:26:59,12 did for improper integrals. 519 00:27:00,89 --> 00:27:06,12 First of all I'm going to have capital S N which is the sum of 520 00:27:06,12 --> 00:27:09,38 a n, n equals 0 to capital N. 521 00:27:09,76 --> 00:27:12,22 And this is what we're calling a partial sum. 522 00:27:12,22 --> 00:27:18,05 523 00:27:18,05 --> 00:27:24,67 And then the full limit which is capital S, if you like, and 524 00:27:24,67 --> 00:27:30,13 N equals 0 to infinity is just the limit as N goes to 525 00:27:30,13 --> 00:27:32,01 infinity of the Sn. 526 00:27:32,01 --> 00:27:36,24 527 00:27:36,24 --> 00:27:39,325 And then we have the same kind of notation that we had before. 528 00:27:39,77 --> 00:27:45,02 Which is there these two choices which is that 529 00:27:45,02 --> 00:27:45,89 if the limit exists. 530 00:27:45,89 --> 00:27:50,45 531 00:27:50,45 --> 00:27:51,48 That's the green choice. 532 00:27:51,83 --> 00:27:52,76 And we say it converges. 533 00:27:54,46 --> 00:28:00,156 So we say the series converges. 534 00:28:00,83 --> 00:28:06,616 And then the other case which is limit does not exist. 535 00:28:06,616 --> 00:28:10,9 536 00:28:10,9 --> 00:28:12,24 And we can say the series diverges. 537 00:28:12,24 --> 00:28:20,56 538 00:28:20,56 --> 00:28:21,06 Question. 539 00:28:22,026 --> 00:28:26,48 AUDIENCE: [INAUDIBLE PHRASE] 540 00:28:26,48 --> 00:28:28,885 PROFESSOR: The question was how did I get to this? 541 00:28:29,29 --> 00:28:31,96 And I will do that next time but in fact of course you've 542 00:28:31,96 --> 00:28:32,885 seen in high school. 543 00:28:33,24 --> 00:28:34,84 Right this a-- Yeah. 544 00:28:35,93 --> 00:28:36,285 Yeah. 545 00:28:36,86 --> 00:28:37,8 We'll do that next time. 546 00:28:40 --> 00:28:42,99 The question was how do we arrive-- sorry I didn't tell 547 00:28:42,99 --> 00:28:44,62 you the question --the question was how do we arrive at this 548 00:28:44,62 --> 00:28:46,202 formula on the right hand side here. 549 00:28:46,59 --> 00:28:48,24 But we'll talk about that next time. 550 00:28:48,24 --> 00:28:53,06 551 00:28:53,06 --> 00:28:53,54 All right. 552 00:28:54,02 --> 00:29:00,17 So here's the basic definition and what we're going to 553 00:29:00,17 --> 00:29:01,38 recognize about series. 554 00:29:02,43 --> 00:29:07,53 And I'm going to give you a few examples and then we'll 555 00:29:07,53 --> 00:29:08,46 do something systematic. 556 00:29:08,46 --> 00:29:12,07 557 00:29:12,07 --> 00:29:15,38 So the first example-- well the first example is the geometric 558 00:29:15,38 --> 00:29:19,43 series But the first example that I'm going to discuss now 559 00:29:19,43 --> 00:29:23,39 and in a little bit of detail is this sum 1 over n squared 560 00:29:23,39 --> 00:29:24,31 N equals 1 to infinity. 561 00:29:24,31 --> 00:29:28,58 562 00:29:28,58 --> 00:29:34,56 It turns out that this series is very analogous-- and we'll 563 00:29:34,56 --> 00:29:38,89 develop this analogy carefully --the integral from 1 to 564 00:29:38,89 --> 00:29:41,065 x d x over x squared. 565 00:29:41,065 --> 00:29:45,88 And we're going to develop this analogy in detail 566 00:29:45,88 --> 00:29:46,89 later in this lecture. 567 00:29:47,92 --> 00:29:50,61 And this one is one of the ones-- so now you have to go 568 00:29:50,61 --> 00:29:52,84 back and actually remember this is one of the ones you 569 00:29:52,84 --> 00:29:53,79 really want to memorize. 570 00:29:54,51 --> 00:29:56,46 And you should especially pay attention to the ones 571 00:29:56,46 --> 00:29:57,72 with an infinity in them. 572 00:29:58,57 --> 00:29:59,465 This one is convergent. 573 00:29:59,465 --> 00:30:03,27 574 00:30:03,27 --> 00:30:04,52 And this series convergent. 575 00:30:04,52 --> 00:30:09,645 Now it turns out that evaluating this is very easy. 576 00:30:11,07 --> 00:30:11,84 This is 1. 577 00:30:12,82 --> 00:30:14,015 It's easy to calculation. 578 00:30:15,31 --> 00:30:17,375 Evaluating this is very tricky. 579 00:30:19,5 --> 00:30:21,23 And Euler did it. 580 00:30:21,61 --> 00:30:24,255 And the answer with pie squared over 6. 581 00:30:26,44 --> 00:30:28,07 That's amazing calculation. 582 00:30:29,05 --> 00:30:32,165 And it was done very early in the history of mathematics. 583 00:30:33,57 --> 00:30:37,4 If you look at another example-- so maybe example 584 00:30:37,4 --> 00:30:40,95 two here -if you look at 1 over n cubed. 585 00:30:42,3 --> 00:30:46,38 n equals-- well you can't start here at 0 by the way. 586 00:30:46,58 --> 00:30:48,675 I get to start where ever I want in these series. 587 00:30:48,93 --> 00:30:49,92 Here I start with 0. 588 00:30:49,92 --> 00:30:50,95 Here I started with 1. 589 00:30:51,22 --> 00:30:55,34 And notice the reason why it was a bad idea to start with 0 590 00:30:55,34 --> 00:30:57,28 was that 1 over 0 is undefined. 591 00:30:58,13 --> 00:31:00,43 Right so I'm just starting where it's convenient for me. 592 00:31:00,43 --> 00:31:04,08 And since I'm interested mostly in the tale behavior it doesn't 593 00:31:04,08 --> 00:31:05,815 matter to me so much where I start. 594 00:31:06,15 --> 00:31:08,37 Although if I want an exact answer I need to start 595 00:31:08,37 --> 00:31:09,71 exactly at n equals 1. 596 00:31:10,165 --> 00:31:10,89 All right. 597 00:31:10,89 --> 00:31:16,765 This one is similar to this integral here. 598 00:31:17,932 --> 00:31:18,7 All right. 599 00:31:18,7 --> 00:31:20,07 Which is convergent again. 600 00:31:20,44 --> 00:31:22,03 So there's a number that you get. 601 00:31:22,39 --> 00:31:26,84 And let's see what is it something like 2 thirds or 602 00:31:26,84 --> 00:31:29,845 something like that for this for this number. 603 00:31:30,19 --> 00:31:30,985 Or a third. 604 00:31:32,44 --> 00:31:33,11 What is it? 605 00:31:33,11 --> 00:31:33,4 No a half. 606 00:31:33,69 --> 00:31:34,52 I guess it's a half. 607 00:31:35 --> 00:31:35,82 This one is a half. 608 00:31:37,11 --> 00:31:39,7 You check that I'm not positive but anyway just doing it 609 00:31:39,7 --> 00:31:41,69 in my head quickly it seems to be a half. 610 00:31:42,15 --> 00:31:43,715 Anyway it's an easy number to calculate. 611 00:31:44,1 --> 00:31:48,94 This one over here stumped mathematicians 612 00:31:48,94 --> 00:31:50,53 basically for all time. 613 00:31:51,87 --> 00:31:55,035 It doesn't have any kind of elementary form like this. 614 00:31:56,25 --> 00:31:58,855 And it was only very recently proved to be rational. 615 00:31:59,91 --> 00:32:02,05 People couldn't even couldn't even decide whether this was 616 00:32:02,05 --> 00:32:03,476 a rational number or not. 617 00:32:05,26 --> 00:32:07,83 But anyway that's been resolved it is an irrational number 618 00:32:07,83 --> 00:32:08,866 which is what people suspected. 619 00:32:09,54 --> 00:32:10,375 Yeah question. 620 00:32:10,65 --> 00:32:14,36 AUDIENCE: [INAUDIBLE] 621 00:32:14,36 --> 00:32:15,496 PROFESSOR: Yeah sorry. 622 00:32:16,35 --> 00:32:25,44 OK I violated a rule of mathematics-- you said 623 00:32:25,44 --> 00:32:26,32 why is this similar? 624 00:32:26,72 --> 00:32:28,835 I thought that similar was something else. 625 00:32:29,14 --> 00:32:30,38 And you're absolutely right. 626 00:32:30,38 --> 00:32:32,846 And I violated a rule of mathematics. 627 00:32:33,52 --> 00:32:37,18 Which is that I uses this symbol for two 628 00:32:37,18 --> 00:32:37,825 different things. 629 00:32:37,825 --> 00:32:41,21 630 00:32:41,21 --> 00:32:42,82 I should have written this symbol here. 631 00:32:42,82 --> 00:32:43,175 All right. 632 00:32:43,53 --> 00:32:44,935 I'll create a new symbol here. 633 00:32:45,26 --> 00:32:48,23 The question of whether this converges or this converges. 634 00:32:48,92 --> 00:32:51,52 These are this the same type of question. 635 00:32:51,92 --> 00:32:54,09 And we'll see why they're the same question it 636 00:32:54,09 --> 00:32:54,73 in a few minutes. 637 00:32:55 --> 00:32:59,78 But in fact the wiggle I use similar I used for the 638 00:32:59,78 --> 00:33:00,75 connection between functions. 639 00:33:02,36 --> 00:33:05,24 The things that are really similar are that 1 over n 640 00:33:05,24 --> 00:33:09,456 resembles 1 over x squared. 641 00:33:10,23 --> 00:33:12,79 So I apologize I didn't-- 642 00:33:12,79 --> 00:33:15,89 AUDIENCE: [INAUDIBLE PHRASE] 643 00:33:15,89 --> 00:33:18,14 PROFESSOR: Oh you thought that this was the 644 00:33:18,14 --> 00:33:19,29 definition of that. 645 00:33:19,29 --> 00:33:20,95 That's actually the reason why these things 646 00:33:20,95 --> 00:33:21,99 correspond so closely. 647 00:33:21,99 --> 00:33:25,26 That is that the Riemann's sum is close to this. 648 00:33:25,56 --> 00:33:26,753 But that doesn't mean they're equal. 649 00:33:27,52 --> 00:33:30,55 The Riemann's sum only works when the delta x goes to 0. 650 00:33:31,95 --> 00:33:33,87 The way that we're going to get a connection between these two 651 00:33:33,87 --> 00:33:37,41 as, we will just a second, is with a Riemann sum with. 652 00:33:38,49 --> 00:33:44,18 What we're going to use is a Riemann's sum 653 00:33:44,18 --> 00:33:45,95 with delta x equals 1. 654 00:33:47,18 --> 00:33:49,825 All right and then that will be the connection between. 655 00:33:50,24 --> 00:33:51,45 Right that's absolutely right. 656 00:33:53,48 --> 00:33:53,74 All right. 657 00:33:53,74 --> 00:33:57,02 658 00:33:57,02 --> 00:34:00,4 So in order to illustrate exactly this idea that 659 00:34:00,4 --> 00:34:01,52 you've just come up with. 660 00:34:01,52 --> 00:34:03,11 And in fact that we're going to use. 661 00:34:03,11 --> 00:34:05,82 We'll do the same thing but we're going to do it on 662 00:34:05,82 --> 00:34:08 the example sum 1 over n. 663 00:34:08 --> 00:34:13,38 664 00:34:13,38 --> 00:34:20,17 So here's example 3 and it's going to be sum 1 over n, 665 00:34:20,17 --> 00:34:21,369 n equals 1 to infinity. 666 00:34:21,369 --> 00:34:24,09 667 00:34:24,09 --> 00:34:28,29 And what we're now going to see is that it corresponds 668 00:34:28,29 --> 00:34:29,34 to this integral here. 669 00:34:29,34 --> 00:34:32,62 670 00:34:32,62 --> 00:34:35,695 And we're going to show therefore that thing diverges. 671 00:34:37,87 --> 00:34:39,995 But we're going to do more carefully. 672 00:34:40,3 --> 00:34:44,52 We're going to do this in some detail so that you see what the 673 00:34:44,52 --> 00:34:47,084 correspondence is between these quantities. 674 00:34:47,39 --> 00:34:50,03 And the same sort of reasoning applies to 675 00:34:50,03 --> 00:34:51,275 these other examples. 676 00:34:51,275 --> 00:34:55,82 677 00:34:55,82 --> 00:34:56,77 So here we go. 678 00:34:59,18 --> 00:35:05,34 I'm going to take the integral and draw the picture 679 00:35:05,34 --> 00:35:06,745 of the Riemann's sum. 680 00:35:07,22 --> 00:35:10,84 So here's the level one and here's the function 681 00:35:10,84 --> 00:35:12,39 y equals 1 over x. 682 00:35:13,84 --> 00:35:15,73 And I'm going to take the Riemann's sum. 683 00:35:15,73 --> 00:35:21,98 684 00:35:21,98 --> 00:35:24,845 With delta x equals 1. 685 00:35:25,2 --> 00:35:27,07 And that's going to be closely connected to 686 00:35:27,07 --> 00:35:29,57 the series that I have. 687 00:35:29,57 --> 00:35:32,16 688 00:35:32,16 --> 00:35:35,51 But now I have to decide whether I want to lower 689 00:35:35,51 --> 00:35:37,82 Riemann's sum or an upper Riemann's sum. 690 00:35:37,82 --> 00:35:40,18 And actually I'm going to check both of them because both 691 00:35:40,18 --> 00:35:41,135 of them are illuminating. 692 00:35:41,135 --> 00:35:44,62 693 00:35:44,62 --> 00:35:46,36 First will do the upper Riemann's sum. 694 00:35:47 --> 00:35:48,7 Now that's this staircase here. 695 00:35:48,7 --> 00:35:51,78 696 00:35:51,78 --> 00:35:54,6 So we'll call this the upper Riemann's sum. 697 00:35:54,6 --> 00:35:58,13 698 00:35:58,13 --> 00:35:59,71 And let's check what it's levels are. 699 00:35:59,71 --> 00:36:00,813 This is not to scale. 700 00:36:01,56 --> 00:36:02,85 This level should be a half. 701 00:36:03,25 --> 00:36:05,61 So if this is 1 and this is 2 and that level was supposed to 702 00:36:05,61 --> 00:36:08,205 be a half then this next level should be a third. 703 00:36:10,34 --> 00:36:12,83 That's how the Riemann's sums are working out. 704 00:36:12,83 --> 00:36:17,04 705 00:36:17,04 --> 00:36:20,265 And now I have the following phenomenon. 706 00:36:21,77 --> 00:36:23,34 Let's cut it off at the n. 707 00:36:24,64 --> 00:36:29,31 So that means that the integral if from 1 to n d x over x. 708 00:36:30,76 --> 00:36:32,96 And the Riemann's sum is something that's 709 00:36:32,96 --> 00:36:33,69 bigger than it. 710 00:36:33,69 --> 00:36:38,84 Because the areas are enclosing the area of the curved region. 711 00:36:40,3 --> 00:36:44,72 And that's going to be the area of the first box which is 1, 712 00:36:44,72 --> 00:36:50,826 plus the area of the second box which is a half plus the 713 00:36:50,826 --> 00:36:52,51 area of the third box which is a third. 714 00:36:54,15 --> 00:36:59,31 All the way up the last one but the last one starts 715 00:36:59,31 --> 00:37:00,425 today n minus 1. 716 00:37:00,84 --> 00:37:03,185 So it has 1 over n minus 1. 717 00:37:03,45 --> 00:37:05,78 There are not n boxes here. 718 00:37:05,78 --> 00:37:07,5 They're only n minus 1 boxes. 719 00:37:07,96 --> 00:37:11,13 Because the distance between 1 n is n minus one. 720 00:37:11,83 --> 00:37:13,555 Right so this is n minus one terms. 721 00:37:13,555 --> 00:37:17,33 722 00:37:17,33 --> 00:37:23,172 However if I use a notation for partial sum. 723 00:37:25,14 --> 00:37:29,92 Which is 1 plus 1 over 2 plus all the way up to n over n 724 00:37:29,92 --> 00:37:32,01 minus one plus 1 over n. 725 00:37:33,53 --> 00:37:35,48 In other words I go out to the n one which is what 726 00:37:35,48 --> 00:37:36,875 I would ordinarily do. 727 00:37:37,37 --> 00:37:42,7 Then this sum that I have here certainly is less than sn. 728 00:37:42,9 --> 00:37:45,68 Because there's one more term there. 729 00:37:47,85 --> 00:37:51,07 And so here I have an integral which is 730 00:37:51,07 --> 00:37:53,9 underneath this sum s n. 731 00:37:53,9 --> 00:38:01,7 732 00:38:01,7 --> 00:38:20,84 Now this is going to allow us to prove conclusively 733 00:38:20,84 --> 00:38:21,68 that the sum diverges. 734 00:38:22,35 --> 00:38:23,34 Why is that? 735 00:38:23,34 --> 00:38:25,213 Because this term here we can calculate. 736 00:38:26 --> 00:38:29,04 This is the log x evaluated at 1 and n. 737 00:38:29,47 --> 00:38:33,77 Which is the same thing as log n minus 0. 738 00:38:34,85 --> 00:38:38,26 All right the quantity log n minus log 1 which is 0. 739 00:38:39,27 --> 00:38:45,458 And so what we have here is that log n is less than s n. 740 00:38:46,892 --> 00:38:51,455 All right and clearly this goes to infinity right. 741 00:38:51,455 --> 00:38:56,337 As n goes to infinity this thing goes to infinity. 742 00:38:57,83 --> 00:38:58,435 So we're done. 743 00:38:58,435 --> 00:39:00,33 All right we've shown divergence. 744 00:39:00,33 --> 00:39:08,73 745 00:39:08,73 --> 00:39:16 The way I'm going to use the lower Riemann's sum is to 746 00:39:16 --> 00:39:21,153 recognize that we've captured the rate appropriately. 747 00:39:21,96 --> 00:39:24,52 That is not only do have a lower bound like this but 748 00:39:24,52 --> 00:39:26,43 I have an upper bound which is very similar. 749 00:39:27,86 --> 00:39:30,44 So if I use the upper-- remind oh sorry --the 750 00:39:30,44 --> 00:39:39,54 lower Riemann's sum again with delta x equals 1. 751 00:39:39,54 --> 00:39:43,76 752 00:39:43,76 --> 00:39:55,53 Then I have that the integral from 1 to n of d x over x is 753 00:39:55,53 --> 00:39:58,26 bigger than-- Well what are the terms going to be if 754 00:39:58,26 --> 00:39:59,095 fit sit them underneath? 755 00:40:00,64 --> 00:40:03,21 If I fit them underneath I'm missing the first term. 756 00:40:03,21 --> 00:40:05,6 That is the box is going to be half height. 757 00:40:05,6 --> 00:40:07,97 It's going to be this lower piece. 758 00:40:08,3 --> 00:40:10,155 So I'm missing this first term. 759 00:40:10,48 --> 00:40:17,87 So it'll be a half plus a third plus it will keep on going. 760 00:40:18,15 --> 00:40:22,15 But now the last one instead of being 1 over n minus 1. 761 00:40:22,15 --> 00:40:23,43 It's going to be 1 over n. 762 00:40:23,84 --> 00:40:26,105 This is again a total of the n minus 1 terms. 763 00:40:27,05 --> 00:40:28,43 This is the lower Riemann's sum. 764 00:40:28,43 --> 00:40:31,19 765 00:40:31,19 --> 00:40:41,25 And now we can recognize that this is exactly equal to s n 766 00:40:41,25 --> 00:40:43,165 minus 1 minus the first term. 767 00:40:43,52 --> 00:40:46,33 So we missed the first term but we got all the rest of them. 768 00:40:47,11 --> 00:40:49,7 So if I put this to the other side remember 769 00:40:49,7 --> 00:40:50,8 this is the log n. 770 00:40:52,432 --> 00:40:53,305 All right. 771 00:40:53,77 --> 00:40:55,73 If I put this to the other side I have the other 772 00:40:55,73 --> 00:40:56,75 side of this bound. 773 00:40:57,13 --> 00:41:05,96 I have that f n is less than if I reverse it log n plus 1. 774 00:41:07,01 --> 00:41:09,04 And so I've trapped it on the other side. 775 00:41:09,04 --> 00:41:10,61 And here I have the lower bound. 776 00:41:10,95 --> 00:41:12,87 So I'm going to combine those together. 777 00:41:13,17 --> 00:41:17,54 So all hold I have this correspondence here. 778 00:41:18,03 --> 00:41:24,52 It is the size of a log n is trapped between the size of f n 779 00:41:24,52 --> 00:41:27,64 which is relatively hard to calculate and understand 780 00:41:27,64 --> 00:41:32,57 exactly is trapped between log n and log n plus 1. 781 00:41:34,26 --> 00:41:34,875 Yeah question. 782 00:41:35,16 --> 00:41:46,28 AUDIENCE: [INAUDIBLE PHRASE] 783 00:41:46,28 --> 00:41:48,1 PROFESSOR: This step here is the step that you're 784 00:41:48,1 --> 00:41:48,92 concerned about. 785 00:41:49,55 --> 00:41:54,58 So this step is a geometric argument which is 786 00:41:54,58 --> 00:41:57,07 analogous to this step. 787 00:41:57,07 --> 00:42:00,865 All right it's the same type of argument. 788 00:42:01,32 --> 00:42:05,28 And in this case it's that the rectangle is on top and so the 789 00:42:05,28 --> 00:42:08,17 area represented on the right hand side is less than the area 790 00:42:08,17 --> 00:42:09,53 represented on this side. 791 00:42:09,84 --> 00:42:12,13 And this is the same type of thing except that the 792 00:42:12,13 --> 00:42:13,09 rectangle is underneath. 793 00:42:14,4 --> 00:42:18,02 So the sum of the areas of the rectangle is less than 794 00:42:18,02 --> 00:42:19,19 the area under the curve. 795 00:42:19,19 --> 00:42:23,84 796 00:42:23,84 --> 00:42:24,24 All right. 797 00:42:24,24 --> 00:42:26,613 So I've now trapped this quantity. 798 00:42:27,38 --> 00:42:34,44 And I'm now going to state the sort of general results. 799 00:42:34,44 --> 00:42:38,42 800 00:42:38,42 --> 00:42:40,42 So here's what's known as integral comparison. 801 00:42:42,23 --> 00:42:45,71 It's this double arrow correspondence in the general 802 00:42:45,71 --> 00:42:54,58 case, for a very general case. 803 00:42:54,58 --> 00:42:57,13 There is actually even more cases where it works. 804 00:42:57,13 --> 00:43:00,823 But this is a good case and convenient. 805 00:43:01,51 --> 00:43:02,845 And this is called integral comparison. 806 00:43:02,845 --> 00:43:07,06 807 00:43:07,06 --> 00:43:11,72 And it comes with hypothesis but it follows the same 808 00:43:11,72 --> 00:43:12,785 argument that I just gave. 809 00:43:13,495 --> 00:43:35,29 If f of x is decreasing and it's positive then the sum f n, 810 00:43:35,29 --> 00:43:39,615 n equals 1 to infinity minus the integral from 1 to infinity 811 00:43:39,615 --> 00:43:45,62 of f of x d x is less than half of 1. 812 00:43:45,62 --> 00:43:50,06 813 00:43:50,06 --> 00:43:51,325 That's basically what we showed. 814 00:43:51,59 --> 00:43:54,14 We showed that the difference between s n and log 815 00:43:54,14 --> 00:43:55,93 n was at most 1. 816 00:43:55,93 --> 00:43:59,66 817 00:43:59,66 --> 00:44:00,06 All right. 818 00:44:00,06 --> 00:44:03,79 819 00:44:03,79 --> 00:44:23,91 And the sum and the integral converge or diverge together. 820 00:44:25,38 --> 00:44:27,75 That is they either both converge or both diverge. 821 00:44:27,75 --> 00:44:30,57 This is the type of test that we like because then we can 822 00:44:30,57 --> 00:44:33,86 just convert the question of convergence over here to this 823 00:44:33,86 --> 00:44:35,8 question of convergence over on the other side. 824 00:44:37,77 --> 00:44:41,69 Now I remind you that it's incredibly hard to 825 00:44:41,69 --> 00:44:42,91 calculate these numbers. 826 00:44:44,87 --> 00:44:46,955 Whereas these numbers are easier to calculate. 827 00:44:47,22 --> 00:44:50,39 Our goal is to reduce things to simpler things. 828 00:44:50,39 --> 00:44:53,96 In this case sums, infinite sums are much harder 829 00:44:53,96 --> 00:44:54,72 than infinite integrals. 830 00:44:54,72 --> 00:45:00,08 831 00:45:00,08 --> 00:45:01,75 All right so that the integral comparison. 832 00:45:03,15 --> 00:45:12,34 And now I have one last bit on comparisons that I 833 00:45:12,34 --> 00:45:13,84 need to tell you about. 834 00:45:13,84 --> 00:45:16,03 And this is very much like what we did with integrals. 835 00:45:16,31 --> 00:45:18,24 Which is so called limit comparison. 836 00:45:18,24 --> 00:45:29,2 837 00:45:29,2 --> 00:45:36,73 The limits comparison says the following if f of n 838 00:45:36,73 --> 00:45:38,27 is similar to g of n. 839 00:45:38,27 --> 00:45:45,67 You will recall that means f of n over g of n tends to 840 00:45:45,67 --> 00:45:47,85 1 as n goes to infinity. 841 00:45:47,85 --> 00:45:51,76 842 00:45:51,76 --> 00:45:54,66 And we're in the positive case. 843 00:45:55 --> 00:45:57,535 So let's just say g n is positive. 844 00:45:57,535 --> 00:46:03,92 845 00:46:03,92 --> 00:46:18,09 Then sum f n sum g n same this as above, either both 846 00:46:18,09 --> 00:46:21,884 converge or both diverge. 847 00:46:21,884 --> 00:46:27,488 848 00:46:27,488 --> 00:46:28,365 All right. 849 00:46:28,77 --> 00:46:31,34 This is just saying that if they behave the same way in the 850 00:46:31,34 --> 00:46:36,63 tail, which is all we really care about, then they have 851 00:46:36,63 --> 00:46:40,82 similar behavior, similar convergence properties. 852 00:46:40,82 --> 00:46:44,62 853 00:46:44,62 --> 00:46:45,69 And let me give you a couple examples. 854 00:46:45,69 --> 00:46:50,37 855 00:46:50,37 --> 00:46:55,36 So here's one example if you take the sum 1 over n 856 00:46:55,36 --> 00:46:57,08 squared plus 1 square root. 857 00:46:57,08 --> 00:47:01,89 858 00:47:01,89 --> 00:47:05,065 This is going to be replaced by something simpler. 859 00:47:05,35 --> 00:47:06,895 Which is the main term here. 860 00:47:07,24 --> 00:47:12,58 Which is 1 over square root of n squared which we recognize as 861 00:47:12,58 --> 00:47:15,1 sum 1 over n which diverges. 862 00:47:15,1 --> 00:47:17,92 863 00:47:17,92 --> 00:47:20,44 So this guy is one of the red guys. 864 00:47:20,44 --> 00:47:24,3 865 00:47:24,3 --> 00:47:30,33 On the red team now we have another example. 866 00:47:30,33 --> 00:47:33,37 867 00:47:33,37 --> 00:47:40,19 Which is let's say the square root of n to the fifth 868 00:47:40,19 --> 00:47:42,28 minus n squared. 869 00:47:43,08 --> 00:47:45,71 Now if you have something where it's negative in the 870 00:47:45,71 --> 00:47:48,4 denominator you kind of do have to watch out that 871 00:47:48,4 --> 00:47:49,54 denominator makes sense. 872 00:47:49,54 --> 00:47:50,24 It isn't 0. 873 00:47:50,57 --> 00:47:53,42 So we're going to be careful and start this at n equals 2. 874 00:47:53,42 --> 00:47:59,34 875 00:47:59,34 --> 00:48:01,72 I don't like 1 over 0 as a term in my series. 876 00:48:01,72 --> 00:48:04,44 So I'm just going to be a little careful about how-- as I 877 00:48:04,44 --> 00:48:05,71 said I was kind of lazy here. 878 00:48:05,71 --> 00:48:08,01 I could have started this one at 0 for instance. 879 00:48:09,93 --> 00:48:10,81 All right. 880 00:48:10,81 --> 00:48:13,83 So here's the picture. 881 00:48:14,11 --> 00:48:18,44 Now this I just replace by it's main term which is 1 over n 882 00:48:18,44 --> 00:48:19,64 to the fifth square root. 883 00:48:20,34 --> 00:48:25,975 Which is sum 1 over n to the five halves which converges. 884 00:48:25,975 --> 00:48:28,705 885 00:48:28,705 --> 00:48:29,52 All right. 886 00:48:29,52 --> 00:48:30,8 The power is bigger than 1. 887 00:48:30,8 --> 00:48:33,96 1 is the divider for these things and it just misses. 888 00:48:33,96 --> 00:48:36,964 This one converges. 889 00:48:38,752 --> 00:48:45,62 All right so these are the typical ways in which these 890 00:48:45,62 --> 00:48:47,05 convergence processes are used. 891 00:48:47,355 --> 00:48:47,87 All right. 892 00:48:47,87 --> 00:48:49,56 So I have one more thing for you. 893 00:48:49,93 --> 00:48:52,295 Which is an advertisement for next time. 894 00:48:52,6 --> 00:48:54,696 And I have this demo here which I will grab. 895 00:48:56,36 --> 00:48:57,85 But you will see this next time. 896 00:48:58,3 --> 00:49:01,5 So here's a question for you to think about overnight but don't 897 00:49:01,5 --> 00:49:04,24 ask friends you have to think about yourself. 898 00:49:04,24 --> 00:49:05,146 So here's the problem. 899 00:49:05,492 --> 00:49:08,18 Here are some blocks which I acquired when 900 00:49:08,18 --> 00:49:09,39 my kids left home. 901 00:49:09,39 --> 00:49:12,19 902 00:49:12,19 --> 00:49:19,395 Anyway yeah that'll happen to you too in about four years. 903 00:49:20,55 --> 00:49:25,775 So now here you are these are blocks. 904 00:49:26,1 --> 00:49:27,93 So now here's the question that we're going to 905 00:49:27,93 --> 00:49:29,13 deal with next time. 906 00:49:29,86 --> 00:49:32,03 I'm going to build it, maybe I'll put it over here 907 00:49:32,03 --> 00:49:33,81 because I want to have some room to head this way. 908 00:49:34,89 --> 00:49:41,32 I want to stack them up so that-- oh didn't work 909 00:49:41,32 --> 00:49:44,26 --going to stack them up in the following way. 910 00:49:44,43 --> 00:49:48,53 I want to do it so that the top one is completely to the 911 00:49:48,53 --> 00:49:50,23 right of the bottom one. 912 00:49:51,54 --> 00:49:53,5 That's the question can I do that? 913 00:49:55,84 --> 00:49:57,06 Can I build this up? 914 00:49:57,87 --> 00:49:59,52 So what's let's see here. 915 00:50:01,98 --> 00:50:04,69 I just seem to be missing-- but anyway what I'm going to do is 916 00:50:04,69 --> 00:50:07,12 I'm going to try to build this and we're going to see how far 917 00:50:07,12 --> 00:50:10,93 we can get with this next time. 918 00:50:10,93 --> 00:50:11,678