1 00:00:00,551 --> 00:00:02,050 So what we're going to attempt to do 2 00:00:02,050 --> 00:00:05,570 is evaluate this sum right over here, 3 00:00:05,570 --> 00:00:08,109 evaluate what this series is, negative 2 4 00:00:08,109 --> 00:00:11,660 over n plus 1 times n plus 2, starting at n equals 2, 5 00:00:11,660 --> 00:00:13,750 all the way to infinity. 6 00:00:13,750 --> 00:00:16,850 And if we wanted to see what this looks like, it starts at n 7 00:00:16,850 --> 00:00:17,400 equals 2. 8 00:00:17,400 --> 00:00:20,430 So when n equals 2, this is negative 2 over 2 9 00:00:20,430 --> 00:00:24,370 plus 1, which is 3, times 2 plus 2, which is 4. 10 00:00:24,370 --> 00:00:28,940 Then when n is equal to 3, this is negative 2 over 3 11 00:00:28,940 --> 00:00:32,560 plus 1, which is 4, times 3 plus 2, which is 5. 12 00:00:32,560 --> 00:00:34,820 And it just keeps going like that, negative 2 13 00:00:34,820 --> 00:00:37,670 over 5 times 6. 14 00:00:37,670 --> 00:00:40,620 And it just keeps going on and on and on. 15 00:00:40,620 --> 00:00:43,880 And now, it looks pretty clear that each successive term 16 00:00:43,880 --> 00:00:45,620 is getting smaller. 17 00:00:45,620 --> 00:00:47,680 And it's getting smaller reasonably fast. 18 00:00:47,680 --> 00:00:51,814 So it's reasonable to assume that even though you have 19 00:00:51,814 --> 00:00:53,480 an infinite number of terms, it actually 20 00:00:53,480 --> 00:00:54,842 might give you a finite value. 21 00:00:54,842 --> 00:00:56,800 But it doesn't jump out at me, at least the way 22 00:00:56,800 --> 00:00:58,470 that I've looked at it right now, 23 00:00:58,470 --> 00:00:59,970 as to what this sum would be, or how 24 00:00:59,970 --> 00:01:02,120 to actually figure out that sum. 25 00:01:02,120 --> 00:01:04,269 So what I want you to do now is pause this video. 26 00:01:04,269 --> 00:01:07,630 And I'm going to give you a hint about how to think about this. 27 00:01:07,630 --> 00:01:12,170 Try to dig up your memories of partial fraction expansion, 28 00:01:12,170 --> 00:01:14,160 or partial fraction decomposition, 29 00:01:14,160 --> 00:01:17,920 to turn this expression into the sum of two fractions. 30 00:01:17,920 --> 00:01:22,280 And that might help us think about what this sum is. 31 00:01:22,280 --> 00:01:24,337 So I'm assuming you've given a go at it. 32 00:01:24,337 --> 00:01:25,920 So let's try to manipulate this thing. 33 00:01:25,920 --> 00:01:28,840 And let's see if we can rewrite this as a sum of two fractions. 34 00:01:28,840 --> 00:01:32,870 So this is negative 2 over-- and I'm 35 00:01:32,870 --> 00:01:35,990 going to do this in two different colors-- n plus 1 36 00:01:35,990 --> 00:01:37,650 times n plus 2. 37 00:01:40,160 --> 00:01:42,870 And we remember from our partial fraction expansion 38 00:01:42,870 --> 00:01:45,710 that we can rewrite this as the sum of two fractions, 39 00:01:45,710 --> 00:01:54,846 as A over n plus 1 plus B over n plus 2. 40 00:01:54,846 --> 00:01:55,970 And why is this reasonable? 41 00:01:55,970 --> 00:01:57,090 Well, if you're adding two fractions, 42 00:01:57,090 --> 00:01:58,620 you want to find a common denominator, which 43 00:01:58,620 --> 00:02:00,650 would be a multiple of the two denominators. 44 00:02:00,650 --> 00:02:03,300 This is clearly a multiple of both of these denominators. 45 00:02:03,300 --> 00:02:05,660 And we learned in partial fraction expansion 46 00:02:05,660 --> 00:02:08,919 that whatever we have up here, especially because the degree 47 00:02:08,919 --> 00:02:13,020 here is lower than the degree down here, whatever we have 48 00:02:13,020 --> 00:02:15,890 up here is going to be a degree lower than what we have here. 49 00:02:15,890 --> 00:02:18,000 So this is a first-degree term in terms of n, 50 00:02:18,000 --> 00:02:20,850 so these are going to be constant terms up here. 51 00:02:20,850 --> 00:02:22,740 So let's figure out what A and B are. 52 00:02:22,740 --> 00:02:25,580 So if we perform the addition-- well, 53 00:02:25,580 --> 00:02:27,590 let's just rewrite both of these with 54 00:02:27,590 --> 00:02:29,400 the same common denominator. 55 00:02:29,400 --> 00:02:34,190 So let's rewrite A over n plus 1, 56 00:02:34,190 --> 00:02:37,720 but let's multiply the numerator and denominator by n plus 2. 57 00:02:37,720 --> 00:02:41,570 So we multiply the numerator by n plus 2 and the denominator 58 00:02:41,570 --> 00:02:42,100 by n plus 2. 59 00:02:42,100 --> 00:02:44,490 I haven't changed the value of this first fraction. 60 00:02:44,490 --> 00:02:50,850 Similarly, let's do the same thing with B over n plus 2. 61 00:02:50,850 --> 00:02:54,470 Multiply the numerator and the denominator by n plus 1, so 62 00:02:54,470 --> 00:02:57,960 n plus 1 over n plus 1. 63 00:02:57,960 --> 00:03:01,276 Once again, I haven't change the value of this fraction. 64 00:03:01,276 --> 00:03:03,400 But by doing this, I now have a common denominator, 65 00:03:03,400 --> 00:03:04,570 and I can add. 66 00:03:04,570 --> 00:03:12,714 So this is going to be equal to n plus 1 times n plus 2 is 67 00:03:12,714 --> 00:03:13,380 our denominator. 68 00:03:15,920 --> 00:03:19,690 And then our numerator-- let me expand it out. 69 00:03:19,690 --> 00:03:21,690 This is going to be, if I distribute the A, 70 00:03:21,690 --> 00:03:25,290 it is An plus 2A. 71 00:03:25,290 --> 00:03:31,730 So let me write that, An plus 2A. 72 00:03:31,730 --> 00:03:40,680 And then let's distribute this B, plus Bn plus B. 73 00:03:40,680 --> 00:03:42,680 Now, what I want to do is I want to rewrite this 74 00:03:42,680 --> 00:03:44,330 so I have all of the n terms. 75 00:03:44,330 --> 00:03:51,070 So for example, An plus Bn-- I could factor an n out. 76 00:03:51,070 --> 00:03:58,730 And I could rewrite that as A plus B times n, those 77 00:03:58,730 --> 00:04:00,350 two terms right over there. 78 00:04:00,350 --> 00:04:03,895 And then these two terms, the 2A plus B, 79 00:04:03,895 --> 00:04:09,080 I could just write it like this, plus 2A plus B. 80 00:04:09,080 --> 00:04:17,550 And, of course, all of that is over n plus 1 times n plus 2. 81 00:04:20,915 --> 00:04:24,020 So how do we solve for A and B? 82 00:04:24,020 --> 00:04:26,650 Well, the realization is that this thing 83 00:04:26,650 --> 00:04:29,070 must be equal to negative 2. 84 00:04:29,070 --> 00:04:31,879 These two things must be equal to each other. 85 00:04:31,879 --> 00:04:33,670 Remember, we're making the claim that this, 86 00:04:33,670 --> 00:04:36,160 which is the same thing as this, is equal to this. 87 00:04:36,160 --> 00:04:38,754 That's the whole reason why we started doing this. 88 00:04:38,754 --> 00:04:40,420 So we're making the claim that these two 89 00:04:40,420 --> 00:04:42,680 things are equivalent. 90 00:04:42,680 --> 00:04:44,470 We're making this claim. 91 00:04:44,470 --> 00:04:47,560 So everything in the numerator must be equal to negative 2. 92 00:04:47,560 --> 00:04:48,710 So how do we do that? 93 00:04:48,710 --> 00:04:52,130 It looks like we have two unknowns here. 94 00:04:52,130 --> 00:04:54,930 To figure out two unknowns, we normally need two equations. 95 00:04:54,930 --> 00:04:56,990 Well, the realization here is, look, 96 00:04:56,990 --> 00:05:00,030 we have an n term on the left-hand side here. 97 00:05:00,030 --> 00:05:01,520 We have no n term here. 98 00:05:01,520 --> 00:05:03,950 So you literally could view this, instead of just viewing 99 00:05:03,950 --> 00:05:05,366 this as negative 2, you could view 100 00:05:05,366 --> 00:05:10,960 this as negative 2 plus 0n, plus 0 times n. 101 00:05:10,960 --> 00:05:11,740 That's not "on." 102 00:05:11,740 --> 00:05:17,599 That's 0-- let me write it this way-- 0 times n. 103 00:05:17,599 --> 00:05:19,140 So when you look at it this way, it's 104 00:05:19,140 --> 00:05:22,430 clear that A plus B is the coefficient on n. 105 00:05:22,430 --> 00:05:24,700 That must be equal to 0. 106 00:05:24,700 --> 00:05:28,340 A plus B must be equal to 0. 107 00:05:28,340 --> 00:05:30,810 And this is kind of bread-and-butter partial 108 00:05:30,810 --> 00:05:32,180 fraction expansion. 109 00:05:32,180 --> 00:05:34,530 We have other videos on that if you need to review that. 110 00:05:34,530 --> 00:05:43,010 And the constant part, 2A plus B, is equal to negative 2. 111 00:05:46,100 --> 00:05:51,020 And so now we have two equations in two unknowns. 112 00:05:51,020 --> 00:05:53,020 And we could solve it a bunch of different ways. 113 00:05:53,020 --> 00:05:55,450 But one interesting way is let's multiply the top equation 114 00:05:55,450 --> 00:05:56,930 by negative 1. 115 00:05:56,930 --> 00:06:01,090 So then this becomes negative A minus B is equal to-- well, 116 00:06:01,090 --> 00:06:03,360 negative 1 times 0 is still 0. 117 00:06:03,360 --> 00:06:05,600 Now we can add these two things together. 118 00:06:05,600 --> 00:06:11,110 And we are left with 2A minus A is A, plus B minus B-- well, 119 00:06:11,110 --> 00:06:13,830 those cancel out. 120 00:06:13,830 --> 00:06:16,320 A is equal to negative 2. 121 00:06:16,320 --> 00:06:20,120 And if A is equal to negative 2, A plus B is 0, 122 00:06:20,120 --> 00:06:21,520 B must be equal to 2. 123 00:06:24,090 --> 00:06:28,000 Negative 2 plus 2 is equal to 0. 124 00:06:28,000 --> 00:06:31,100 We solved for A. And then I substituted it back up here. 125 00:06:31,100 --> 00:06:34,860 So now we can rewrite all of this right over here. 126 00:06:34,860 --> 00:06:37,830 We can rewrite it as the sum-- and actually, 127 00:06:37,830 --> 00:06:39,200 let me do a little bit instead. 128 00:06:39,200 --> 00:06:43,020 Let me just write it as a finite sum as opposed 129 00:06:43,020 --> 00:06:44,200 to an infinite sum. 130 00:06:44,200 --> 00:06:47,450 And then we can just take the limit as we go to infinity. 131 00:06:47,450 --> 00:06:49,190 So let me rewrite it like this. 132 00:06:49,190 --> 00:06:53,700 So this is the sum from n equals 2-- instead to infinity, 133 00:06:53,700 --> 00:06:56,540 I'll just say to capital N. And then later, we 134 00:06:56,540 --> 00:07:00,750 could take the limit as this goes to infinity of-- well, 135 00:07:00,750 --> 00:07:03,850 instead of writing this, I can write this right over here. 136 00:07:03,850 --> 00:07:06,370 So A is negative 2. 137 00:07:06,370 --> 00:07:11,110 So it's negative 2 over n plus 1. 138 00:07:11,110 --> 00:07:17,820 And then B is 2, plus B over n plus 2. 139 00:07:17,820 --> 00:07:20,610 So once again, I've just expressed this as a finite sum. 140 00:07:20,610 --> 00:07:23,210 Later, we can take the limit as capital N approaches 141 00:07:23,210 --> 00:07:25,020 infinity to figure out what this thing is. 142 00:07:25,020 --> 00:07:27,870 Oh, sorry, and B-- let me not write B anymore. 143 00:07:27,870 --> 00:07:33,450 We now know that B is 2 over n plus 2. 144 00:07:33,450 --> 00:07:37,850 Now, how does this actually go about helping us? 145 00:07:37,850 --> 00:07:39,330 Well, let's do what we did up here. 146 00:07:39,330 --> 00:07:42,260 Let's actually write out what this is going to be equal to. 147 00:07:42,260 --> 00:07:46,900 This is going to be equal to-- when n is 2, 148 00:07:46,900 --> 00:07:54,410 this is negative 2/3, so it's negative 2/3, plus 2/4. 149 00:08:00,160 --> 00:08:02,810 So that's n equals-- let me do it down here, because I'm 150 00:08:02,810 --> 00:08:04,400 about to run out of real estate. 151 00:08:04,400 --> 00:08:06,640 That is when n is equal to 2. 152 00:08:06,640 --> 00:08:10,120 Now, what about when n is equal to 3? 153 00:08:10,120 --> 00:08:21,920 When n is equal to 3, this is going to be negative 2/4 154 00:08:21,920 --> 00:08:22,610 plus 2/5. 155 00:08:28,770 --> 00:08:30,530 What about when n is equal to 4? 156 00:08:30,530 --> 00:08:33,890 I think you might see a pattern that's starting to form. 157 00:08:33,890 --> 00:08:34,640 Let's do one more. 158 00:08:34,640 --> 00:08:42,090 When n is equal to 4, well, then, this 159 00:08:42,090 --> 00:08:46,710 is going to be negative 2/5-- let 160 00:08:46,710 --> 00:08:53,170 me do that same blue color-- negative 2/5 plus 2/6. 161 00:08:57,640 --> 00:09:00,460 And we're just going to keep going. 162 00:09:00,460 --> 00:09:02,520 Let me scroll down to get some space-- we're 163 00:09:02,520 --> 00:09:05,190 going to keep going all the way until the N-th term. 164 00:09:08,960 --> 00:09:13,680 So plus dot dot dot plus our capital N-th term, 165 00:09:13,680 --> 00:09:24,110 which is going to be negative 2 over capital N plus 1 plus 2 166 00:09:24,110 --> 00:09:27,560 over capital N plus 2. 167 00:09:27,560 --> 00:09:29,310 So I think you might see the pattern here. 168 00:09:29,310 --> 00:09:33,466 Notice, from our first when n equals 2, we got the 2/4. 169 00:09:33,466 --> 00:09:35,590 But then when n equals 3, you had the negative 2/4. 170 00:09:35,590 --> 00:09:37,450 That cancels with that. 171 00:09:37,450 --> 00:09:39,090 When n equals 3, you had 2/5. 172 00:09:39,090 --> 00:09:42,690 Then that cancels when n equals 4 with the negative 2/5. 173 00:09:42,690 --> 00:09:47,170 So the second term cancels with-- the second part, 174 00:09:47,170 --> 00:09:50,380 I guess, for each n, for each index, 175 00:09:50,380 --> 00:09:53,040 cancels out with the first part for the next index. 176 00:09:53,040 --> 00:09:55,420 And so that's just going to keep happening all the way 177 00:09:55,420 --> 00:09:59,730 until n is equal to capital N. 178 00:09:59,730 --> 00:10:02,300 And so this is going to cancel out with the one right 179 00:10:02,300 --> 00:10:03,110 before it. 180 00:10:03,110 --> 00:10:06,540 And all we're going to be left with 181 00:10:06,540 --> 00:10:13,790 is this term and this term right over here. 182 00:10:13,790 --> 00:10:15,950 So let's rewrite that. 183 00:10:15,950 --> 00:10:19,090 So we get-- let's get more space here. 184 00:10:19,090 --> 00:10:26,290 This thing can be rewritten as the sum from lowercase n 185 00:10:26,290 --> 00:10:30,850 equals 2 to capital N of negative 2 186 00:10:30,850 --> 00:10:37,020 over n plus 1 plus 2 over n plus 2 187 00:10:37,020 --> 00:10:39,440 is equal to-- well, everything else in the middle canceled 188 00:10:39,440 --> 00:10:39,940 out. 189 00:10:39,940 --> 00:10:43,740 We're just left with negative 2/3 190 00:10:43,740 --> 00:10:50,460 plus 2 over capital N plus 2. 191 00:10:50,460 --> 00:10:53,110 So this was a huge simplification right over here. 192 00:10:53,110 --> 00:10:57,000 And remember, our original sum that we wanted to calculate, 193 00:10:57,000 --> 00:11:00,510 that just has a limit as capital N goes to infinity. 194 00:11:00,510 --> 00:11:04,620 So let's just take the limit as capital N goes to infinity. 195 00:11:04,620 --> 00:11:06,170 So let me write it this way. 196 00:11:06,170 --> 00:11:07,878 Well, actually, let me write it this way. 197 00:11:07,878 --> 00:11:10,810 The limit-- so we can write it this way. 198 00:11:10,810 --> 00:11:15,350 The limit as capital N approaches 199 00:11:15,350 --> 00:11:20,030 infinity is going to be equal to the limit as capital N 200 00:11:20,030 --> 00:11:22,239 approaches infinity of-- well, we just 201 00:11:22,239 --> 00:11:23,280 figured out what this is. 202 00:11:23,280 --> 00:11:33,240 This is negative 2/3 plus 2 over capital N plus 2. 203 00:11:33,240 --> 00:11:36,180 Well, as n goes to infinity, this negative 2/3 204 00:11:36,180 --> 00:11:37,600 doesn't get impacted at all. 205 00:11:37,600 --> 00:11:40,375 This term right over here, 2 over an ever larger number, 206 00:11:40,375 --> 00:11:42,000 over an infinitely large number-- well, 207 00:11:42,000 --> 00:11:43,520 that's going to go to 0. 208 00:11:43,520 --> 00:11:47,990 And we're going to be left with negative 2/3. 209 00:11:47,990 --> 00:11:48,750 And we're done. 210 00:11:48,750 --> 00:11:54,740 We were able to figure out the sum of this infinite series. 211 00:11:54,740 --> 00:11:58,460 So this thing right over here is equal to negative 2/3. 212 00:11:58,460 --> 00:12:01,320 And this type of series is called a telescoping series-- 213 00:12:01,320 --> 00:12:02,920 telescoping, I should say. 214 00:12:02,920 --> 00:12:04,380 This is a telescoping series. 215 00:12:09,270 --> 00:12:12,130 And a telescoping series is a general term. 216 00:12:12,130 --> 00:12:14,500 So if you were to take its partial sums, 217 00:12:14,500 --> 00:12:18,430 it has this pattern right over here, where, in each term, 218 00:12:18,430 --> 00:12:20,020 you're starting to cancel things out. 219 00:12:20,020 --> 00:12:23,270 So what you're left with is just a fixed number 220 00:12:23,270 --> 00:12:25,520 of terms at the end. 221 00:12:25,520 --> 00:12:27,130 But either way, this was a pretty-- 222 00:12:27,130 --> 00:12:29,421 it's a little bit hairy, but it was a pretty satisfying 223 00:12:29,421 --> 00:12:30,640 problem.