1 00:00:00,320 --> 00:00:02,908 Let's not start with the zero of term. 2 00:00:02,908 --> 00:00:07,155 [SOUND] 3 00:00:07,155 --> 00:00:13,970 How do I sum this geometric series? It's the sum 4 00:00:15,360 --> 00:00:20,750 k goes from m to infinity of r to the k. 5 00:00:22,100 --> 00:00:27,180 So this starts out with a term r to the m and then the next 6 00:00:27,180 --> 00:00:32,538 term is r to the m plus 1, and it keeps on going like that. 7 00:00:32,538 --> 00:00:37,850 It's a geometric series but the first term is r to the m, not r to the 0. 8 00:00:37,850 --> 00:00:43,860 I remember how to sum the series that starts not with m but with 0. 9 00:00:43,860 --> 00:00:52,310 Well, in that case, to evaluate the sum k goes from 0 to infinity of r to the k. 10 00:00:52,310 --> 00:00:57,020 We already studied that, and that's 1 over 1 minus r, 11 00:00:57,020 --> 00:01:00,210 provided the absolute value of r is less than 1. 12 00:01:00,210 --> 00:01:07,030 I want to somehow change that 0 back into an m. 13 00:01:08,830 --> 00:01:11,570 there's a general result that I can invoke. 14 00:01:11,570 --> 00:01:16,860 Some constant times the series, which adds up all the a sub 15 00:01:16,860 --> 00:01:22,710 k's is equal to the series that adds up c times 16 00:01:22,710 --> 00:01:23,749 the a sub k's. 17 00:01:24,870 --> 00:01:27,660 This is of course assuming that the series makes sense. 18 00:01:27,660 --> 00:01:28,041 Right? 19 00:01:28,041 --> 00:01:32,680 I'm assuming that this series converges in order to assert this equality. 20 00:01:34,220 --> 00:01:37,650 So, in this case, I'll multiply by r to the m. 21 00:01:37,650 --> 00:01:38,840 So let me write that down. 22 00:01:38,840 --> 00:01:43,140 I'm going to multiply both sides by r to the m. 23 00:01:43,140 --> 00:01:48,620 So the r to the m times the sum k goes from 0 to infinity 24 00:01:48,620 --> 00:01:54,690 of r to the k is r to the m over 1 minus r. 25 00:01:54,690 --> 00:01:57,250 Now I can use this result. 26 00:01:57,250 --> 00:02:02,860 Remember that this result is assuming that this series converges. 27 00:02:02,860 --> 00:02:06,970 So I'm always going to be working with the assumption that the absolute value 28 00:02:06,970 --> 00:02:12,460 of r is less than 1 in order to guarantee that this series converges. 29 00:02:12,460 --> 00:02:13,970 Oh, wait, but yeah, let's apply the result. 30 00:02:13,970 --> 00:02:18,668 So if I apply the result in this case, what do I get? 31 00:02:18,668 --> 00:02:23,588 Well, here, r to the m is playing the role of c, 32 00:02:23,588 --> 00:02:28,550 and r to the k is a sub k. So this can be replaced 33 00:02:28,550 --> 00:02:34,410 by this. Which means this is equal to what? 34 00:02:34,410 --> 00:02:39,560 It's equal to the sum k goes from 0 to infinity of 35 00:02:39,560 --> 00:02:46,970 r to the m times r to the k. Which I could rewrite as the sum 36 00:02:46,970 --> 00:02:52,730 k goes from 0 to infinity of r to the m plus k. 37 00:02:52,730 --> 00:02:57,360 The bounds on the series can be rewritten. Well, what do I mean by that? 38 00:02:57,360 --> 00:03:00,330 Well, let me write out what this series looks like. 39 00:03:00,330 --> 00:03:02,820 I'm just going to write out the first few terms of the series. 40 00:03:02,820 --> 00:03:05,090 The k equals 0 term is 41 00:03:05,090 --> 00:03:08,940 r to the m. The k equals 1 term is r to to 42 00:03:08,940 --> 00:03:12,750 the m plus 1. The k equals 2 term is r 43 00:03:12,750 --> 00:03:16,750 to the m plus 2, and so on. So, I could 44 00:03:16,750 --> 00:03:21,430 rewrite this series as the series that I'm interested in. 45 00:03:21,430 --> 00:03:30,090 This series ends up just being the sum k goes from m to infinity of r to the k. 46 00:03:32,300 --> 00:03:37,800 And if I go all the way back, right. I figured out that that series 47 00:03:37,800 --> 00:03:42,640 has this value of course assuming that the absolute value of r is less than 1. 48 00:03:42,640 --> 00:03:48,750 But all told then, I can conclude that this series 49 00:03:48,750 --> 00:03:53,668 is equal to r to the m over 1 minus 50 00:03:53,668 --> 00:03:58,820 r, as long as the absolute value of r is less than 1. 51 00:03:58,820 --> 00:04:01,480 So that's a useful formula, but even more useful than 52 00:04:01,480 --> 00:04:04,700 the formula is the method that we used for deriving it. 53 00:04:04,700 --> 00:04:08,650 We used this fact to derive that formula. 54 00:04:08,650 --> 00:04:10,880 But this fact, while it looks like the 55 00:04:10,880 --> 00:04:14,650 distributive law is more than just the distributive law. 56 00:04:14,650 --> 00:04:17,570 What I mean is that the distributive law says this. 57 00:04:17,570 --> 00:04:24,750 It says that C times a sub 0 plus a sub 1 plus dot, dot, dot plus a sub n 58 00:04:24,750 --> 00:04:28,180 is equal to C times a sub 0 plus C times a 59 00:04:28,180 --> 00:04:31,920 sub 1 plus dot, dot, dot plus C times a sub n. 60 00:04:31,920 --> 00:04:34,000 It's something about finite sums. 61 00:04:34,000 --> 00:04:36,120 I could even write it using summation notation. 62 00:04:36,120 --> 00:04:42,886 The distributive law says that C times the sum k goes from 0 to n of 63 00:04:42,886 --> 00:04:49,770 a sub k is equal to the sum of k goes from 0 to n of C times a 64 00:04:49,770 --> 00:04:50,580 sub k. 65 00:04:50,580 --> 00:04:53,130 That's what the distributive law is telling me. 66 00:04:53,130 --> 00:04:55,880 But contrast that with this statement. 67 00:04:55,880 --> 00:04:56,110 All right? 68 00:04:56,110 --> 00:04:58,090 This isn't just a statement about finite 69 00:04:58,090 --> 00:05:01,020 sums, this is a statement about infinite series. 70 00:05:01,020 --> 00:05:03,290 So how do I justify something like that? 71 00:05:03,290 --> 00:05:05,470 Well, it's more than just the distributive law. 72 00:05:05,470 --> 00:05:07,360 Let me show you. 73 00:05:07,360 --> 00:05:15,040 So, I could write down C times the limit as n approaches infinity of the sum 74 00:05:15,040 --> 00:05:20,760 k goes from 0 to n of a sub k. And by definition, this is C 75 00:05:20,760 --> 00:05:27,140 times the infinite series k goes from 0 to infinity of a sub k. 76 00:05:27,140 --> 00:05:29,320 But I know something about limits. 77 00:05:29,320 --> 00:05:32,560 A constant multiple of a limit is the limit of 78 00:05:32,560 --> 00:05:35,490 that constant multiple, times whatever I'm taking the limit of. 79 00:05:35,490 --> 00:05:35,670 Right? 80 00:05:35,670 --> 00:05:40,450 So this thing here is equal to the limit as 81 00:05:40,450 --> 00:05:48,730 n approaches infinity of C times the sum k goes from 0 to n of a sub k. 82 00:05:50,060 --> 00:05:54,720 And now I can apply the distributive law because this is just a finite sum. 83 00:05:54,720 --> 00:05:56,890 It's a finite sum where n is maybe very big. 84 00:05:56,890 --> 00:05:58,470 But it still is a finite sum. 85 00:05:58,470 --> 00:06:05,570 So this is the limit n goes to infinity of the sum k goes from 86 00:06:05,570 --> 00:06:11,770 0 to n of C times a sub k. So this equality is the distributive law. 87 00:06:11,770 --> 00:06:17,850 Now I've got the limit of this sum, and that is the infinite series k goes 88 00:06:17,850 --> 00:06:24,130 from 0 to infinity of C times a sub k. So let's look at what happened here. 89 00:06:24,130 --> 00:06:25,010 All right? 90 00:06:25,010 --> 00:06:28,120 This is just the definition of the infinite series. 91 00:06:28,120 --> 00:06:31,860 The fact that these are equal is a property about limits. 92 00:06:31,860 --> 00:06:35,050 The fact that these are equal is the distributive law. 93 00:06:35,050 --> 00:06:39,100 And the fact that these are equal is the definition of infinite series. 94 00:06:39,100 --> 00:06:44,940 So the fact that a constant multiple times an infinite series is the infinite series 95 00:06:44,940 --> 00:06:47,220 of that constant multiple of the terms is 96 00:06:47,220 --> 00:06:49,950 more than just the distributive law at play. 97 00:06:49,950 --> 00:06:56,100 It's really the distributive law plus a fact about limits. 98 00:06:59,295 --> 00:07:05,289 [SOUND]