1 00:00:00,000 --> 00:00:00,410 2 00:00:00,410 --> 00:00:01,040 Welcome back. 3 00:00:01,040 --> 00:00:03,459 So where we left off in the last video, I'd shown you 4 00:00:03,459 --> 00:00:06,019 this thing called the geometric series. 5 00:00:06,019 --> 00:00:07,809 And, you know, we could have some base a. 6 00:00:07,809 --> 00:00:08,619 It could be any number. 7 00:00:08,619 --> 00:00:12,799 It could be 1/2, it could be 10. 8 00:00:12,800 --> 00:00:13,870 But that's just-- but some number. 9 00:00:13,869 --> 00:00:15,769 And we keep taking it to increasing exponents, and we 10 00:00:15,769 --> 00:00:18,739 sum them up, and this is called a geometric series. 11 00:00:18,739 --> 00:00:23,809 And so I want to figure out the sum of a geometric series of, 12 00:00:23,809 --> 00:00:26,939 you know, when I have some base a, and I go up to some 13 00:00:26,940 --> 00:00:29,440 number a to the n. 14 00:00:29,440 --> 00:00:31,520 What-- is this a to the-- why did I write 15 00:00:31,519 --> 00:00:32,689 a to n minus 2 there? 16 00:00:32,689 --> 00:00:37,390 That should be a to the big N. 17 00:00:37,390 --> 00:00:39,509 My brain must have been malfunctioning in 18 00:00:39,509 --> 00:00:40,929 the previous video. 19 00:00:40,929 --> 00:00:42,520 That always happens when I start running out of time. 20 00:00:42,520 --> 00:00:43,740 But anyway. 21 00:00:43,740 --> 00:00:44,490 Let's go back to this. 22 00:00:44,490 --> 00:00:48,170 So I defined s as this geometric sum. 23 00:00:48,170 --> 00:00:50,160 Now I'm going to define another sum. 24 00:00:50,159 --> 00:00:53,169 And that sum I'm going to define as a times s. 25 00:00:53,170 --> 00:00:57,969 26 00:00:57,969 --> 00:01:00,969 And that equals-- well, that's just going to be a times 27 00:01:00,969 --> 00:01:04,109 this exact sum, right? 28 00:01:04,109 --> 00:01:06,540 And that's the same a as this a, right? 29 00:01:06,540 --> 00:01:08,400 That a is the same as this a. 30 00:01:08,400 --> 00:01:11,620 So what's a times this whole thing? 31 00:01:11,620 --> 00:01:15,350 Well, it's the a times a to the zero is-- let me 32 00:01:15,349 --> 00:01:16,369 write it down for you. 33 00:01:16,370 --> 00:01:19,609 So this'll be a because I just distribute the a, right? a 34 00:01:19,609 --> 00:01:25,459 times a to the zero, plus a times a to the 1, plus a times 35 00:01:25,459 --> 00:01:30,879 a squared, plus all the way a times a to the n minus one, 36 00:01:30,879 --> 00:01:34,259 plus a times a to the n. 37 00:01:34,260 --> 00:01:36,260 I just took an a and I distributed it along 38 00:01:36,260 --> 00:01:39,540 this whole sum. 39 00:01:39,540 --> 00:01:41,200 But what is this equal to? 40 00:01:41,200 --> 00:01:43,810 Well, this is equal to a times a to the zero. 41 00:01:43,810 --> 00:01:51,219 That's a one-- a to the first power-- plus a squared, plus a 42 00:01:51,219 --> 00:01:55,420 cubed, plus a to the n, right? 43 00:01:55,420 --> 00:01:57,769 Because you just add the exponents, a to the n. 44 00:01:57,769 --> 00:02:00,599 Plus a to the n plus 1. 45 00:02:00,599 --> 00:02:03,799 So this is as. 46 00:02:03,799 --> 00:02:07,450 And we saw before that s is just our original sum. 47 00:02:07,450 --> 00:02:10,530 48 00:02:10,530 --> 00:02:16,949 That is just a to the zero, plus a to the 1, plus a 49 00:02:16,949 --> 00:02:19,379 squared, plus up, up, up, up. 50 00:02:19,379 --> 00:02:26,120 All the way to plus a to the n, right? 51 00:02:26,120 --> 00:02:29,680 So let me ask you a question. 52 00:02:29,680 --> 00:02:33,980 What happens if I subtract this from that? 53 00:02:33,979 --> 00:02:36,119 What happens? 54 00:02:36,120 --> 00:02:41,840 If I say, as minus s. 55 00:02:41,840 --> 00:02:45,979 Well, I subtracted this from here, on the left hand side. 56 00:02:45,979 --> 00:02:48,031 What happens on the right hand side? 57 00:02:48,032 --> 00:02:50,120 Well, all of these become negative, right? 58 00:02:50,120 --> 00:02:51,770 Let me do it in a bold color. 59 00:02:51,770 --> 00:02:54,240 This becomes-- because I'm subtracting-- negative, 60 00:02:54,240 --> 00:02:55,740 negative, these are all negatives. 61 00:02:55,740 --> 00:02:56,250 Negative. 62 00:02:56,250 --> 00:02:57,479 Negative. 63 00:02:57,479 --> 00:03:00,269 Well, a to the first, minus a to the first. 64 00:03:00,270 --> 00:03:02,735 That crosses out. a squared minus a squared crosses 65 00:03:02,735 --> 00:03:04,530 out. a to the third, it'll all cross out. 66 00:03:04,530 --> 00:03:06,969 All the way up to a to the n, right? 67 00:03:06,969 --> 00:03:07,919 So what are we left with? 68 00:03:07,919 --> 00:03:11,919 We're just left with minus a to the zero, right? 69 00:03:11,919 --> 00:03:13,599 We're just left with that term. 70 00:03:13,599 --> 00:03:15,400 And we're just left with that term. 71 00:03:15,400 --> 00:03:19,224 Plus a to the n plus 1. 72 00:03:19,224 --> 00:03:20,799 And of course, what's a to the zero? 73 00:03:20,800 --> 00:03:22,240 That's just 1. 74 00:03:22,240 --> 00:03:32,240 So we have a times s minus s is equal to a to 75 00:03:32,240 --> 00:03:36,110 the n plus 1 minus 1. 76 00:03:36,110 --> 00:03:37,850 And now let's distribute the s out. 77 00:03:37,849 --> 00:03:44,180 So we get s times a minus 1 is equal to a to the n 78 00:03:44,180 --> 00:03:48,950 plus 1 minus 1, right? 79 00:03:48,949 --> 00:03:50,500 And then what do we get? 80 00:03:50,500 --> 00:03:52,599 Well, we can just divide both sides by a minus 1. 81 00:03:52,599 --> 00:03:55,819 Let me erase some of this stuff on top. 82 00:03:55,819 --> 00:03:58,889 83 00:03:58,889 --> 00:04:05,699 I think I can safely erase all of this, really. 84 00:04:05,699 --> 00:04:07,869 Well, I don't want to erase that much. 85 00:04:07,870 --> 00:04:09,240 I want to erase this stuff. 86 00:04:09,240 --> 00:04:13,830 87 00:04:13,830 --> 00:04:15,360 That's good enough. 88 00:04:15,360 --> 00:04:16,889 OK. 89 00:04:16,889 --> 00:04:24,149 So I have just-- dividing both sides of this equation by a 90 00:04:24,149 --> 00:04:31,519 minus 1, I get s is equal to a to the n plus 1 minus 91 00:04:31,519 --> 00:04:37,560 1 over a minus 1. 92 00:04:37,560 --> 00:04:41,280 So where did that get us? 93 00:04:41,279 --> 00:04:46,269 We defined the geometric series as equal to the sum. 94 00:04:46,269 --> 00:04:51,699 From k is equal to 0, to n of a to the k. 95 00:04:51,699 --> 00:04:54,170 And now we've just derived a formula for what that 96 00:04:54,170 --> 00:04:55,770 sum ends up being. 97 00:04:55,769 --> 00:05:03,990 Equals a to the n plus 1 minus 1 over a minus 1. 98 00:05:03,990 --> 00:05:05,600 And why is this useful? 99 00:05:05,600 --> 00:05:10,939 We now know, if I were to say, well, what is-- let me clean 100 00:05:10,939 --> 00:05:12,490 up all of this, as well. 101 00:05:12,490 --> 00:05:17,400 Let me clean up all of this and we can-- OK. 102 00:05:17,399 --> 00:05:22,259 So if I said, you figure out the sum of, I don't know, the 103 00:05:22,259 --> 00:05:28,276 powers of 3 up to 3 to the, I don't know, 3 to 104 00:05:28,276 --> 00:05:29,550 the tenth power. 105 00:05:29,550 --> 00:05:32,759 So, you know, 3. 106 00:05:32,759 --> 00:05:37,170 So 3 to the zero, plus 3 to the one, plus 3 squared, plus all 107 00:05:37,170 --> 00:05:39,280 the way to 3 to the tenth. 108 00:05:39,279 --> 00:05:43,799 So this is the same thing as the sum of k equals zero 109 00:05:43,800 --> 00:05:48,660 to 10, of 3 to the k. 110 00:05:48,660 --> 00:05:49,520 Right? 111 00:05:49,519 --> 00:05:54,079 So this formula we just figured out, a is 3 and n is 10. 112 00:05:54,079 --> 00:05:58,620 So this sum is just going to be equal to 3 to the eleventh 113 00:05:58,620 --> 00:06:05,480 power minus 1 over 3 minus 1. 114 00:06:05,480 --> 00:06:08,140 Which equals-- well, I don't know what 3 to 115 00:06:08,139 --> 00:06:09,419 the eleventh power is. 116 00:06:09,420 --> 00:06:11,350 Minus 1 over 2. 117 00:06:11,350 --> 00:06:13,129 So that's kind of useful. 118 00:06:13,129 --> 00:06:15,050 That is a number. 119 00:06:15,050 --> 00:06:17,460 Although you'd have to memorize your exponent tables to the 120 00:06:17,459 --> 00:06:18,549 eleventh power to do that. 121 00:06:18,550 --> 00:06:19,879 But I think you get the idea. 122 00:06:19,879 --> 00:06:23,709 This is especially useful if we were dealing with-- well, if 123 00:06:23,709 --> 00:06:27,000 the base was a power of ten, it would be very, very easy. 124 00:06:27,000 --> 00:06:31,259 But what I actually want to do now is I want to take this and 125 00:06:31,259 --> 00:06:34,509 say, well, what happens if n goes to infinity? 126 00:06:34,509 --> 00:06:36,269 Let me show you. 127 00:06:36,269 --> 00:06:37,439 So what happens? 128 00:06:37,439 --> 00:06:41,350 So there's two types of series that we can take-- that's 129 00:06:41,350 --> 00:06:42,750 not what I wanted to do. 130 00:06:42,750 --> 00:06:45,250 There are two types of series that we can take that we 131 00:06:45,250 --> 00:06:47,449 can find the sums of. 132 00:06:47,449 --> 00:06:51,219 There's finite series, and infinite series. 133 00:06:51,220 --> 00:06:56,310 And in order for an infinite series to come up to a sum 134 00:06:56,310 --> 00:06:58,949 that's not infinity, they need to-- what we say-- 135 00:06:58,949 --> 00:07:00,149 they need to converge. 136 00:07:00,149 --> 00:07:02,870 And if you think about what has to happen for them to converge, 137 00:07:02,870 --> 00:07:06,519 every next digit has to essentially get smaller and 138 00:07:06,519 --> 00:07:09,259 smaller and smaller, as we go towards infinity. 139 00:07:09,259 --> 00:07:12,909 So let's say that a is a fraction. 140 00:07:12,910 --> 00:07:13,700 a is 1/2. 141 00:07:13,699 --> 00:07:17,870 So how does a geometric series look like if we have 1/2 there? 142 00:07:17,870 --> 00:07:22,649 So let's say that we're taking the geometric series from k 143 00:07:22,649 --> 00:07:25,909 is equal to 0 to infinity. 144 00:07:25,910 --> 00:07:27,160 So this is neat. 145 00:07:27,160 --> 00:07:29,890 We're going to take an infinite sum, an infinite number of 146 00:07:29,889 --> 00:07:34,379 terms, and let's see if we can actually get an actual number. 147 00:07:34,379 --> 00:07:35,949 You know, we take an infinite thing, add it up, and it 148 00:07:35,949 --> 00:07:38,199 actually adds up to a finite thing. 149 00:07:38,199 --> 00:07:39,990 This has always amazed me. 150 00:07:39,990 --> 00:07:43,129 And the base now is going to be 1/2. 151 00:07:43,129 --> 00:07:45,409 It's 1/2 and it's going to be 1/2 to the k power. 152 00:07:45,410 --> 00:07:45,770 So this is going to be what? 153 00:07:45,769 --> 00:07:50,219 1/2 to the zero, plus 1/2, plus-- what's 1/2 squared? 154 00:07:50,220 --> 00:07:54,860 Plus 1/4, plus 1/8, plus 1/16. 155 00:07:54,860 --> 00:07:59,680 So as you see, each term is getting a lot, lot smaller. 156 00:07:59,680 --> 00:08:03,900 It's getting half of the previous term. 157 00:08:03,899 --> 00:08:06,799 Well, let's say, what happens if this wasn't infinity? 158 00:08:06,800 --> 00:08:09,879 What happens if this was n? 159 00:08:09,879 --> 00:08:13,480 Well, then we'd get plus 1 over 2 to the n, right? 160 00:08:13,480 --> 00:08:16,140 1/2 to the n is the same thing as 1 over 2 to the n. 161 00:08:16,139 --> 00:08:19,079 And if we look at the formula we figured out, we would say, 162 00:08:19,079 --> 00:08:26,339 well, that is just equal to 1/2 to the n plus 1, minus 163 00:08:26,339 --> 00:08:32,959 1, over 1/2 minus one. 164 00:08:32,960 --> 00:08:34,850 And that would be our answer. 165 00:08:34,850 --> 00:08:36,550 We'd have to know what n is. 166 00:08:36,549 --> 00:08:39,279 But now we want to know what happens if we go to infinity. 167 00:08:39,279 --> 00:08:41,139 So this is essentially a limit problem. 168 00:08:41,139 --> 00:08:45,740 What happens-- what's the limit, as n goes to infinity, 169 00:08:45,740 --> 00:08:52,980 of 1/2 to the n plus one minus 1 over 1/2 minus 1? 170 00:08:52,980 --> 00:08:55,330 Well, all of these are constant terms, so nothing happens. 171 00:08:55,330 --> 00:08:58,139 So what happens as this term, right here, goes to infinity? 172 00:08:58,139 --> 00:09:01,220 What's 1/2 to the infinity power? 173 00:09:01,220 --> 00:09:03,000 Well, that's zero. 174 00:09:03,000 --> 00:09:04,590 That's an unbelievably small number. 175 00:09:04,590 --> 00:09:08,790 Take 1/2 to arbitrarily large exponents, this just goes to 0. 176 00:09:08,789 --> 00:09:10,269 And so what are we left with? 177 00:09:10,269 --> 00:09:16,059 We're just left with this equals minus 1 over 1/2 minus 178 00:09:16,059 --> 00:09:19,274 1, or we could multiply the top and the bottom by negative 1. 179 00:09:19,274 --> 00:09:22,211 And we get 1 over 1 minus 1/2. 180 00:09:22,211 --> 00:09:27,319 Which equals 1 over 1/2, which is equal to 2. 181 00:09:27,320 --> 00:09:28,810 I find that amazing. 182 00:09:28,809 --> 00:09:34,089 If I add 0 plus 1/2 plus 1/4 plus 1/8 plus 1/16 and I never 183 00:09:34,090 --> 00:09:37,410 stop-- I go to infinity-- and not infinity, but I go to 1 184 00:09:37,409 --> 00:09:41,329 over essentially 2 to the infinity-- I end up with 185 00:09:41,330 --> 00:09:43,080 this neat and clean number. 186 00:09:43,080 --> 00:09:43,530 2. 187 00:09:43,529 --> 00:09:45,350 And this might be a little project for you, to actually 188 00:09:45,350 --> 00:09:48,070 draw it out into like maybe a pie and see what happens as 189 00:09:48,070 --> 00:09:51,310 you keep adding smaller and smaller pieces to the pie. 190 00:09:51,309 --> 00:09:54,549 But it never ceases to amaze me, that I added an infinite 191 00:09:54,549 --> 00:09:55,519 number of terms, right? 192 00:09:55,519 --> 00:09:56,389 This was infinity. 193 00:09:56,389 --> 00:09:58,529 And I got a finite number. 194 00:09:58,529 --> 00:09:59,929 I got a finite number. 195 00:09:59,929 --> 00:10:01,629 Anyway, we ran out of time. 196 00:10:01,629 --> 00:10:03,131 See you soon.