1 00:00:00,000 --> 00:00:00,720 2 00:00:00,720 --> 00:00:03,240 Let's learn a little bit about sequences and series. 3 00:00:03,240 --> 00:00:04,020 So what's a sequence? 4 00:00:04,019 --> 00:00:06,979 Well, a sequence is just a bunch of numbers in some order. 5 00:00:06,980 --> 00:00:11,630 You know, the most difficult sequence is 1, 2, 3, 4. 6 00:00:11,630 --> 00:00:13,370 You get the point. 7 00:00:13,369 --> 00:00:14,479 And what's a series? 8 00:00:14,480 --> 00:00:17,550 Well, it's often represented-- it's just a sum of sequences, 9 00:00:17,550 --> 00:00:18,980 a sum of a sequence. 10 00:00:18,980 --> 00:00:23,350 So, for example, the arithmetic sequence-- sorry the arithmetic 11 00:00:23,350 --> 00:00:26,920 series is just the sum of the arithmetic sequence. 12 00:00:26,920 --> 00:00:31,630 So 1 plus 2 plus 3 plus-- we could keep going 13 00:00:31,629 --> 00:00:33,439 until maybe some number. 14 00:00:33,439 --> 00:00:36,344 This is called the arithmetic series. 15 00:00:36,344 --> 00:00:42,750 16 00:00:42,750 --> 00:00:45,369 Nothing too fancy here. 17 00:00:45,369 --> 00:00:48,859 But before we move forward, let's get a notation for how we 18 00:00:48,859 --> 00:00:51,630 can represent these sums without necessarily having to 19 00:00:51,630 --> 00:00:54,429 write out all of the digits or having to keep doing this dot, 20 00:00:54,429 --> 00:00:57,140 dot, dot, plus notation. 21 00:00:57,140 --> 00:00:59,475 And that notation is Sigma notation. 22 00:00:59,475 --> 00:01:03,320 23 00:01:03,320 --> 00:01:06,739 That's an upper case Sigma. 24 00:01:06,739 --> 00:01:08,509 And how do you use Sigma notation? 25 00:01:08,510 --> 00:01:09,490 Well, let's say I wanted to represent this 26 00:01:09,489 --> 00:01:11,019 arithmetic series. 27 00:01:11,019 --> 00:01:13,589 So I would say, well, let's add up a bunch of-- 28 00:01:13,590 --> 00:01:14,939 let's call them k's. 29 00:01:14,939 --> 00:01:16,810 This is an arbitrary variable. 30 00:01:16,810 --> 00:01:19,480 And we'll start at k equals 1. 31 00:01:19,480 --> 00:01:22,240 We'll start at k equals 1, and we'll go to k 32 00:01:22,239 --> 00:01:24,849 is equal to big N. 33 00:01:24,849 --> 00:01:27,794 And this is the exact same thing, so we first make k equal 34 00:01:27,795 --> 00:01:32,099 to 1, and then we add it to k is equal to 2 plus 3, and we go 35 00:01:32,099 --> 00:01:38,549 all the way until N minus 1 and then plus N. 36 00:01:38,549 --> 00:01:45,769 So this is the Sigma notation for the arithmetic series. 37 00:01:45,769 --> 00:01:48,109 Before I move on, I think this is a good time just to like 38 00:01:48,109 --> 00:01:50,730 learn a little bit more about the arithmetic series. 39 00:01:50,730 --> 00:01:54,010 We'll actually focus on this one and the geometric series 40 00:01:54,010 --> 00:01:56,560 because those are the two that you'll see most often. 41 00:01:56,560 --> 00:01:59,010 And then once you learn calculus, I'll show you the 42 00:01:59,010 --> 00:02:02,040 power and Taylor series, which is exact-- the Taylor series is 43 00:02:02,040 --> 00:02:05,250 a specific version of a power series. 44 00:02:05,250 --> 00:02:08,689 But let's play around with this arith-- I keep wanting to say 45 00:02:08,689 --> 00:02:11,175 arith-MET-ic, but a-RITH-metic, either way-- series. 46 00:02:11,175 --> 00:02:14,770 47 00:02:14,770 --> 00:02:17,170 So let's call the sum S. 48 00:02:17,169 --> 00:02:23,359 Let's say that this is equal to the sum from k is equal to 1 to 49 00:02:23,360 --> 00:02:30,880 N of k, which is equal to, just like we said, 1 plus 2 plus 3. 50 00:02:30,879 --> 00:02:33,799 And we'll just keep adding them, dot, dot, dot, to a bunch 51 00:02:33,800 --> 00:02:39,200 of numbers, to big N minus 1 plus big N, right? 52 00:02:39,199 --> 00:02:41,139 Fair enough. 53 00:02:41,139 --> 00:02:42,939 Now, bear with me a second. 54 00:02:42,939 --> 00:02:47,430 I'm just going to write that same exact sum again, but 55 00:02:47,430 --> 00:02:48,640 I'm just going to write it in reverse order. 56 00:02:48,639 --> 00:02:52,519 And I think it's intuitive to you that it doesn't matter what 57 00:02:52,520 --> 00:02:54,170 order I add up numbers in. 58 00:02:54,169 --> 00:02:55,579 They'll add up to the same number. 59 00:02:55,580 --> 00:02:58,290 2 plus 1 is the same thing as 1 plus 2, right? 60 00:02:58,289 --> 00:03:00,891 So let me write this exact same sum, but I'll write 61 00:03:00,891 --> 00:03:01,799 it in reverse order. 62 00:03:01,800 --> 00:03:09,040 So that's the same thing as N plus N minus 1, plus N minus 63 00:03:09,039 --> 00:03:13,319 2, plus-- and the pluses keep going-- plus 2 plus 1. 64 00:03:13,319 --> 00:03:16,189 This is the exact sum, just in the reverse order. 65 00:03:16,189 --> 00:03:19,960 And I did that for a reason because now I'm going to add 66 00:03:19,960 --> 00:03:21,250 both sides of this equation. 67 00:03:21,250 --> 00:03:23,430 I'm going to take-- S plus S. 68 00:03:23,430 --> 00:03:25,450 Well, that's just 2S. 69 00:03:25,449 --> 00:03:30,869 And that's going to equal this sum plus this sum. 70 00:03:30,870 --> 00:03:34,340 I wrote this so that the sum becomes clean. 71 00:03:34,340 --> 00:03:35,490 And why do I say that? 72 00:03:35,490 --> 00:03:38,000 Well, let's add up corresponding terms. 73 00:03:38,000 --> 00:03:40,599 We could have added up any terms, but-- so since they all 74 00:03:40,599 --> 00:03:43,344 have to add up, let's just add the 1 plus the N, then we'll 75 00:03:43,344 --> 00:03:47,000 add the 2 plus the N minus 1, then we'll add the 3 plus the 76 00:03:47,000 --> 00:03:48,819 N minus 2, and so forth. 77 00:03:48,819 --> 00:03:50,560 And I think you'll see in a second, or maybe you already 78 00:03:50,560 --> 00:03:51,789 realize why I'm doing this. 79 00:03:51,789 --> 00:03:56,489 One plus the N, the 2 plus the N minus 1, the 3 and the N 80 00:03:56,490 --> 00:04:00,159 minus 2, all the way to the N minus 1 and the 2, 81 00:04:00,159 --> 00:04:02,310 the N and the 1. 82 00:04:02,310 --> 00:04:03,830 What's 1 plus N? 83 00:04:03,830 --> 00:04:07,500 Well, that's just N plus 1, right? 84 00:04:07,500 --> 00:04:10,409 What's 2 plus N minus 1? 85 00:04:10,409 --> 00:04:14,060 Well, that's also N plus 1, right? 86 00:04:14,060 --> 00:04:16,670 What's 3 plus N minus 2? 87 00:04:16,670 --> 00:04:17,705 I think you could guess. 88 00:04:17,704 --> 00:04:19,909 It's N minus 1. 89 00:04:19,910 --> 00:04:21,140 And we just keep doing that. 90 00:04:21,139 --> 00:04:23,889 And what's N minus 2 plus 2? 91 00:04:23,889 --> 00:04:25,599 Sorry, this is a plus. 92 00:04:25,600 --> 00:04:27,780 N plus 1. 93 00:04:27,779 --> 00:04:29,349 And what's N plus 1? 94 00:04:29,350 --> 00:04:32,310 Well, that's just N plus 1, of course. 95 00:04:32,310 --> 00:04:34,189 So my question to you is how many of these 96 00:04:34,189 --> 00:04:36,019 N plus 1's are there? 97 00:04:36,019 --> 00:04:37,409 Well, there are N of them, right? 98 00:04:37,410 --> 00:04:40,550 Each N plus 1 corresponds to each of these terms, 99 00:04:40,550 --> 00:04:41,530 so there are N of these. 100 00:04:41,529 --> 00:04:45,000 So instead of just adding N plus 1 N times, we could say 101 00:04:45,000 --> 00:04:51,579 that this is just N times N plus 1. 102 00:04:51,579 --> 00:04:55,484 So we have 2 times the sum is equal to N times N plus 1, and 103 00:04:55,485 --> 00:04:58,310 we could divide both sides by 2, and we get the sum is equal 104 00:04:58,310 --> 00:05:04,569 to N times N plus 1 over 2. 105 00:05:04,569 --> 00:05:07,800 Now, why is this neat, or why is this cool at all? 106 00:05:07,800 --> 00:05:11,210 Well, first of all, we found out a way to sum this 107 00:05:11,209 --> 00:05:12,329 Sigma notation up. 108 00:05:12,329 --> 00:05:16,669 We got kind of a well-defined formula. 109 00:05:16,670 --> 00:05:19,350 And what makes this especially cool is you can use this 110 00:05:19,350 --> 00:05:23,650 for low-end parlor tricks. 111 00:05:23,649 --> 00:05:24,529 What do I mean by that? 112 00:05:24,529 --> 00:05:28,759 Well, you can go up to someone and you can say, well, how 113 00:05:28,759 --> 00:05:32,490 quickly do you think I can add up the numbers between 1 and-- 114 00:05:32,490 --> 00:05:35,000 what am I doing-- oh, between 1 and 100? 115 00:05:35,000 --> 00:05:36,480 And, you know, people will say, oh, it will take you a little 116 00:05:36,480 --> 00:05:37,780 time: 1 plus 2 plus 3. 117 00:05:37,779 --> 00:05:40,719 And you say, well, it takes me no time at all because 118 00:05:40,720 --> 00:05:42,630 this is what I can do. 119 00:05:42,629 --> 00:05:47,339 So the sum-- and I just want to show you that you can use 120 00:05:47,339 --> 00:05:50,469 different variables from B equals 1, we're taking the 121 00:05:50,470 --> 00:05:55,100 variable B, to 100, right? 122 00:05:55,100 --> 00:05:56,720 That's the sum from 1 to 100. 123 00:05:56,720 --> 00:05:59,160 And we figured out what that formula is. 124 00:05:59,160 --> 00:06:07,230 It's going to be 100 times 101 over 2. 125 00:06:07,230 --> 00:06:09,340 Well, what's 100 times 101? 126 00:06:09,339 --> 00:06:13,250 It's just going to be 101 with two zeroes, right? 127 00:06:13,250 --> 00:06:20,029 10,100 over 2, and that equals 5,050. 128 00:06:20,029 --> 00:06:20,729 That's pretty neat. 129 00:06:20,730 --> 00:06:24,400 Instead of having to say 1 plus 2 plus 3 plus blah, blah, blah, 130 00:06:24,399 --> 00:06:29,394 blah, blah, blah, blah, blah, plus 98 plus 99 plus 100, this 131 00:06:29,394 --> 00:06:31,589 would take you some time, and there's a very good chance you 132 00:06:31,589 --> 00:06:32,859 would make a careless mistake. 133 00:06:32,860 --> 00:06:35,290 We could just plug into this formula, which we proved and 134 00:06:35,290 --> 00:06:39,680 hopefully you understood, and say that equals 5,050. 135 00:06:39,680 --> 00:06:41,560 You could do even something more impressive: the 136 00:06:41,560 --> 00:06:43,939 sum from 1 to 1,000. 137 00:06:43,939 --> 00:06:45,149 What's the sum from 1 to 1,000? 138 00:06:45,149 --> 00:06:48,899 Well, our formula, remember, was N times N plus 1 over 2. 139 00:06:48,899 --> 00:06:53,714 So if N is equal to 1,000, then what's our sum? 140 00:06:53,714 --> 00:07:02,224 It's 1,000 times 1,001 over 2, which is equal to-- well, we'll 141 00:07:02,225 --> 00:07:09,290 just add three zeroes to this: 1,001, one, two, three. 142 00:07:09,290 --> 00:07:11,390 Sorry, I think that was my first burp ever on 143 00:07:11,389 --> 00:07:13,029 one of these videos. 144 00:07:13,029 --> 00:07:17,299 I should re-record it, but I'm going to move forward. 145 00:07:17,300 --> 00:07:19,280 That kind of disconcerted me a little bit. 146 00:07:19,279 --> 00:07:20,619 I'd eaten too much. 147 00:07:20,620 --> 00:07:24,819 Anyway, divided by 2, and what is that? 148 00:07:24,819 --> 00:07:28,810 Let's see, it'll be 500-- let's see, this is a million. 149 00:07:28,810 --> 00:07:32,829 Half of a million is 500,000. 150 00:07:32,829 --> 00:07:37,509 500,500. 151 00:07:37,509 --> 00:07:41,189 And that would have taken you forever to do manually. 152 00:07:41,189 --> 00:07:43,810 But based on this formula we just got, you know how to 153 00:07:43,810 --> 00:07:46,449 do it very, very quickly. 154 00:07:46,449 --> 00:07:48,620 So that's the arithmetic series. 155 00:07:48,620 --> 00:07:49,620 But let's do another one. 156 00:07:49,620 --> 00:07:54,519 This is another typical series that you might see. 157 00:07:54,519 --> 00:07:56,909 Actually, this one you'll see a lot in your life, especially if 158 00:07:56,910 --> 00:08:03,730 you go into finance or really a whole series of scientific-- 159 00:08:03,730 --> 00:08:06,890 this shows up a lot, and this is called the geometric series. 160 00:08:06,889 --> 00:08:10,529 And the geometric series is-- essentially you take x. 161 00:08:10,529 --> 00:08:14,439 And I'll do it generally where I just take a variable x, 162 00:08:14,439 --> 00:08:15,879 and I say-- well, no, no. 163 00:08:15,879 --> 00:08:16,740 Let me just not take an x. 164 00:08:16,740 --> 00:08:19,019 Let me just take some number. 165 00:08:19,019 --> 00:08:29,299 So let's say some number a to the k from-- I don't know. 166 00:08:29,300 --> 00:08:33,940 Let's say from k is equal to 0 to k is equal to N. 167 00:08:33,940 --> 00:08:34,660 What does that mean? 168 00:08:34,659 --> 00:08:40,949 Well, that means a to the 0, right, k is 0, plus a 1 plus a 169 00:08:40,950 --> 00:08:45,780 squared plus a to the third plus-- and you could keep 170 00:08:45,779 --> 00:08:53,569 going-- plus a to the N minus 1 plus a to the N minus 2. 171 00:08:53,570 --> 00:08:55,560 This is called the geometric series. 172 00:08:55,559 --> 00:08:58,209 And it might not be obvious to you, but this type of growth, 173 00:08:58,210 --> 00:09:00,460 where you keep increasing the exponent, this is called 174 00:09:00,460 --> 00:09:02,392 geometric growth. 175 00:09:02,392 --> 00:09:04,360 So how do you take the sum of this? 176 00:09:04,360 --> 00:09:06,300 Well, let's see if we can do a similar trick, although this 177 00:09:06,299 --> 00:09:09,009 trick will involve one more step. 178 00:09:09,009 --> 00:09:13,639 So let's call the sum S. 179 00:09:13,639 --> 00:09:23,059 Let's call it the sum from k equals 0 to N, a to the k. 180 00:09:23,059 --> 00:09:24,949 And that, of course, is equal to what I just wrote. 181 00:09:24,950 --> 00:09:27,570 I probably didn't have to do it like this. 182 00:09:27,570 --> 00:09:32,310 a squared plus bup, bup, bup, bup, plus a to the N minus 183 00:09:32,309 --> 00:09:37,250 1, plus a to the N minus 2. 184 00:09:37,250 --> 00:09:42,039 Now let's define another sum, and I'm going to call that aS. 185 00:09:42,039 --> 00:09:44,579 186 00:09:44,580 --> 00:09:46,120 Actually, I'm about to run out of time, so I'll continue 187 00:09:46,120 --> 00:09:48,429 this in the next video.