1 00:00:00,280 --> 00:00:03,590 In this segment, we will discuss what a sequence is and 2 00:00:03,590 --> 00:00:06,480 what it means for a sequence to converge. 3 00:00:06,480 --> 00:00:13,580 So a sequence is nothing but some collection of elements 4 00:00:13,580 --> 00:00:17,650 that are coming out of some set, and that collection of 5 00:00:17,650 --> 00:00:21,690 elements is indexed by the natural numbers. 6 00:00:21,690 --> 00:00:27,530 We often use the notation, and we say that we have a sequence 7 00:00:27,530 --> 00:00:31,510 ai, or sometimes we use the notation that we have a 8 00:00:31,510 --> 00:00:34,830 sequence of this kind to emphasize the fact that it's a 9 00:00:34,830 --> 00:00:37,780 sequence and not just a single number. 10 00:00:37,780 --> 00:00:43,260 And what we mean by this is that we have i, an index that 11 00:00:43,260 --> 00:00:48,470 runs over the natural numbers, which is the set of positive 12 00:00:48,470 --> 00:00:57,110 integers, and each ai is an element of some set. 13 00:00:57,110 --> 00:01:00,460 In many cases, the set is going to be just the real 14 00:01:00,460 --> 00:01:03,890 line, in which case we're dealing with a sequence of 15 00:01:03,890 --> 00:01:05,099 real numbers. 16 00:01:05,099 --> 00:01:09,030 But it is also possible that the set over which our 17 00:01:09,030 --> 00:01:15,000 sequence takes values is Euclidean space n-dimensional 18 00:01:15,000 --> 00:01:16,710 space, in which case we're dealing with 19 00:01:16,710 --> 00:01:18,230 a sequence of vectors. 20 00:01:18,230 --> 00:01:22,300 But it also could be any other kind of set. 21 00:01:22,300 --> 00:01:24,410 Now, the definition that I gave you may 22 00:01:24,410 --> 00:01:25,950 still be a little vague. 23 00:01:25,950 --> 00:01:28,460 You may wonder how a mathematician would define 24 00:01:28,460 --> 00:01:30,170 formally a sequence. 25 00:01:30,170 --> 00:01:36,830 Formally, what a sequence is, is just a function that, to 26 00:01:36,830 --> 00:01:44,700 any natural number, associates an element of S. In 27 00:01:44,700 --> 00:01:50,920 particular, if we evaluate the function f at some argument i, 28 00:01:50,920 --> 00:01:54,570 this gives us the ith element of the sequence. 29 00:01:54,570 --> 00:01:56,590 So that's what a sequence is. 30 00:01:56,590 --> 00:01:59,780 Now, about sequences, we typically care whether a 31 00:01:59,780 --> 00:02:05,500 sequence converges to some number a, and we often use 32 00:02:05,500 --> 00:02:06,360 this notation. 33 00:02:06,360 --> 00:02:09,940 But to make it more precise, you also add 34 00:02:09,940 --> 00:02:11,630 this notation here. 35 00:02:11,630 --> 00:02:15,520 And we read this as saying that as i converges to 36 00:02:15,520 --> 00:02:18,460 infinity, the sequence ai converges to a 37 00:02:18,460 --> 00:02:20,360 certain number a. 38 00:02:20,360 --> 00:02:25,380 A more formal mathematical notation would be the limit as 39 00:02:25,380 --> 00:02:32,470 i goes to infinity of ai is equal to a certain number, a. 40 00:02:32,470 --> 00:02:35,370 But what exactly does this mean? 41 00:02:35,370 --> 00:02:37,329 What does it mean for a sequence to converge? 42 00:02:37,329 --> 00:02:39,550 What is the formal definition? 43 00:02:39,550 --> 00:02:41,180 It is as follows. 44 00:02:41,180 --> 00:02:47,050 Let us plot the sequence as a function of i. 45 00:02:47,050 --> 00:02:52,220 So this is the i-axis, and here we plot entries of ai. 46 00:02:52,220 --> 00:02:56,670 For a sequence to converge to a certain number a, we need to 47 00:02:56,670 --> 00:02:58,820 the following to happen. 48 00:02:58,820 --> 00:03:06,070 If we draw a small band around that number a, what we want is 49 00:03:06,070 --> 00:03:11,440 that the elements of the sequence, as i increases, 50 00:03:11,440 --> 00:03:16,140 eventually get inside this band and stay 51 00:03:16,140 --> 00:03:19,020 inside that band forever. 52 00:03:19,020 --> 00:03:22,579 Now, let us turn this into a more precise statement. 53 00:03:22,579 --> 00:03:25,600 What we mean is the following. 54 00:03:25,600 --> 00:03:31,920 If I give you some positive number epsilon, and I'm going 55 00:03:31,920 --> 00:03:35,790 to use that positive number epsilon to define a band 56 00:03:35,790 --> 00:03:37,820 around the number a. 57 00:03:37,820 --> 00:03:41,810 So it's this band here. 58 00:03:41,810 --> 00:03:44,150 If I give you a positive number epsilon, and therefore, 59 00:03:44,150 --> 00:03:48,890 this way, have defined a certain band, there exists a 60 00:03:48,890 --> 00:03:53,430 time after which the entries will get the inside the band. 61 00:03:53,430 --> 00:03:56,390 In this picture, it would be this time. 62 00:03:56,390 --> 00:03:58,175 So there exists a time-- 63 00:04:03,940 --> 00:04:07,470 let's call that time i0-- 64 00:04:07,470 --> 00:04:24,490 so i0 is here such that after that time, what we have is 65 00:04:24,490 --> 00:04:26,430 that the element of the sequence is 66 00:04:26,430 --> 00:04:28,560 within epsilon of a. 67 00:04:34,540 --> 00:04:38,130 So this is the formal definition of convergence of a 68 00:04:38,130 --> 00:04:40,120 sequence to a certain number a. 69 00:04:40,120 --> 00:04:42,890 The definition may look formidable and difficult to 70 00:04:42,890 --> 00:04:45,230 parse, but what it says in plain 71 00:04:45,230 --> 00:04:46,980 English is pretty simple. 72 00:04:46,980 --> 00:04:52,150 No matter what kind of band I take around my limit a, 73 00:04:52,150 --> 00:04:55,570 eventually, the sequence will be inside this band and will 74 00:04:55,570 --> 00:04:58,050 stay inside there. 75 00:04:58,050 --> 00:05:01,700 Convergence of sequences has some very nice properties that 76 00:05:01,700 --> 00:05:03,400 you're probably familiar with. 77 00:05:03,400 --> 00:05:06,580 For example, if I tell you that a certain sequence 78 00:05:06,580 --> 00:05:10,670 converges to a number a and another sequence converges to 79 00:05:10,670 --> 00:05:17,800 a number b, then we will have that ai plus bi, which is a 80 00:05:17,800 --> 00:05:18,650 new sequence-- 81 00:05:18,650 --> 00:05:21,340 the ith element of the sequence is this sum-- 82 00:05:21,340 --> 00:05:24,010 will converge to a plus b. 83 00:05:24,010 --> 00:05:28,720 Or similarly, ai times bi, which is another sequence, 84 00:05:28,720 --> 00:05:31,180 converges to a times b. 85 00:05:31,180 --> 00:05:41,880 And if, in addition, g is a continuous function, then g of 86 00:05:41,880 --> 00:05:46,840 ai will converge to g of a. 87 00:05:46,840 --> 00:05:51,890 So for example, if the ais converge to a, then the 88 00:05:51,890 --> 00:05:56,700 sequence ai squared is going to converge to a squared.