1 00:00:01,370 --> 00:00:04,480 So we looked at the formal definition of what it means 2 00:00:04,480 --> 00:00:08,210 for a sequence to converge, but as a practical matter, how 3 00:00:08,210 --> 00:00:13,080 can we tell whether a given sequence converges or not? 4 00:00:13,080 --> 00:00:16,309 There are two criteria that are the most commonly used for 5 00:00:16,309 --> 00:00:20,690 that purpose, and it's useful to be aware of them. 6 00:00:20,690 --> 00:00:24,210 The first one deals with the case where we have a sequence 7 00:00:24,210 --> 00:00:28,500 of numbers that keep increasing, or at least, they 8 00:00:28,500 --> 00:00:30,380 do not go down. 9 00:00:30,380 --> 00:00:37,210 In that case, those numbers may go up forever without any 10 00:00:37,210 --> 00:00:41,180 bound, so if you look at any particular value, there's 11 00:00:41,180 --> 00:00:44,740 going to be a time at which the sequence has 12 00:00:44,740 --> 00:00:46,700 exceeded that value. 13 00:00:46,700 --> 00:00:49,350 In that case, we say that the sequence 14 00:00:49,350 --> 00:00:51,680 converges to infinity. 15 00:00:51,680 --> 00:00:55,810 But if this is not the case, if it does not converge to 16 00:00:55,810 --> 00:00:58,800 infinity, which means that the entries of the 17 00:00:58,800 --> 00:01:01,180 sequence are bounded-- 18 00:01:01,180 --> 00:01:04,230 they do not grow arbitrarily large-- 19 00:01:04,230 --> 00:01:09,360 then, in that case, it is guaranteed that the sequence 20 00:01:09,360 --> 00:01:12,760 will converge to a certain number. 21 00:01:12,760 --> 00:01:16,180 This is not something that we will attempt to prove, but it 22 00:01:16,180 --> 00:01:19,900 is a useful fact to know. 23 00:01:19,900 --> 00:01:24,580 Another way of establishing convergence is to derive some 24 00:01:24,580 --> 00:01:29,060 bound on the distance or our sequence from the number that 25 00:01:29,060 --> 00:01:32,440 we suspect to be the limit. 26 00:01:32,440 --> 00:01:37,039 If that distance becomes smaller and smaller, if we can 27 00:01:37,039 --> 00:01:41,550 manage to bound that distance by a certain number and that 28 00:01:41,550 --> 00:01:47,789 number goes down to 0, then it is guaranteed that since this 29 00:01:47,789 --> 00:01:50,900 distance goes down to 0, that the sequence, ai, 30 00:01:50,900 --> 00:01:52,850 converges to a. 31 00:01:52,850 --> 00:01:55,400 And there's a variation of this argument, which is the 32 00:01:55,400 --> 00:02:00,330 so-called sandwich argument, and it goes as follows. 33 00:02:00,330 --> 00:02:05,050 If we have a certain sequence that converges to some number, 34 00:02:05,050 --> 00:02:10,090 a, and we have another sequence that converges to 35 00:02:10,090 --> 00:02:15,070 that same number, a, and our sequence is somewhere 36 00:02:15,070 --> 00:02:20,750 in-between, then our sequence must also converge to that 37 00:02:20,750 --> 00:02:23,620 particular number, a. 38 00:02:23,620 --> 00:02:29,040 So these are the usual ways of quickly saying something about 39 00:02:29,040 --> 00:02:32,970 the convergence of a given sequence, and we will be often 40 00:02:32,970 --> 00:02:37,160 using this type of argument in this class, but without making 41 00:02:37,160 --> 00:02:42,190 a big fuss about them, or without even referring to 42 00:02:42,190 --> 00:02:43,990 these facts in an explicit manner.