1 00:00:00,340 --> 00:00:02,866 I want to know where a sequence is heading. 2 00:00:02,866 --> 00:00:08,869 [MUSIC] 3 00:00:08,869 --> 00:00:13,015 Let's think about a specific sequence. Here's an example to think about. 4 00:00:13,015 --> 00:00:16,985 The sequence a sub n defined by this formula. 5 00:00:16,985 --> 00:00:21,280 The numerator's 6n plus 2, and the denominator's 3n plus 3. 6 00:00:21,280 --> 00:00:23,520 We can list off the first few terms. 7 00:00:23,520 --> 00:00:29,428 For instance, if I want to compute a sub 1, I just plug in 1 for n. 8 00:00:29,428 --> 00:00:34,100 I can get 6 times 1 plus 2 divided by 3 times 9 00:00:34,100 --> 00:00:38,020 1 plus 3. That numerator is 8. 10 00:00:38,020 --> 00:00:40,750 That denominator is 6. 11 00:00:40,750 --> 00:00:46,340 And I could re-write 8 6ths as 4 3rds in lowest terms. 12 00:00:46,340 --> 00:00:48,390 And I could keep on going by plugging in 2. 13 00:00:48,390 --> 00:00:53,540 I find that a sub 2 is 14 9ths, a sub 3 is 5 3rds. 14 00:00:53,540 --> 00:00:55,380 A sub 4 is 26 15ths. 15 00:00:55,380 --> 00:00:59,550 A sub 5 is 16 9ths. And I could keep on going. 16 00:00:59,550 --> 00:01:03,130 Those fractions, really aren't providing me with much insight. 17 00:01:03,130 --> 00:01:05,950 What's the eventual value of the sequence? 18 00:01:05,950 --> 00:01:07,860 Where is the sequence heading? 19 00:01:07,860 --> 00:01:11,920 Might get a better sense, by plugging in, 1000. 20 00:01:11,920 --> 00:01:18,344 Say, what's the 1000th term? Well that's 6 21 00:01:18,344 --> 00:01:24,530 times 1000 plus 2 over 3 times 1000 plus 3. 22 00:01:24,530 --> 00:01:30,610 That's 6002 over 3003. Well, that's 23 00:01:31,710 --> 00:01:37,560 awfully close to 2. I can go even farther out in the sequence. 24 00:01:37,560 --> 00:01:41,400 A sub a million. Is what? 25 00:01:42,430 --> 00:01:50,190 It's 6 times a million plus 2 divided by 3 times 26 00:01:50,190 --> 00:01:55,954 a million plus 3. And that works out to be 27 00:01:55,954 --> 00:02:01,582 6,000,002 divided by 28 00:02:01,582 --> 00:02:07,940 3,000,003. And that's insanely close to 2. 29 00:02:07,940 --> 00:02:10,470 Alright, the limit of a sub n as 30 00:02:10,470 --> 00:02:14,620 n approaches infinity is 2, to express this idea. 31 00:02:14,620 --> 00:02:15,180 Okay, 32 00:02:15,180 --> 00:02:20,420 so what does that really mean? Well I promise that a sub n, is as close 33 00:02:20,420 --> 00:02:25,860 as you want to 2. Provided that n is large enough. 34 00:02:25,860 --> 00:02:27,670 For better or for worse, these things are 35 00:02:27,670 --> 00:02:30,990 usually written out with fewer words, and more symbols. 36 00:02:30,990 --> 00:02:33,245 Instead of saying as close as you want to 37 00:02:33,245 --> 00:02:38,040 2, I'll say that a sub n is within epsilon. 38 00:02:38,040 --> 00:02:41,120 Some small positive number of 2. 39 00:02:41,120 --> 00:02:46,330 And how large is large enough? Well instead of saying large enough, I'll 40 00:02:46,330 --> 00:02:52,210 just that n is at least as big as some index big N. 41 00:02:52,210 --> 00:02:53,770 So I can say it like this. 42 00:02:53,770 --> 00:02:59,920 So for every epsilon greater than 0, there is an index, big N. 43 00:02:59,920 --> 00:03:02,610 So that, whenever little n is bigger than or 44 00:03:02,610 --> 00:03:07,290 equal to big N, a sub n is within epsilon 45 00:03:07,290 --> 00:03:08,190 of 2. 46 00:03:08,190 --> 00:03:11,670 The idea here, is that this epsilon is measuring how 47 00:03:11,670 --> 00:03:14,420 close you want a sub n to be to 2. 48 00:03:14,420 --> 00:03:20,730 And I'm telling you that, no matter how close you want a sub n to be to 2. 49 00:03:20,730 --> 00:03:23,540 If you go out far enough in the sequence, 50 00:03:23,540 --> 00:03:27,100 all the terms after that are actually that close. 51 00:03:27,100 --> 00:03:29,370 Instead of writing within epsilon of 2, 52 00:03:29,370 --> 00:03:32,430 you'll normally see it with absolute value. 53 00:03:32,430 --> 00:03:38,740 So instead of saying within epsilon of 2, I'll say that the distance between 54 00:03:38,740 --> 00:03:43,090 a sub n and 2, is less than epsilon. Put it all together. 55 00:03:43,090 --> 00:03:48,550 The same that the limit of a sub n, as n approaches infinity equals 2. 56 00:03:48,550 --> 00:03:52,780 Means for every epsilon greater than 0, no matter how close you want 57 00:03:52,780 --> 00:03:57,590 to be to the limiting value of 2, there's some index n, so that 58 00:03:57,590 --> 00:04:00,650 whenever you're farther out in the sequence than big N. 59 00:04:00,650 --> 00:04:03,340 Whenever little n is bigger than or equal to Big N. 60 00:04:03,340 --> 00:04:07,100 Then the absolute value of a sub n minus 2, which is measuring the 61 00:04:07,100 --> 00:04:11,890 distance between a sub n and 2, that absolute value is less than epsilon. 62 00:04:11,890 --> 00:04:15,850 Of course, there's nothing really special about the 2 in this. 63 00:04:15,850 --> 00:04:18,630 So in general, to say that the limit of a sub 64 00:04:18,630 --> 00:04:22,640 n equals L, means that for every epsilon, there's a whole number 65 00:04:22,640 --> 00:04:23,470 big N. 66 00:04:23,470 --> 00:04:27,510 So that whenever little n is bigger than or equal to big N, the absolute value of a 67 00:04:27,510 --> 00:04:36,108 sub n minus that limiting value L, is less than epsilon. 68 00:04:36,108 --> 00:04:42,696 [SOUND]