WEBVTT 00:00:00.508 --> 00:00:04.975 One type of series where we can actually come up with a formula 00:00:04.975 --> 00:00:07.991 for the sequence of partial sums 00:00:07.991 --> 00:00:10.024 is called a geometric series. 00:00:10.024 --> 00:00:12.974 You might remember learning about geometric sequences 00:00:12.974 --> 00:00:14.640 in the previous section. 00:00:14.640 --> 00:00:18.774 Basically, it's the same idea except the terms are being added together. 00:00:18.774 --> 00:00:24.774 Every term is being multiplied by a common ratio, which we call r, 00:00:24.774 --> 00:00:31.474 and that means the formula for our generic nth term would be ‘a’, 00:00:31.474 --> 00:00:33.324 which was the first term, 00:00:33.324 --> 00:00:36.790 and then times r to the n minus 1st power, 00:00:36.790 --> 00:00:41.724 so that would be the formula for our series for each individual term. 00:00:41.724 --> 00:00:44.907 But the idea here is, in some cases, 00:00:44.907 --> 00:00:49.073 we can find a formula for the nth partial sum, 00:00:49.073 --> 00:00:52.040 for the sum of the first n terms, and based on that, 00:00:52.040 --> 00:00:56.207 we can determine whether this series will converge or diverge. 00:00:56.207 --> 00:00:59.657 A couple of cases to get out of the way first, 00:00:59.657 --> 00:01:05.490 if r is equal to 1, what would this thing look like? 00:01:05.490 --> 00:01:11.307 Well, what it would look like is just ‘a’ and then plus another ‘a’ 00:01:11.307 --> 00:01:18.725 and then plus another ‘a’ and continuing in that fashion forever. 00:01:18.725 --> 00:01:22.258 Now, if I just added up the first n terms here, 00:01:22.258 --> 00:01:28.058 I would have a total of n a’s all being added together. 00:01:28.058 --> 00:01:31.506 That would be n times ‘a’ [na]. 00:01:31.506 --> 00:01:37.906 But if we take a limit as n approaches infinity for n times ‘a’, 00:01:37.906 --> 00:01:41.456 that limit is going to be infinite. 00:01:41.456 --> 00:01:46.356 So if r is equal to 1, this series is going to diverge. 00:01:46.356 --> 00:01:48.272 Hopefully, that makes sense; 00:01:48.272 --> 00:01:52.809 if you add up an infinite number of terms and they're all staying the same, 00:01:52.809 --> 00:01:56.022 your sum is not going to approach a value. 00:01:56.022 --> 00:01:58.589 Basically, the only way this wouldn't diverge 00:01:58.589 --> 00:02:00.139 is if ‘a’ is equal to 0, 00:02:00.139 --> 00:02:02.839 and most of the time, we're going to ignore that case 00:02:02.839 --> 00:02:07.555 because it's really not a very interesting version of a geometric series. 00:02:07.555 --> 00:02:13.449 Next up, we're going to look at the case: if r is not equal to 1. 00:02:13.449 --> 00:02:16.416 In this case, the sum of the first n terms 00:02:16.416 --> 00:02:19.215 would just be all of these terms added together. 00:02:19.215 --> 00:02:21.032 So this is our nth term. 00:02:21.032 --> 00:02:24.882 But actually, we can do something interesting algebraically here, 00:02:24.882 --> 00:02:29.750 if directly below this, I take r times all of the terms, 00:02:29.750 --> 00:02:32.972 because if I do that, I take r times this term, 00:02:32.972 --> 00:02:36.018 I'm going to get ‘a’ times r. 00:02:36.018 --> 00:02:39.950 If I take r times this term, I'll get ‘a’ times r squared. 00:02:39.950 --> 00:02:42.890 I'm going to continue in this fashion, 00:02:42.890 --> 00:02:46.836 and this is actually going to be the second-to-last term 00:02:46.836 --> 00:02:51.965 because that's what I would get if I took the previous term times r. 00:02:51.965 --> 00:02:57.269 That means, if I take this times r, I'm going to get ‘a’ times r to the n. 00:02:57.715 --> 00:03:01.766 And it turns out, since both of these sums are finite, 00:03:01.766 --> 00:03:07.133 (there's a set number of terms), I can combine them together. 00:03:07.133 --> 00:03:10.133 But surprisingly, I'm actually going to subtract them. 00:03:10.133 --> 00:03:14.966 I'm going to get Sn minus r times Sn. 00:03:14.966 --> 00:03:18.949 My goal here is to come up with a formula for just Sn. 00:03:18.949 --> 00:03:21.732 But the interesting thing that happens is, 00:03:21.732 --> 00:03:25.149 if we were to subtract all of these terms from these terms, 00:03:25.149 --> 00:03:27.846 almost all of the terms are going to cancel out. 00:03:27.846 --> 00:03:30.839 The only ones that are going to be left are the very first term 00:03:30.839 --> 00:03:36.185 and the very last term of this one. 00:03:36.185 --> 00:03:38.786 Since we're solving for Sn, 00:03:38.786 --> 00:03:46.285 it hopefully makes sense to factor that out from this side of the equation. 00:03:46.285 --> 00:03:48.116 There's also a common factor of ‘a’ 00:03:48.116 --> 00:03:53.749 that we could factor out from this side of the equation. 00:03:53.749 --> 00:03:59.606 Solving for Sn, we can get this by itself by dividing by 1 minus r. 00:03:59.606 --> 00:04:02.323 The reason we don't have to worry about dividing by 0 00:04:02.323 --> 00:04:04.849 is because we're working under the assumption 00:04:04.849 --> 00:04:17.316 that r is not equal to 1 in this case, which gives us this formula. 00:04:21.530 --> 00:04:23.582 What we're looking at here 00:04:23.582 --> 00:04:28.842 is a formula for the sum of the first n terms of a geometric series 00:04:28.842 --> 00:04:32.224 as long as our common ratio is not 1. 00:04:32.224 --> 00:04:33.957 What I'm interested in, though, 00:04:33.957 --> 00:04:37.141 is if the infinite series is going to converge or not. 00:04:37.141 --> 00:04:39.109 Since we have a formula for Sn, 00:04:39.109 --> 00:04:42.659 now we're going to take the limit of this as n approaches infinity 00:04:42.659 --> 00:04:54.283 to see what's going to happen to the sum. 00:04:54.283 --> 00:04:56.937 One thing you'll maybe notice here is, 00:04:56.937 --> 00:05:01.786 there's only one part of this entire formula that has an n in it, 00:05:01.786 --> 00:05:04.154 and it's this part right here. 00:05:04.154 --> 00:05:08.587 This is actually something we talked about in Section 11.1. 00:05:08.587 --> 00:05:16.954 This right here is only going to converge if the absolute value of r is less than 1. 00:05:16.954 --> 00:05:19.769 Think about it: If this is greater than 1, 00:05:19.769 --> 00:05:22.554 then multiplying it by itself multiple times 00:05:22.554 --> 00:05:26.037 will cause it to basically blow up to infinity. 00:05:26.037 --> 00:05:29.637 And if it's less than negative 1, the same problem is going to happen, 00:05:29.637 --> 00:05:32.970 except the sign will be alternating from positive to negative. 00:05:32.970 --> 00:05:38.303 So this is only going to converge if the absolute value of r is less than 1. 00:05:38.303 --> 00:05:42.470 Any other value for r, and this thing is going to diverge, 00:05:42.470 --> 00:05:45.783 except of course, 1, which we've already ruled out. 00:05:45.783 --> 00:05:51.613 Basically, this sequence, which stands for the sequence of partial sums, 00:05:51.613 --> 00:06:00.797 converges if the absolute value of r is less than 1; 00:06:00.797 --> 00:06:09.930 and it's going to diverge if the absolute value of r is greater than or equal to 1. 00:06:10.912 --> 00:06:13.358 We've already excluded this case, 00:06:13.358 --> 00:06:17.174 but if r was negative 1, we would run into the same problem. 00:06:17.174 --> 00:06:19.074 It would still diverge. 00:06:19.074 --> 00:06:21.107 What does it converge to? 00:06:21.107 --> 00:06:24.141 Well, if the absolute value of r is less than 1, 00:06:24.141 --> 00:06:29.561 then this term right here is going to approach 0 as n grows. 00:06:29.561 --> 00:06:31.890 And that means, what we're going to be left with 00:06:31.890 --> 00:06:36.373 is just ‘a’ in the numerator divided by 1 minus r, 00:06:36.373 --> 00:06:40.190 and that winds up being the sum of a geometric series. 00:06:40.190 --> 00:06:46.257 If the absolute value of r is less than 1, 00:06:46.257 --> 00:06:54.924 then this series (the geometric series) is going to have a sum, 00:06:54.924 --> 00:07:02.957 and that sum is going to equal ‘a’ divided by 1 minus r. 00:07:02.957 --> 00:07:04.974 If this condition is not met, 00:07:04.974 --> 00:07:07.691 then the geometric series is going to diverge.