WEBVTT 99:59:59.999 --> 99:59:59.999 1520 11.2 #3 Geometric Series https://youtu.be/DvBMYCz9M8s https://youtu.be/UiAUGF-eDjk 99:59:59.999 --> 99:59:59.999 One type of series where we can actually come up with a formula 99:59:59.999 --> 99:59:59.999 for the sequence of partial sums 99:59:59.999 --> 99:59:59.999 is called a geometric series. 99:59:59.999 --> 99:59:59.999 You might remember learning about geometric sequences 99:59:59.999 --> 99:59:59.999 in the previous section. 99:59:59.999 --> 99:59:59.999 Basically, it's the same idea except the terms are being added together. 99:59:59.999 --> 99:59:59.999 Every term is being multiplied by a common ratio, which we call r, 99:59:59.999 --> 99:59:59.999 and that means the formula for our generic nth term would be ‘a’, 99:59:59.999 --> 99:59:59.999 which was the first term, 99:59:59.999 --> 99:59:59.999 and then times r to the n minus 1st power, 99:59:59.999 --> 99:59:59.999 so that would be the formula for our series for each individual term. 99:59:59.999 --> 99:59:59.999 But the idea here is, in some cases, 99:59:59.999 --> 99:59:59.999 we can find a formula for the nth partial sum, 99:59:59.999 --> 99:59:59.999 for the sum of the first n terms, and based on that, 99:59:59.999 --> 99:59:59.999 we can determine whether this series will converge or diverge. 99:59:59.999 --> 99:59:59.999 A couple of cases to get out of the way first, 99:59:59.999 --> 99:59:59.999 if r is equal to 1, what would this thing look like? 99:59:59.999 --> 99:59:59.999 Well, what it would look like is just ‘a’ and then plus another ‘a’ 99:59:59.999 --> 99:59:59.999 and then plus another ‘a’ and continuing in that fashion forever. 99:59:59.999 --> 99:59:59.999 Now, if I just added up the first n terms here, 99:59:59.999 --> 99:59:59.999 I would have a total of n a’s all being added together. 99:59:59.999 --> 99:59:59.999 That would be n times ‘a’ [na]. 99:59:59.999 --> 99:59:59.999 But if we take a limit as n approaches infinity for n times ‘a’, 99:59:59.999 --> 99:59:59.999 that limit is going to be infinite. 99:59:59.999 --> 99:59:59.999 So if r is equal to 1, this series is going to diverge. 99:59:59.999 --> 99:59:59.999 Hopefully, that makes sense; 99:59:59.999 --> 99:59:59.999 if you add up an infinite number of terms and they're all staying the same, 99:59:59.999 --> 99:59:59.999 your sum is not going to approach a value. 99:59:59.999 --> 99:59:59.999 Basically, the only way this wouldn't diverge 99:59:59.999 --> 99:59:59.999 is if ‘a’ is equal to 0, 99:59:59.999 --> 99:59:59.999 and most of the time, we're going to ignore that case 99:59:59.999 --> 99:59:59.999 because it's really not a very interesting version of a geometric series. 99:59:59.999 --> 99:59:59.999 Next up, we're going to look at the case: if r is not equal to 1. 99:59:59.999 --> 99:59:59.999 In this case, the sum of the first n terms 99:59:59.999 --> 99:59:59.999 would just be all of these terms added together. 99:59:59.999 --> 99:59:59.999 So this is our nth term. 99:59:59.999 --> 99:59:59.999 But actually, we can do something interesting algebraically here 99:59:59.999 --> 99:59:59.999 if directly below this, I take r times all of the terms, 99:59:59.999 --> 99:59:59.999 because if I do that, I take r times this term, 99:59:59.999 --> 99:59:59.999 I'm going to get ‘a’ times r. 99:59:59.999 --> 99:59:59.999 If I take r times this term, I'll get ‘a’ times r squared. 99:59:59.999 --> 99:59:59.999 I'm going to continue in this fashion, 99:59:59.999 --> 99:59:59.999 and this is actually going to be the second-to-last term 99:59:59.999 --> 99:59:59.999 because that's what I would get if I took the previous term times r. 99:59:59.999 --> 99:59:59.999 That means, if I take this times r, I'm going to get ‘a’ times r to the n. 99:59:59.999 --> 99:59:59.999 And it turns out, since both of these sums are finite, 99:59:59.999 --> 99:59:59.999 (there's a set number of terms), I can combine them together. 99:59:59.999 --> 99:59:59.999 But surprisingly, I'm actually going to subtract them. 99:59:59.999 --> 99:59:59.999 I'm going to get Sn minus r times Sn. 99:59:59.999 --> 99:59:59.999 My goal here is to come up with a formula for just Sn. 99:59:59.999 --> 99:59:59.999 But the interesting thing that happens is, 99:59:59.999 --> 99:59:59.999 if we were to subtract all of these terms from these terms, 99:59:59.999 --> 99:59:59.999 almost all of the terms are going to cancel out. 99:59:59.999 --> 99:59:59.999 The only ones that are going to be left are the very first term 99:59:59.999 --> 99:59:59.999 and the very last term of this one. 99:59:59.999 --> 99:59:59.999 Since we're solving for Sn, 99:59:59.999 --> 99:59:59.999 it hopefully makes sense to factor that out from this side of the equation. 99:59:59.999 --> 99:59:59.999 There's also a common factor of ‘a’ 99:59:59.999 --> 99:59:59.999 that we could factor out from this side of the equation. 99:59:59.999 --> 99:59:59.999 Solving for Sn, we can get this by itself by dividing by 1 minus r. 99:59:59.999 --> 99:59:59.999 The reason we don't have to worry about dividing by 0 99:59:59.999 --> 99:59:59.999 is because we're working under the assumption 99:59:59.999 --> 99:59:59.999 that r is not equal to 1 in this case, which gives us this formula. 99:59:59.999 --> 99:59:59.999 What we're looking at here 99:59:59.999 --> 99:59:59.999 is a formula for the sum of the first n terms of a geometric series 99:59:59.999 --> 99:59:59.999 as long as our common ratio is not 1. 99:59:59.999 --> 99:59:59.999 What I'm interested in, though, 99:59:59.999 --> 99:59:59.999 is if the infinite series is going to converge or not. 99:59:59.999 --> 99:59:59.999 Since we have a formula for Sn, 99:59:59.999 --> 99:59:59.999 now we're going to take the limit of this as n approaches infinity 99:59:59.999 --> 99:59:59.999 to see what's going to happen to the sum. 99:59:59.999 --> 99:59:59.999 One thing you'll maybe notice here is, 99:59:59.999 --> 99:59:59.999 there's only one part of this entire formula that has an n in it, 99:59:59.999 --> 99:59:59.999 and it's this part right here. 99:59:59.999 --> 99:59:59.999 This is actually something we talked about in Section 11.1. 99:59:59.999 --> 99:59:59.999 This right here is only going to converge if the absolute value of r is less than 1. 99:59:59.999 --> 99:59:59.999 Think about it: If this is greater than 1, 99:59:59.999 --> 99:59:59.999 then multiplying it by itself multiple times 99:59:59.999 --> 99:59:59.999 will cause it to basically blow up to infinity. 99:59:59.999 --> 99:59:59.999 And if it's less than negative 1, the same problem is going to happen, 99:59:59.999 --> 99:59:59.999 except the sign will be alternating from positive to negative. 99:59:59.999 --> 99:59:59.999 This is only going to converge if the absolute value of r is less than 1. 99:59:59.999 --> 99:59:59.999 Any other value for r, and this thing is going to diverge, 99:59:59.999 --> 99:59:59.999 except of course, 1, which we've already ruled out. 99:59:59.999 --> 99:59:59.999 Basically, this sequence, which stands for the sequence of partial sums, 99:59:59.999 --> 99:59:59.999 converges if the absolute value of r is less than 1, 99:59:59.999 --> 99:59:59.999 and it's going to diverge if the absolute value of r is greater than or equal to 1. 99:59:59.999 --> 99:59:59.999 We've already excluded this case, 99:59:59.999 --> 99:59:59.999 but if r was negative 1, we would run into the same problem. 99:59:59.999 --> 99:59:59.999 It would still diverge. 99:59:59.999 --> 99:59:59.999 What does it converge to? 99:59:59.999 --> 99:59:59.999 Well, if the absolute value of r is less than 1, 99:59:59.999 --> 99:59:59.999 then this term right here is going to approach 0 as n grows. 99:59:59.999 --> 99:59:59.999 And that means, what we're going to be left with 99:59:59.999 --> 99:59:59.999 is just ‘a’ in the numerator divided by 1 minus r, 99:59:59.999 --> 99:59:59.999 and that winds up being the sum of a geometric series. 99:59:59.999 --> 99:59:59.999 If the absolute value of r is less than 1, 99:59:59.999 --> 99:59:59.999 then this series (the geometric series) is going to have a sum, 99:59:59.999 --> 99:59:59.999 and that sum is going to equal ‘a’ divided by 1 minus r. 99:59:59.999 --> 99:59:59.999 If this condition is not met, 99:59:59.999 --> 99:59:59.999 then the geometric series is going to diverge.