9:59:59.000,9:59:59.000 1520 11.2 #3 Geometric Series[br]https://youtu.be/DvBMYCz9M8s[br]https://youtu.be/UiAUGF-eDjk 9:59:59.000,9:59:59.000 One type of series where we can [br]actually come up with a formula 9:59:59.000,9:59:59.000 for the sequence of partial sums 9:59:59.000,9:59:59.000 is called a geometric series. 9:59:59.000,9:59:59.000 You might remember learning [br]about geometric sequences 9:59:59.000,9:59:59.000 in the previous section. 9:59:59.000,9:59:59.000 Basically, it's the same idea except [br]the terms are being added together. 9:59:59.000,9:59:59.000 Every term is being multiplied [br]by a common ratio, which we call r, 9:59:59.000,9:59:59.000 and that means the formula for our generic nth term would be ‘a’, 9:59:59.000,9:59:59.000 which was the first term, 9:59:59.000,9:59:59.000 and then times r to the [br]n minus 1st power, 9:59:59.000,9:59:59.000 so that would be the formula for [br]our series for each individual term. 9:59:59.000,9:59:59.000 But the idea here is, in some cases, 9:59:59.000,9:59:59.000 we can find a formula [br]for the nth partial sum, 9:59:59.000,9:59:59.000 for the sum of the first n terms, [br]and based on that, 9:59:59.000,9:59:59.000 we can determine whether [br]this series will converge or diverge. 9:59:59.000,9:59:59.000 A couple of cases to get [br]out of the way first, 9:59:59.000,9:59:59.000 if r is equal to 1, [br]what would this thing look like? 9:59:59.000,9:59:59.000 Well, what it would look like is [br]just ‘a’ and then plus another ‘a’ 9:59:59.000,9:59:59.000 and then plus another ‘a’ and [br]continuing in that fashion forever. 9:59:59.000,9:59:59.000 Now, if I just added up [br]the first n terms here, 9:59:59.000,9:59:59.000 I would have a total of n a’s [br]all being added together. 9:59:59.000,9:59:59.000 That would be n times ‘a’ [na]. 9:59:59.000,9:59:59.000 But if we take a limit as n [br]approaches infinity for n times ‘a’, 9:59:59.000,9:59:59.000 that limit is going to be infinite. 9:59:59.000,9:59:59.000 So if r is equal to 1, [br]this series is going to diverge. 9:59:59.000,9:59:59.000 Hopefully, that makes sense; 9:59:59.000,9:59:59.000 if you add up an infinite number of [br]terms and they're all staying the same, 9:59:59.000,9:59:59.000 your sum is not going [br]to approach a value. 9:59:59.000,9:59:59.000 Basically, the only way [br]this wouldn't diverge 9:59:59.000,9:59:59.000 is if ‘a’ is equal to 0, 9:59:59.000,9:59:59.000 and most of the time, [br]we're going to ignore that case 9:59:59.000,9:59:59.000 because it's really not a very interesting [br]version of a geometric series. 9:59:59.000,9:59:59.000 Next up, we're going to look [br]at the case: if r is not equal to 1. 9:59:59.000,9:59:59.000 In this case, the sum of the first n terms 9:59:59.000,9:59:59.000 would just be all of these [br]terms added together. 9:59:59.000,9:59:59.000 So this is our nth term. 9:59:59.000,9:59:59.000 But actually, we can do something [br]interesting algebraically here 9:59:59.000,9:59:59.000 if directly below this, [br]I take r times all of the terms, 9:59:59.000,9:59:59.000 because if I do that, [br]I take r times this term, 9:59:59.000,9:59:59.000 I'm going to get ‘a’ times r. 9:59:59.000,9:59:59.000 If I take r times this term, [br]I'll get ‘a’ times r squared. 9:59:59.000,9:59:59.000 I'm going to continue in this fashion, 9:59:59.000,9:59:59.000 and this is actually going to [br]be the second-to-last term 9:59:59.000,9:59:59.000 because that's what I would get [br]if I took the previous term times r. 9:59:59.000,9:59:59.000 That means, if I take this times r, [br]I'm going to get ‘a’ times r to the n. 9:59:59.000,9:59:59.000 And it turns out, since both [br]of these sums are finite, 9:59:59.000,9:59:59.000 (there's a set number of terms),[br]I can combine them together. 9:59:59.000,9:59:59.000 But surprisingly, [br]I'm actually going to subtract them. 9:59:59.000,9:59:59.000 I'm going to get Sn minus r times Sn. 9:59:59.000,9:59:59.000 My goal here is to come up [br]with a formula for just Sn. 9:59:59.000,9:59:59.000 But the interesting thing that happens is, 9:59:59.000,9:59:59.000 if we were to subtract all of [br]these terms from these terms, 9:59:59.000,9:59:59.000 almost all of the terms [br]are going to cancel out. 9:59:59.000,9:59:59.000 The only ones that are going [br]to be left are the very first term 9:59:59.000,9:59:59.000 and the very last term of this one. 9:59:59.000,9:59:59.000 Since we're solving for Sn, 9:59:59.000,9:59:59.000 it hopefully makes sense to factor [br]that out from this side of the equation. 9:59:59.000,9:59:59.000 There's also a common factor of ‘a’ 9:59:59.000,9:59:59.000 that we could factor out [br]from this side of the equation. 9:59:59.000,9:59:59.000 Solving for Sn, we can get this [br]by itself by dividing by 1 minus r. 9:59:59.000,9:59:59.000 The reason we don't have [br]to worry about dividing by 0 9:59:59.000,9:59:59.000 is because we're working [br]under the assumption 9:59:59.000,9:59:59.000 that r is not equal to 1 in this case, [br]which gives us this formula. 9:59:59.000,9:59:59.000 What we're looking at here 9:59:59.000,9:59:59.000 is a formula for the sum of the first [br]n terms of a geometric series 9:59:59.000,9:59:59.000 as long as our common ratio is not 1. 9:59:59.000,9:59:59.000 What I'm interested in, though, 9:59:59.000,9:59:59.000 is if the infinite series is [br]going to converge or not. 9:59:59.000,9:59:59.000 Since we have a formula for Sn, 9:59:59.000,9:59:59.000 now we're going to take the limit [br]of this as n approaches infinity 9:59:59.000,9:59:59.000 to see what's going to happen to the sum. 9:59:59.000,9:59:59.000 One thing you'll maybe notice here is, 9:59:59.000,9:59:59.000 there's only one part of this[br] entire formula that has an n in it, 9:59:59.000,9:59:59.000 and it's this part right here. 9:59:59.000,9:59:59.000 This is actually something [br]we talked about in Section 11.1. 9:59:59.000,9:59:59.000 This right here is only going to converge [br]if the absolute value of r is less than 1. 9:59:59.000,9:59:59.000 Think about it: If this is greater than 1, 9:59:59.000,9:59:59.000 then multiplying it [br]by itself multiple times 9:59:59.000,9:59:59.000 will cause it to basically [br]blow up to infinity. 9:59:59.000,9:59:59.000 And if it's less than negative 1, [br]the same problem is going to happen, 9:59:59.000,9:59:59.000 except the sign will be alternating [br]from positive to negative. 9:59:59.000,9:59:59.000 This is only going to converge [br]if the absolute value of r is less than 1. 9:59:59.000,9:59:59.000 Any other value for r, [br]and this thing is going to diverge, 9:59:59.000,9:59:59.000 except of course, 1, [br]which we've already ruled out. 9:59:59.000,9:59:59.000 Basically, this sequence, which stands[br] for the sequence of partial sums, 9:59:59.000,9:59:59.000 converges if the absolute [br]value of r is less than 1, 9:59:59.000,9:59:59.000 and it's going to diverge if the absolute [br]value of r is greater than or equal to 1. 9:59:59.000,9:59:59.000 We've already excluded this case, 9:59:59.000,9:59:59.000 but if r was negative 1, [br]we would run into the same problem. 9:59:59.000,9:59:59.000 It would still diverge. 9:59:59.000,9:59:59.000 What does it converge to? 9:59:59.000,9:59:59.000 Well, if the absolute [br]value of r is less than 1, 9:59:59.000,9:59:59.000 then this term right here is [br]going to approach 0 as n grows. 9:59:59.000,9:59:59.000 And that means, [br]what we're going to be left with 9:59:59.000,9:59:59.000 is just ‘a’ in the numerator [br]divided by 1 minus r, 9:59:59.000,9:59:59.000 and that winds up being [br]the sum of a geometric series. 9:59:59.000,9:59:59.000 If the absolute value of r is less than 1, 9:59:59.000,9:59:59.000 then this series (the geometric series) [br]is going to have a sum, 9:59:59.000,9:59:59.000 and that sum is going to equal [br]‘a’ divided by 1 minus r. 9:59:59.000,9:59:59.000 If this condition is not met, 9:59:59.000,9:59:59.000 then the geometric series [br]is going to diverge.[br]