Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16...

Question

Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

Accepted Answer

Welcome back, I'm so glad you're here. We have been given an infinite geometric series. 25 then negative 125 over two and so on. And we are to find the sum. So to find the sum of an infinite geometric series. We first start by finding the ratio to find the ratio we take any two consecutive terms in our series and we take the second of those two and divide it by the one immediately preceding it. So I'll choose negative 125 over to the second term and divide it by the one immediately proceeding it. So we'll divide that by 25. We know this is the same as negative over two times one, 25th. And we can do some reduction here. 25 divided by 25 is one, negative 125 divided by 25 is negative five. So multiplying that straight across, our ratio is going to be -5/2. But hold on a second, there's a rule for finding the sum of an infinite geometric series that the ratio has to be in between negative one and one and negative five halves or negative 2.5 does not follow that rule. What does that mean? Well, let's take a look here and see what would happen if we did add up these numbers. So if we took 25 plus negative 125 over two, that would give us negative 37.5 And then we took that number and added 625 over four. This third term in our series, then we would go back up to 118.75 and then if we took 118.75 and added the fourth term here negative 3125 over eight, that would get us back down to negative 271.875. So when we keep going in this series we're going to keep going back and forth between positive and negative and we're going to get bigger and bigger and bigger. We could not find a sum for that. This series does what we call diverges. It does not converge to a sum. It diverges and therefore the answer is c no solution. Well done. We'll catch you on the next one.

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