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JH
Numerade Educator

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Problem 24 Easy Difficulty

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 0}^{\infty} \frac {3^{n + 1}}{(-2)^n} $

Answer

The geometric series diverges

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Video Transcript

it's determine whether or not this geometric series converges and if it's commercial, will go in and find that some as well. So here we can rewrite the Siri's and equal zero to infinity. Let me go ahead and pull out one of the factors of three here, so I'm going to do here is use the fact that three the M plus one, it's just three and times three. So I'm gonna pull on a three there I have my three to the end still and then on the bottom I still have that minus two to the end and then once more I'll use another laud iPhone. It's three and then up. What I'LL do here is all used the fact that eight of them over B to the end is a over B to the end. So using that here with a equals three B equals negative, too. We can write it in this form and now we see that this Siri's look something like an equal zero to infinity a are the end. So this is usually how geometric series are reading the signal notation so we could see that the A is three. And here we see that the R is negative. Three halves, so for geometric series. So let's write the following. We have since the absolute value our which in our case is positive three over too satisfies this and really it could even be equal to one. Any time. Your absolute value are is one or more, which is what we have here. The series will diverge. So let's say the Siri's is diversion. Therefore, we don't have to find the sum so we could ignore the second sentence so divergent and that's my final answer.