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# Determine whether the series is convergent or divergent. If it is convergent, find its sum.$\displaystyle \sum_{n = 1}^{\infty} 3^{n + 1} 4^{-n}$

## The series converges to 9

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##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

let's determine whether the Siri's converges or diverges. And then if it's conversion, then we'LL go ahead and find the sun. So here, let's just go ahead and rewrite this from one to infinity and then let's write. This is three times three end over for the end, and then I'LL write this as three times. Then we have three over four to the end. Now what kind of Siri's is this? She a metric, and we could see that R. R is three over four, and since three over four satisfies this inequality here in converges. And now, since the commercials will go ahead and find the song. So for a geometric series, the formula is to do the first term of the Siri's. So this is the term corresponding to plugging in, and I should go back here plugging in that and value that's given under the summation the smallest one and then one minus R. So in our problems the first term here, go ahead and plug in and equals one, and you get three times three over for and then divided by one minus tree over for so up, top nine over four and the denominator we have one over four, and then it gives us a final answer of nine. That will be the sun of this geometric series that was originally given to us, and that's your final answer.

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