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JH
Numerade Educator

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Problem 15 Easy Difficulty

Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n \pi^n}{( - 3)^{n-1}} $

Answer

Diverges

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Video Transcript

tease the ratio test to see whether the Siri's conversions now the ratio test involves this term here. Let's call it a n and pi to the end Power overnegative three to the N minus one. Now for the ratio test, we're interested in the value of the following limit. So let's go ahead and write this out. So this is good right now working on the numerator, the and plus one. So that's this whole term. Over here we still have the absolute value. And now for the denominator, there's our end. We just use this formula up here and notice we could drop the absolute values. And then we could also ignore the negative sign on the inside. So we have n plus one over n three and minus one over three end, and then we have pie and plus one over part of them. Now, really? Here We only have one left that needs to be evaluated and plus one over n and the limit that goes toe one. We write this down here, then we have, after doing some cancellation. Here we have one over three, and then after doing some cancellation with the pies, we have pie, and that's equal toe just pie over three. However, this is bigger than one. So we conclude that the Siri's converges excuse me, diverges by the ratio test, and that's our final answer.