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

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^{n-1} \frac {3^n}{2^n n^3} $

Diverges

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Missouri State University

Oregon State University

University of Nottingham

let's use the ratio test to show that the Siri's diverges. So here this is our A end, and the ratio test requires that we look at the limit and goes to infinity. Absolute value a N plus one over. And so let's evaluate this limit here. We don't even have to bother writing the negative one because we're taking absolute value. So we have three and plus one, two and plus one. And then So this is an plus one up here. No, than in the denominator, we have an All right. So now let's go ahead and write. This is a product So three and plus one to end and cubed three and two and plus one and then and plus one cute. Now let's go ahead and buy as much as we can and then also take a limit. So here we could cancel and on the threes. So we have three left over. Similarly, with their tattoos, you could take off end of those. You have one left in the bottom. And then if you look at the limit of end, cubed over and tell us one cute, that's just one. So this limit is equal to three over, too. That's a dangerous and won. So from there we can conclude that the Siri's diverges by the ratio test. So let's go ahead and circle that up there. Diversion, I'll say, by the ratio test, and that's our final answer.