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Test the series for convergence or divergence.

$ \displaystyle \sum_{k = 1}^{\infty} \frac {2^{k-1} 3^{k+1}}{k^k} $

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

for this problem we'LL try using the root test for a lot of problems. You can get away with using, uh, different techniques to figure it out. The ratio test, for example, would also lead you to the answer here. But we'll do a problem. Ah, using the retest. So the root tests we take this thing and then we look at the K through and then similar to the ratio test. What we would like to happen would be for this to be less than one. Okay. And then this should be it one okay. And then similar to the ratio test, it's important that the stuff in here is positive. If it wasn't, then we would just, you know, put the absolute value signs around there. So we're looking at the absolute value of the terms in consideration, and we're taking the K through and limit is kay goes to infinity. Okay, so this thing, the exponents well, distribute over these guys. So this is going toe Simplify some. You will have to. That k over K is one in minus one. Over. Kay. Sorry. That's should be a k. And then similarly here we have k over kay, which is one and that we have plus one over Kay Cannon Down at the bottom, we have ke the k to the one over cape hours That's going to make a to the k o ver que So that's just k to the one. Okay, but the limit conduce skin.