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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n5^{2n}}{10^{n+1}} $

Two words: Divergence test

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Oregon State University

University of Michigan - Ann Arbor

Boston College

Let's use the ratio test to show that the Siri's diverges. So here's our a n and we'LL look at the limit and goes to infinity Absolute value and plus one over an. So let's go ahead and write out these fractions. And here we could drop the absolute value sign because all the ends there are positive for any end. So here replace and within plus one and to be safe, I just go ahead and use parentheses so we don't make any of those common mistakes. And then we'LL divide thiss I just am which is given by the original formula. So at this point, we can just rewrite this now take the green fraction and flip that over, multiply and then let's cancel out as much as we can. If we take off and plus one of these tens, we'LL have one left on the bottom. And if we cross off to any of these fives, we'LL have two left on the top. And then here there was a mistake. There should be a five square come back and erase that That's a five square. We still have a ten on the bottom and then end and to take the limited and plus one over end. That's just goes toe one. And so we have twenty five over ten, which is bigger than one. So by the route the ratio test, we conclude that the Siri's divergence