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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n e^{-n} $

converges

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Missouri State University

University of Nottingham

Idaho State University

Boston College

Let's test this series for convergence or divergence here. Let's note that this is alternating and then we have bee n equals either the minus end, which is one over e to the end, and that's bigger than zero. That's important to know, because you need bigger than or equal to zero in order to use the alternating serious test. Let's abbreviate this with a S t. So we have two conditions here. One, we need this limit to be zero. So we have one over either the end and that's equal to zero. Since either the end goes to infinity and one more condition here we need B n plus one to be less than bn, so decreasing. So in our case, that means one over E and plus one is less than or equal to one over e. And so this can be simplified. Yeah, and that's equivalent to And that's true. So the lemon zero the beans are decreasing, the beings are positive. It's an all star rating series. So this series convergence by the alternating series test A s t slow. Yeah. Yeah, and that's our final answer