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

$ \displaystyle \sum_{k = 1}^{\infty} ke^{- k} $

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Oregon State University

Baylor University

University of Michigan - Ann Arbor

let's use the ratio test to see whether or not the Siri's converges. So the ratio test requires this expression a k equals K E to the minus K. So let's look at the limit. Is king goes to infinity? Absolute value a K plus one over a k So the numerator Let's do this in red, they have K plus one e to the minus K minus one. What the denominator? Okay, we know what that is from up here, such as plug that in and we could drop the absolute values here because all the numbers involved, they're positive except the exploring here. But he too, the exponents still positive. So let's look at this fraction over here. That's just one over easy. And then also we have K plus one over again. However, if we take the limit of this, you could do low pitch house rule, for example, and you will get that this limit is equal to one. So here when we multiply these two numbers together, we're just getting one times one overeat, which is one overeat. That's less than one. So we conclude that the Siri's conversions bye the ratio test, and that's our final answer