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Determine whether the series is convergent or divergent.

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

convergent

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

for this problem. We are going to use the ratio test and first I'm just going to simplify the ratio and then I'll take the limit. So we want to know is what does next term that is the K plus one term look like when it's divided by case turns. So I'm going to distribute on the numerator and split the fraction up so I can simplify things down. So I get Kay need the negative K minus one over K to the negative k. Let's eat the negative K minus one over. Okay, even negative K. Now the left hand side reduces to just mhm the negative one and the right hand side. It becomes each the negative one over K. Now, if we take the limit, this K goes to infinity of this we say that either the negative one over K goes to zero. And so we are left with just the two, the negative one and either the negative one is less than one. So the series is convergent by the ratio test