I would like to compute the value of this series:
\begin{equation*} \sum_{n = 0}^{+ \infty} n . e^{- \alpha n} \end{equation*}
Where $\alpha$ is a constant.
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3This may give you some ideas. – David Mitra May 19 '15 at 15:12
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The answer to your question is in the related links: http://math.stackexchange.com/questions/525021/how-to-compute-infinite-series-sum-n-0-infty-ne-n?rq=1 – user222031 May 19 '15 at 15:14
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$\sum_{n = 0}^{ \infty} n e^{- a n} $ can be written $\sum_{n = 0}^{ \infty} n (e^{- a})^ n =\sum_{n = 0}^{ \infty} n x^ n $ where $x = e^{-a} $.
This is a well-known sum that is asked here often.
Ways of evaluating this include differentiating $\sum_{n = 0}^{ \infty} x^ n $ and multiplying the sum by $1-x$.
Have at it.
marty cohen
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