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If $$\sum_{n=1}^{\infty}\frac{a_n}{e^n}$$ is convergent


show that $$\sum_{n=1}^{\infty}\frac{S_n}{e^n}$$ is convergent.

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Just to clarify, $e=\ln(1)$ here, right? Its a constant? – icurays1 Nov 14 '12 at 16:43
@icurays1 Rather $\ln e=1$. – Hagen von Eitzen Nov 14 '12 at 16:49
Ha, whoops. Coffee hasn't kicked in yet. Yes, that's what I meant. – icurays1 Nov 14 '12 at 16:51
yes, $e$ means euler constant – Laura Nov 15 '12 at 0:45
up vote 3 down vote accepted

Use Abel's identity to show that:


Now the answer to your question follows very easily because the limit of the right hand side exists.

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+1 for nice argument! I want to comment that though the convergence of $S_{k} e^{-k}$ follows from Cesaro-Stolz theorem together with the observation $a_n = o(e^n)$, it still seems to deserve a justification when it comes to elementary level analysis. – Sangchul Lee Nov 14 '12 at 16:53

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