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Is there a way to find precise asymptotics or better bounds of series such as $\sum_{n=1}^{\infty}x^{n+1/n}/n!>e^x$ ?

Or $\sum_{n=1}^{\infty}x^{\sqrt n}/e^n$?

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    @Administrator Well, note that for x>1 $\root n \of x \to {1^ + }$ (it tends to 1 from values greater than 1) and for 0 $\root n \of x \to {1^- }$2012-04-24

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In the first case, the sums $S(x)$ are such that $\exp(-x)S(x)\to1$ when $x\to+\infty$. In the second case, the sums $T(x)$ are such that $\exp(-(\log x)^2/4)T(x)\to2\sqrt{\pi}$ when $x\to+\infty$.

A nice probabilistic interpretation helps in the first case while some raw real analysis techniques inspired by Laplace's method solve the second case. The two questions do not have much in common. No idea what "better bounds" means.

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    And the absence of comment by the OP is puzzlin$g$.2014-07-29