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