Value of $\sum\limits^{\infty}_{n=1}\frac{\ln n}{n^{1/2}\cdot 2^n}$ Here is a series:
$$\displaystyle \sum^{\infty}_{n=1}\dfrac{\ln n}{n^{\frac12}\cdot 2^n}$$
It is convergent by d'Alembert's law. Can we find the sum of this series ?

$f(n)=\dfrac{\ln n}{n^{1/2}\cdot 2^n}>0~\forall~n>1\\e^nf(e^n)=e^{n/2}\dfrac{n}{2^n}$
$[e^nf(e^n)]^{1/n}=\dfrac{\sqrt e n^{1/n}}{2}\to\sqrt e/2<1$
$\displaystyle \sum^{\infty}_{n=1}e^nf(e^n)$ is convergent implies $\displaystyle \sum^{\infty}_{n=1}f(n)$ is convergent.
