What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$?
The sum is not known to possess a closed form expression. In general, no series of the form $\displaystyle\sum_{n}a^{\large n^b}$ does, for $a\neq0$ and $b\not\in\{0,1,2\}$. However, $~\displaystyle\sum_{n=1}^\infty n~a^{\sqrt n}~\simeq~\int_0^\infty x~a^{\sqrt x}~dx$, which, after letting $x=t^2$, can be evaluated using the expression of the $\Gamma$ function, yielding $~\dfrac{12}{\ln^4a}$