I am interested in series of the form
$$S(k)=\sum_{n=k}^\infty e^{-an^\alpha},$$ where $a>0$ and $0<\alpha\leq1$ are fixed parameters. Clearly, this series converges, i.e. $S(k)\to 0$ for $k\to\infty$. My questions are: Is some special function representation of $S$ known? And: Do there exist useful bounds, e.g. $S(k) \leq c\,e^{-a'k^\alpha}$ for all integers $k$, for suitable $c,a'>0$?