Anyone familiar with this exotic infinite sum? I'm assuming $n$ and $x$ are positive integers as well as $n-x>1$ (for convergence).
$$\lim_{m\to\infty}\sum_{s=x}^m\frac{s!}{s^n(s-x)!}$$
One can make arguments using Stirling's approximation and eventually trading sums for integrals, but things get messy. I think I've got an upper bound but it blows up exponentially as $n$ grows which isn't very useful because it should decrease as $n$ increases.
Even directing me to some literature would be much appreciated.
We have $$\frac{s!}{(s-x)!} < s^x,$$ so the infinite sum is bounded by
$$\sum_{s=x}^\infty s^{x-n}=\zeta (n-x,x),$$ where the zeta function is the Hurwitz zeta.