I'm looking for an expression that yields the above sum.
It's a straight and simple question, feel free to move this post to any other section, if it's not in the appropriate section, I know this site can be pretty strict with questions:
Is $\sum_{k=1}^{\infty}\frac{e^{-k}}{k^2}$ known?
Making the problem more general $$\sum_{k=1}^{\infty}\frac{e^{-k x}}{k^n}=\text{Li}_n\left(e^{-x}\right)$$where appears the polylogarithm function which is a special function.