How to solve $\sum_{k=1}^{\infty} ke^{-k}$? I am interested in
$$\sum_{k=1}^{\infty} ke^{-k}$$
And I can get the closed form on Wolfram Alpha, $\frac{e}{(e-1)^2}$, but I am curious how to derive it. It doesn't appear to be a typical geometric series.

By elementary means:
$$S:=a+a^2+a^2+a^3+a^3+a^3+a^4+a^4+a^4+a^4+\cdots\\
=(a+a^2+a^3+a^4+\cdots)+a(a+a^2+a^2+a^3+a^3+a^3+\cdots),$$
hence for $|a|<1$,
$$S=\frac a{1-a}+aS.$$
