#51 If Floor(x) denotes the greatest integer not exceeding x, then Integral(Floor(x)*Exp(-x))dx from 0 to infinity = ?

The answer turns out to be 1/(e-1).

I can get it to Sum(n*Exp(-n)*(1-1/e)) from n = 0 to infinity, which Mathematica confirms is 1/(e-1), I just have no idea to find the sum of that infinite series by hand. Off the top of my head, I can only think of the sum of a geometric series or p-series where p = 2.

Any help would be appreciated.