calculus - How to Evaluate $int _0^{infty} lfloor x rfloor e^{-x} dx$?

How to Evaluate $\int _0^{\infty} \lfloor x \rfloor e^{-x} dx$ , where $\lfloor x \rfloor$ is the largest integer less than x?

I am thinking of using something like $\int_0^{\infty} (e^{-2}+2e^{-3}+....+ne^{-n-1}+.....)dx$ and then use $\int ne^{-n-1}= -e^{-n}$ But the integral, AFAIK, is only finitely-additive, so now what?