calculus - Evaluate the integral $int_0^{infty} lfloor x rfloor e^{-x}mathrm dx$

I'd like some help with the following integral:

$$\int_0^\infty \lfloor x \rfloor e^{-x}\mathrm dx .$$

Thanks.

Answer

This reduces to a series $\displaystyle \sum_{n=0}^{\infty} \int_n^{n+1}\!\! n e^{-x}\;dx$. The integrals are easy to evaluate and so is the series.