TITLE: Evaluate the integral $\int_0^{\infty} \lfloor x \rfloor e^{-x}\mathrm dx$
QUESTION [6 upvotes]: I'd like some help with the following integral:
$$\int_0^\infty \lfloor x \rfloor e^{-x}\mathrm dx .$$
Thanks.

REPLY [6 votes]: No need for any explicit summing. (Note that we have $\lfloor x+n \rfloor = \lfloor x \rfloor +n$ for any integer $n$.)
Let $I = \int_0^\infty \lfloor x \rfloor e^{-x} dx $. Letting $t=x+1$, we obtain
$I = \int_1^\infty (\lfloor t \rfloor -1)e e^{-t} dt = e\int_1^\infty \lfloor t \rfloor e^{-t} dt - e \int_1^\infty e^{-t} dt = e (I-e^{-1})$.
Solving gives
$I={1 \over e-1}$.