# Math Help - Integral of a floor function

1. ## Integral of a floor function

Greetings.

A bit of help would be much appreciated on the following integral:

$\int_{0}^{\infty} \lfloor x \rfloor e^{-x} dx$

Edit:

Ah this is just a geometric series: $\int_{n-1}^{n} n e^{-x} dx$

2. Note $\int_{j}^{j+1}{e^{-x}\,dx}=(e-1)e^{-j-1},$ so $\int_{0}^{\infty }{\left\lfloor x \right\rfloor e^{-x}\,dx}=\sum\limits_{j=0}^{\infty }{\int_{j}^{j+1}{\left\lfloor x \right\rfloor e^{-x}\,dx}}=(e-1)\sum\limits_{j=0}^{\infty }{je^{-j-1}}.$

Computation of that series will give ya the answer.

3. Good catch it looks like we cross posted.