Sum of an integral

$\begin{aligned} S & = \sum_{n=1}^\infty \int_n^{n+1} xe^{-x} dx & \small \color{#3D99F6} \text{By integration by parts} \\ & = \sum_{n=1}^\infty \left(- xe^{-x} \bigg|_n^{n+1} + \int_n^{n+1} e^{-x} dx \right) \\ & = \sum_{n=1}^\infty \left[ \frac {x+1}{e^x} \right]_{n+1}^n \\ & = \sum_{n=1}^\infty \left(\frac {n+1}{e^n} - \frac {n+2}{e^{n+1}} \right) \\ & = \boxed{\dfrac 2e} \end{aligned}$