Pi and Gamma

$\begin{aligned} L & = \lim_{n \to \infty} \sum_{k=1}^n \frac {\pi \displaystyle \color{#3D99F6} \int_0^\infty x^k e^{-x} dx}{n k!} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac {\pi \color{#3D99F6} \Gamma (k+1)}{n k!} & \small \color{#3D99F6} \text{where }\Gamma (\cdot) \text{ denotes the gamma function.} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac {\pi \color{#3D99F6} k!}{n k!} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac \pi n \\ & = \lim_{n \to \infty} \pi \\ & = \boxed \pi \end{aligned}$