For $n \in \mathbb{N}$, we set $I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx$.
Show that for $n \geqslant 1$, we have
$$I_{n} = \left(\frac{n}{e}\right)^{n} \sqrt{n} \int_{-\sqrt{n}}^{+\infty} \left(1 + \frac{x}{\sqrt{n}}\right)^{n} e^{-x\sqrt{n}}\, dx.$$