Verify that, for all $n \in \mathbb{N}^*$, $$n! \left(\frac{2}{n}\right)^n P\left(S_n > n\right) = \mathrm{e}^{-n/2} \sum_{k=1}^{\infty} \frac{n! n^k}{(n+k)!} \left(\frac{1}{2}\right)^k.$$