sequences and series - Weird limit $lim limits_{nmathoptoinfty}frac{1}{e^n}sum limits_{kmathop=0}^nfrac{n^k}{k!}$

$$\lim \limits_{n\mathop\to\infty}\frac{1}{e^n}\sum \limits_{k\mathop=0}^n\frac{n^k}{k!}$$

I thought this limit was obviously $1$ at first but approximations on Mathematica tells me it's $1/2$. Why is this?

Answer

In this answer, it is shown that
$$\begin{align} e^{-n}\sum_{k=0}^n\frac{n^k}{k!} &=\frac{1}{n!}\int_n^\infty e^{-t}\,t^n\,\mathrm{d}t\\ &=\frac12+\frac{2/3}{\sqrt{2\pi n}}+O(n^{-1}) \end{align}$$