limits - Is the sequences${S_n}$ convergent?

Let

$$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$

Is the sequences$\{S_n\}$ convergent?

The following is my answer,but this is not correct. please give some hints.

For all $x\in\mathbb{R}$,

$$\lim_{n\rightarrow\infty}\sum_{k=0}^n\frac{x^k}{k!}=e^x.$$
then

$$\lim_{n\rightarrow\infty}e^{-n}\sum_{k=0}^n\frac{n^k}{k!}=1.$$