What is the solution to the exponential series limit problem?

[juantheron]

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$

 

[HallsofIvy]

Spoiler

Suppose the "n" in $$\frac{n^k}{k!}$$ were "x". Do you recognize $$\sum_{k= 0}^n \frac{x^k}{k!}$$ as a partial sum for the power series $$\sum_{k=0}^\infty \frac{x^k}{k!}= e^x$$?

 

[SatyaDas]

Spoiler

Suppose the "n" in $$\frac{n^k}{k!}$$ were "x". Do you recognize $$\sum_{k= 0}^n \frac{x^k}{k!}$$ as a partial sum for the power series $$\sum_{k=0}^\infty \frac{x^k}{k!}= e^x$$?

So, the limit should be just 1, right?

 

[I like Serena]

The problem was posted in the challenge forum. Please provide full solutions. And please hide them including any hints between spoiler tags.