MHB What is the solution to the exponential series limit problem?

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Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$
 
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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?
 
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HallsofIvy said:
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?
 
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The problem was posted in the challenge forum. Please provide full solutions. And please hide them including any hints between spoiler tags.
 

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