What is the solution to the exponential series limit problem?

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SUMMARY

The limit problem evaluated is $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$. The solution reveals that this limit converges to 1, utilizing the properties of the Poisson distribution. The discussion highlights the application of Stirling's approximation for factorials to simplify the expression, confirming the convergence through rigorous mathematical analysis.

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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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