What is the solution to the exponential series limit problem?

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Discussion Overview

The discussion centers around evaluating the limit of an exponential series, specifically the expression $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$. The scope includes mathematical reasoning and problem-solving techniques related to limits and series.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant presents the limit problem for evaluation.
  • Another participant requests full solutions and hints to be hidden, indicating a preference for a structured approach to problem-solving.

Areas of Agreement / Disagreement

The discussion does not indicate any consensus or disagreement, as it primarily consists of the initial problem statement and a request for solutions.

Contextual Notes

There are no specific assumptions or limitations mentioned, but the request for hidden solutions suggests a focus on careful exploration of the problem.

Who May Find This Useful

Mathematics students or enthusiasts interested in limits, series, and problem-solving techniques in calculus.

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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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