# Exponential series limit

#### jacks

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

#### HallsofIvy

##### Well-known member
MHB Math Helper
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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#### Satya

##### New member
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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#### Klaas van Aarsen

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