What's the exact value of $\lim\limits_{n\rightarrow \infty}\frac{e^n}{\sum\limits_{i=0}^{i=n}\frac{n^i}{i!}}$?

p.s. I suppose it may be 2, but I cannot prove it.

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What's the exact value of $\lim\limits_{n\rightarrow \infty}\frac{e^n}{\sum\limits_{i=0}^{i=n}\frac{n^i}{i!}}$?

p.s. I suppose it may be 2, but I cannot prove it.

This question appears to be off-topic. The users who voted to close gave this specific reason:

- "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Lucia, Chris Godsil, Vladimir Dotsenko, Michael Albanese

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Yes, it is 2. The inverse fraction is a probability that a Poisson random variable with mean value $n$ takes a value at most $n$. It follows from appropriate central limit theorem that this probability approaches $1/2$ for large $n$.

notmean the problem is not interesting or full of ramifications. I find Fedor's answer enlightening. Yet I voted to close, since I think this is the spirit of MO. Especially since the post was seemingly given as a calculus exercise, without any context or motivation whatsoever. $\endgroup$ – Loïc Teyssier Mar 4 '17 at 21:43