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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1$\begingroup$ I do protest again against this put on hold ! It is an interesting problem with long history! It was first proposed at MGU Olympiad in 1976. It is also in deep connection with famous unusual Ramanujan's inequality and further Szego results. $\endgroup$– SergeiCommented Mar 4, 2017 at 17:54
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$\begingroup$ @Sergei: I do believe that old MGU Olympiad problems are off-topic here. Which does not mean 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 TeyssierCommented Mar 4, 2017 at 21:43
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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 $n$.
answered Mar 4, 2017 at 15:01