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Evaluate :

$$ \lim_{n\to\infty}\left(\dfrac{1}{e^{n}}\displaystyle \sum_{r=0}^{n} \dfrac{n^r}{r!}\right) $$

Numerical calculation suggests that the limit should be $\dfrac{1}{2}$. I tried using Squeeze Theorem and Stolz - Cesaro Theorem, but to no avail.

An elementary solution is appreciated.
Thanks in advance.

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Henry
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    You will find many interesting answers here: http://math.stackexchange.com/questions/160248/evaluating-lim-limits-n-to-infty-e-n-sum-limits-k-0n-fracnkk – Olivier Oloa Jul 25 '16 at 20:04
  • @OlivierOloa Thanks a lot! – Henry Jul 25 '16 at 20:05
  • There are no elementary solutions there though :( Please don't close my question, I'd like to see if there are elementary proofs of this. – Henry Jul 25 '16 at 20:10
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    @Jack D'Aurizio There are no elementary solutions there though :( Please don't close my question, I'd like to see if there are elementary proofs of this. – Henry Jul 25 '16 at 20:12
  • @Henry: to use the Poisson distribution and the Central Limit Theorem is the usual way. An "elementary solution" strongly depends on what you meant by "elementary". I think that Sangchul Lee's answer in the other topic is elementary enough. – Jack D'Aurizio Jul 25 '16 at 20:32

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