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Is the sequence $\frac{e^nn!}{n^n}$ convergent?

All I tried is to calculate - $$\lim \frac{a_{n+1}}{a_n}=1$$ but it isn't helping...

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    $\begingroup$ Do you know the Stirling approximation to the factorial? $\endgroup$
    – thedude
    Commented Nov 13, 2017 at 12:41
  • $\begingroup$ @thedude no. I will check. $\endgroup$ Commented Nov 13, 2017 at 12:44
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    $\begingroup$ See here also. $\endgroup$ Commented Nov 13, 2017 at 12:47
  • $\begingroup$ @thedude cool!!! $\endgroup$ Commented Nov 13, 2017 at 12:48
  • $\begingroup$ By very elementary means we can show there exists $ K>0$ such that $(\frac {n}{e})^n K\sqrt n\; \cdot (n!)^{-1} \to 1$ as $n\to \infty.$ Stirling's Formula is that $K=\sqrt {2\pi}\;$, which is difficult. $\endgroup$ Commented Nov 13, 2017 at 15:52

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Look up Stirling's Formula. It should be clear from here.

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