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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1Do you know the Stirling approximation to the factorial? – thedude Nov 13 '17 at 12:41
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@thedude no. I will check. – Yarden Sharabi Nov 13 '17 at 12:44
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1See here also. – David Mitra Nov 13 '17 at 12:47
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@thedude cool!!! – Yarden Sharabi Nov 13 '17 at 12:48
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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. – DanielWainfleet Nov 13 '17 at 15:52
