How to prove that $\lim\frac{\sum^{n}_{i = 1}{i^n}}{n^n} = e / (e - 1)$

Asked Sep 06 '22 at 11:14

Active Sep 06 '22 at 11:14

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It is easy to show that $\displaystyle\sum^{n}_{i=1}i^n/n^n < e/(e - 1)$, the question is to show the other direction.

asked Sep 06 '22 at 11:14

HWC

You might try to bound (a) $e^{i-n} - i^n/n^n$ and (b) the effect of a finite sum – Henry Sep 06 '22 at 11:26
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@geetha290krm I think it is supposed to be $\lim\limits_{ n \to \infty}$ – Henry Sep 06 '22 at 11:27

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