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Find $$\lim_{n\to\infty} \sum_{k=1}^{n}\left( \frac{k}{n}\right)^{n}$$

I can't compare it with similar series and I can't change it to Riemann's sum.


marked as duplicate by YuiTo Cheng, Thomas Shelby, postmortes, Leucippus, Theo Bendit 28 mins ago

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  • $\begingroup$ I'm embarrassed to say that I've answered this question, although the answer I gave there is quite different. $\endgroup$ – robjohn Mar 21 '17 at 11:16

For each $k\ge0$, $[n\gt k]\left(1-\frac kn\right)^n$ is non-decreasing in $n$, where $[\dots]$ are Iverson brackets. Therefore, by monotone convergence $$ \begin{align} \lim_{n\to\infty}\sum_{k=1}^n\left(\frac kn\right)^n &=\lim_{n\to\infty}\sum_{k=0}^{n-1}\left(1-\frac kn\right)^n\\ &=\sum_{k=0}^\infty e^{-k}\\[6pt] &=\frac{e}{e-1} \end{align} $$

  • $\begingroup$ Sorry I couldn't find the question before, thanks a lot $\endgroup$ – Laurence Mar 21 '17 at 12:13

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