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This question already has an answer here:

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.

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ 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
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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 $$ \begin{align} \lim_{n\to\infty}\sum_{k=1}^n\left(\frac kn\right)^n &=\lim_{n\to\infty}\sum_{k=0}^{n-1}\left(1-\frac kn\right)^n\\ &=\sum_{k=0}^\infty e^{-k}\\[6pt] &=\frac{e}{e-1} \end{align} $$

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

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