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

Maybe, we can make use of Tannery's Theorem, or dominated convergence theorem, to exchange the order of the limit and summation:

\begin{align*} \lim_{n \to \infty}\sum_{k=1}^n\left(\frac{k}{n}\right)^k&=\lim_{n \to \infty}\sum_{k=0}^{n-1}\left[\left(1-\frac{k}{n}\right)^{n}\right]^{\frac{n-k}{n}}=\sum_{k=0}^{\infty}\lim_{n \to \infty}\left[\left(1-\frac{k}{n}\right)^{n}\right]^{\frac{n-k}{n}}=\sum_{k=0}^{\infty} e^{-k}=\frac{e}{e-1} \end{align*}

This is correct? How to verify that it satisfy the conditons of the theorem?

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