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I can show that the following limit exists but I am having difficulties to find it. It is $$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$ Can someone please help me?

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    How could you show that the limit exist?2012-06-28
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    @Siminore: The sum is very similar to $\int_{0}^{1}x^xdx$. Isn't it?2012-06-28
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    Numerically, for $n=1000$, I get 1.58098.2012-06-28
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    @Siminore: That happens to be only a little shy of the correct answer, $\dfrac{1}{1 - 1/e}$.2012-06-28
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    Probably. But the result is not $\int_0^1 x^x\, dx$.2012-06-28

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