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?
How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
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calculus
sequences-and-series
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0How could you show that the limit exist? – 2012-06-28
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1@Siminore: The sum is very similar to $\int_{0}^{1}x^xdx$. Isn't it? – 2012-06-28
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0Numerically, for $n=1000$, I get 1.58098. – 2012-06-28
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0@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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0Probably. But the result is not $\int_0^1 x^x\, dx$. – 2012-06-28