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Possible Duplicate:
“Closed” form for $\sum \frac{1}{n^n}$

Using Weierstrass theorem (any monotonic and bounded sequence is convergent), we can prove that the sequence $u_{n}=\sum_{k=1}^{n}\frac{1}{k^k}$ with $n$ a positive integer is convergent. But can we actually find its limit ?

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    Actually, I think this is a duplicate of 21330. Ismail writes "the sequence $u_n$ ... is convergent. But can we actually find its limit?" This seems to me to be asking abut the infinite sum, which is the subject of question 21330.2011-11-27

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There is the famous sophmore's dream identity: $\sum_{k=1}^{\infty} \frac{1}{k^k} = \int_0^1 \frac{dx}{x^x}.$

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"Find" in the sense of "having a closed form expression"? Likely, no, under most reasonable interpretations of "closed form expression". One can of course calculate the limit approximately (it is roughly 1.2913...) - this is pretty easy since the series converges very fast.