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Possible Duplicate:
Finite Sum of Power?

Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula for any value of $x$?

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    This seems to be the same question as [this one](http://math.stackexchange.com/questions/155166/finite-sum-of-power).2012-08-31

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For arbitrary natural $x\in\mathbb{N}$, the general formula is given by Faulhaber's Formula, which is $\sum_{k=1}^n k^x = \frac{1}{x+1} \sum_{i=0}^x (-1)^i{x+1 \choose i} B_i \cdot n^{x+1-i}$ where $B_i$ are the Bernoulli Numbers with $B_1 = -\frac{1}{2}$.

I am not too convinced that there exists a nice closed form expression for arbitrary $x$.

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    @Sasha: You can probably also use the Hurwitz zeta function and the zeta function, but that again is somewhat tautological.2012-08-31
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I already answered a question similar to this on this website. See here. You can use the zeta function and the Hurwitz zeta function to get a closed form formula to your sum,

$ \sum_{k=1}^n k^x = \zeta(-x) + \zeta(-x,n+1) \,. $

See reference 1 and reference 2.

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    Funny how that analytic continuation stuff works here. ;)2016-11-29