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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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hasExams
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2 Answers2

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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 $x$.

EuYu
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  • that's more scarier :(( ... is there any formula without summation? – hasExams Aug 31 '12 at 03:30
  • @testuser If there is one, then I don't know about it. But I really doubt that there exists anything much nicer than this expression. You can already see in the list you provided that the expressions are getting quite large by the $10$ sum. This is simply a more compact way of expressing the inevitably larger sums that occur with higher powers. – EuYu Aug 31 '12 at 03:32
  • perhaps so ... thank you for answer – hasExams Aug 31 '12 at 03:34
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    Technically there is a "formula" without summation using the Bernoulli polynomials, specifically $$\sum_{k=0}^{x} k^p = \frac{B_{p+1}(x+1)-B_{p+1}(0)}{p+1},$$ but in some sense that is cheating, because the Bernoulli polynomial by definition is just the above sum! – Eric Naslund Aug 31 '12 at 03:40
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    @testuser Of course the right-hand-side is related to Bernoulli polynomials $$ \sum_{k=1}^n k^m = \frac{1}{m+1} \left(B_{m+1}(n+1)-B_{m+1}(1)\right)$$ – Sasha Aug 31 '12 at 03:40
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    Also the sum defines what is known as generalized harmonic numbers: $$ \sum_{k=1}^n k^m = H_n^{(-m)} $$ This is somewhat a tautology, though. – Sasha Aug 31 '12 at 03:42
  • Waaw ... I didn't know that – hasExams Aug 31 '12 at 03:43
  • @Sasha: You can probably also use the Hurwitz zeta function and the zeta function, but that again is somewhat tautological. – Eric Naslund Aug 31 '12 at 04:17
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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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Mhenni Benghorbal
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