Sorry for the remedial math but:
Can someone tell me how to get a closed form for
$$\sum_{k=1}^n k^x$$
For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?
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The general formula is fairly complicated: $$\sum_{k=1}^n{k^x} = {1\over x+1}\sum_{k=0}^x{{x+1\choose k}B_k n^{x+1-k}}$$ Where $B_k$ are the Bernoulli numbers. This is commonly called "Faulhaber's formula", but that is a misattribution. The Donald Knuth paper "Johann Faulhaber and sums of powers" explains the history in some detail, including what Faulhaber did and did not know. In particular, he did not know the general formula above, although he did work on many special cases. For the special case of $x=2$ we have: $$\sum_{k=1}^n{k^2} = {\frac 16}(2n^3+3n^2+n)$$ |
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