Faulhaber's formula

In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers

\sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}

as a polynomial in n. In modern notation, Faulhaber's formula is

\sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{r=0}^{p}{\binom {p+1}{r}}B_{r}n^{p-r+1}.

Here, ${\textstyle {\binom {p+1}{r}}}$ is the binomial coefficient "p + 1 choose r", and the B_j are the Bernoulli numbers with the convention that ${\textstyle B_{1}=+{\frac {1}{2}}}$ .

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