Faulhaber's Formula

In a 1631 edition of Academiae Algebrae, J. Faulhaber published the general formula for the Sum of $p$th Powers of the first $n$ Positive Integers,

$$\sum_{k=1}^n k^p = {1\over p+1}\sum_{i=1}^{p+1} (-1)^{\delta_{ip}}{p+1\choose i}B_{p+1-i} n^i,$$ (1)

where $\delta_{ip}$ is the Kronecker Delta, ${n\choose i}$ is a Binomial Coefficient, and $B_i$ is the $i$th Bernoulli Number. Computing the sums for $p=1$, ..., 10 gives

$$\sum_{k=1}^n k = {\textstyle{1\over 2}}(n^2+n)$$ (2)

$$\sum_{k=1}^n k^2 = {\textstyle{1\over 6}}(2n^3+3n^2+n)$$ (3)

$$\sum_{k=1}^n k^3 = {\textstyle{1\over 4}}(n^4+2n^3+n^2)$$ (4)

$$\sum_{k=1}^n k^4 = {\textstyle{1\over 30}}(6n^5+15n^4+10n^3-n)$$ (5)

$$\sum_{k=1}^n k^5 = {\textstyle{1\over 12}}(2n^6+6n^5+5n^4-n^2)$$ (6)

$$\sum_{k=1}^n k^6 = {\textstyle{1\over 42}}(6n^7+21n^6+21n^5-7n^3+n)$$ (7)

$$\sum_{k=1}^n k^7 = {\textstyle{1\over 24}}(3n^8+12n^7+14n^6-7n^4+2n^2)$$ (8)

$$\sum_{k=1}^n k^8 = {\textstyle{1\over 90}}(10n^9+45n^8+60n^7-42n^5+20n^3-3n)$$ (9)

$$\sum_{k=1}^n k^9 = {\textstyle{1\over 20}}(2n^{10}+10n^9+15n^8-14n^6+10n^4-3n^2)$$ (10)

$$\sum_{k=1}^n k^{10} = {\textstyle{1\over 66}}(6n^{11}+33n^{10}+55n^9-66n^5-33n^3+5n).$$ (11)

See also Power, Sum

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.