Faulhaber's Formula

Theorem
Let $n$ and $p$ be positive integers.

Then:


 * $\displaystyle \sum_{k \mathop = 1}^n k^p = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \left({-1}\right)^i \binom {p + 1} i B_i n^{p + 1 - i}$

where $B_n$ denotes the $n$th Bernoulli number.

Proof
Let $x \ge 0$.

We also have:

By equating coefficients, we find that: