Faulhaber's Formula

Theorem

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

Then:

$\displaystyle \sum_{k \mathop = 1}^n k^p = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}$

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

Proof

Let $x \ge 0$.

 $\displaystyle \sum_{k \mathop = 0}^{n - 1} e^{k x}$ $=$ $\displaystyle \sum_{k \mathop = 0}^{n - 1} \sum_{p \mathop = 0}^\infty \frac {\paren {k x}^p} {p!}$ $\quad$ Power Series Expansion for Exponential Function $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p} \frac {x^p} {p!}$ $\quad$ rearrangement is valid by Tonelli's Theorem $\quad$

We also have:

 $\displaystyle \sum_{k \mathop = 0}^{n - 1} e^{k x}$ $=$ $\displaystyle \frac {1 - e^{n x} } {1 - e^x}$ $\quad$ Sum of Infinite Geometric Progression $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {e^{n x} - 1} x \frac x {e^x - 1}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{p \mathop = 0}^\infty \frac {n^{p + 1} x^p} {\paren {p + 1}!} \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}$ $\quad$ Definition of Bernoulli Numbers $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{p \mathop = 0}^\infty \sum_{i \mathop = 0}^p \frac {n^{p + 1 - i} x^{p - i} } {\paren {p + 1 - i}!} \frac {B_i x^i} {i!}$ $\quad$ Definition of Cauchy Product $\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}$ $\quad$ $\quad$

Equating coefficients:

 $\displaystyle \sum_{k \mathop = 0}^{n - 1} k^p$ $=$ $\displaystyle \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i}$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{k \mathop = 1}^n k^p$ $=$ $\displaystyle \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}$ $\quad$ as $B_1 = -\dfrac 1 2$ and Odd Bernoulli Numbers Vanish $\quad$

$\blacksquare$

Source of Name

This entry was named for Johann Faulhaber.