Faulhaber's Formula
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Theorem
Let $n, p \in \Z_{>0}$ be (strictly) positive integers.
Then:
| \(\ds \sum_{k \mathop = 1}^n k^p\) | \(=\) | \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n^{p + 1} } {p + 1} - \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots\) |
where:
- $B_i$ denotes the $i$th Bernoulli number.
Proof
Let $x \ge 0$.
| \(\ds \sum_{k \mathop = 0}^{n - 1} e^{k x}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^{n - 1} \sum_{p \mathop = 0}^\infty \frac {\paren {k x}^p}
{p!}\)
|
Power Series Expansion for Exponential Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p} \frac {x^p} {p!}\) | rearrangement is valid by Tonelli's Theorem |
We also have:
| \(\ds \sum_{k \mathop = 0}^{n - 1} e^{k x}\) | \(=\) | \(\ds \frac {1 - e^{n x} } {1 - e^x}\) | Sum of Geometric Sequence | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{n x} - 1} x \frac x {e^x - 1}\) | multiplying numerator and denominator by $x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 x \paren {\sum_{p \mathop = 0}^\infty \frac {\paren {n x }^p} {p!} - 1 } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}\) | Definition of Bernoulli Numbers and Power Series Expansion for Exponential Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 x \paren {\sum_{p \mathop = 1}^\infty \frac {\paren {n x }^p} {p!} } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}\) | Factorial of $0$ and Zeroth Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 x \paren {\sum_{p \mathop = 0}^\infty \frac {\paren {n x }^{p + 1} } {\paren {p + 1}! } } \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}\) | Translation of Index Variable of Summation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{p \mathop = 0}^\infty \frac {n^{p + 1} x^p} {\paren {p + 1}!} \sum_{p \mathop = 0}^\infty \frac {B_p x^p} {p!}\) | Power of Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{p \mathop = 0}^\infty \sum_{i \mathop = 0}^p \frac {n^{p + 1 - i} x^{p - i} } {\paren {p + 1 - i}!} \frac {B_i x^i} {i!}\) | Definition of Cauchy Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{p \mathop = 0}^\infty \paren {\sum_{i \mathop = 0}^p \dfrac {p + 1} {p + 1} \dfrac {p!} {p!} \dfrac 1 {\paren {p + 1 - i}! i!} B_i n^{p + 1 - i} } x^p\) | multiplying by $1$ and Product of Powers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}\) | Definition of Binomial Coefficient |
Equating coefficients:
| \(\ds \sum_{k \mathop = 0}^{n - 1} k^p\) | \(=\) | \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^{n - 1} k^p + n^p\) | \(=\) | \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} + \paren {\frac 1 {p + 1} \binom {p + 1} 1 n^p }\) | adding $n^p$ to both sides and Binomial Coefficient with One | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^n k^p\) | \(=\) | \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}\) | as $B_1 = -\dfrac 1 2$ and Odd Bernoulli Numbers Vanish |
$\blacksquare$
Source of Name
This entry was named for Johann Faulhaber.