Faulhaber's Formula

Theorem

Let $n, p \in \Z_{>0}$ be (strictly) positive integers.


Then:

\(\ds \sum_{k \mathop = 1}^n k^p\) \(=\) \(\ds 1^p + 2^p + \cdots + n^p\)
\(\ds \) \(=\) \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}\)
\(\ds \) \(=\) \(\ds \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \paren {-1}^i \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}\)
\(\ds \) \(=\) \(\ds \frac {n^{p + 1} } {p + 1} + \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots\)

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
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^{n - 1} e^{k x}\) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p} \frac {x^p} {p!}\)


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
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \paren {\dfrac 1 {p + 1} \sum_{i \mathop = 0}^p \dfrac {\paren {p + 1}! } {\paren {p + 1 - i}! i!} B_i n^{p + 1 - i} } \dfrac {x^p} {p!}\) Definition of Factorial
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}\) Definition of Binomial Coefficient
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^{n - 1} e^{k x}\) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \paren {\frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} } \frac {x^p} {p!}\)


Equating coefficients:

\(\ds \sum_{k \mathop = 0}^{n - 1} k^p\) \(=\) \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i}\) from $\paren 1$ and $\paren 2$ above
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^{n - 1} k^p + n^p\) \(=\) \(\ds \frac 1 {p + 1} \sum_{i \mathop = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} + \paren {\frac 1 {p + 1} \binom {p + 1} 1 n^p}\) 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

$\blacksquare$


Also presented as

Some sources present in the form:

\(\ds \sum_{k \mathop = 1}^n k^p\) \(=\) \(\ds \frac {\map {B_{p + 1} } {n + 1} - B_{p + 1} } {p + 1}\)

where:

$B_{p + 1}$ denotes a Bernoulli number
$\map {B_{p + 1} } {n + 1}$ denotes a Bernoulli polynomial.


can also be expressed using the archaic form of the Bernoulli numbers as:

\(\ds \sum_{k \mathop = 1}^n k^p\) \(=\) \(\ds \frac {n^{p + 1} } {p + 1} + \frac {n^p} 2 + \frac { {B_1}^* p n^{p - 1} } {2!} + \frac { {B_2}^* p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots\)


Source of Name

This entry was named for Johann Faulhaber.


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Sums of Powers of Positive Integers: $19.8$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 21$: Series of Constants: Sums of Powers of Positive Integers: $21.8.$
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