Faulhaber's Formula/Also presented as

Faulhaber's Formula: Also presented as

Some sources present Faulhaber's formula 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.


Faulhaber's formula can also be expressed using the archaic form of the Bernoulli numbers as:

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


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: $21.8.$
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bernoulli polynomial
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