Sum of Powers of Positive Integers
Theorem
Let $n, p \in \Z_{>0}$ be (strictly) positive integers.
Then:
| \(\displaystyle \sum_{k \mathop = 1}^n k^p\) | \(=\) | \(\displaystyle 1^p + 2^p + \cdots + n^p\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {n^{p + 1} } {p + 1} + \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \left({p - 1}\right) \left({p - 2}\right) n^{p - 3} } {4!} + \cdots\) | $\quad$ | $\quad$ |
where:
- $B_k$ are the Bernoulli numbers
- $p^{\underline k}$ is the $k$th falling factorial of $p$.
Proof
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Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Sums of Powers of Positive Integers: $19.8$
