Series representation of $m=1^k + 2^k + \ldots n^k$.

Question

Is there any simple form for following question?

$m=1^k+2^k+\cdots+n^k$

Answer 1

See Faulhaber's formula, for instance.

Answer 2

As a follow-up to Lucian's answer, you might also like to take a look at the proof of Faulhaber's formula on planetmath to see how those Bernoulli numbers manage to sneak into the discussion, since it looks like the Wikipedia article doesn't go into much detail on that point.

Short answer: It's because Bernoulli numbers are defined as the coefficients in the exponential generating function $$\frac{x}{e^x - 1} = \sum_{k = 0}^\infty B_k \frac{x^k}{k!}.$$

The crazy cool thing is that this generating function, and slight variations, crop up surprisingly often in various higher math contexts, so the Bernoulli numbers do too!

Here's just a sampling:

• (Ir)regular primes and Kummer's work on Fermat's Last Theorem
• Special values of Riemann zeta: at negative integers, at positive even integers
• Euler-Maclaurin summation
• Todd classes in algebraic topology and algebraic geometry. (This and Hirzebruch-Riemann-Roch, it seems, is why Bernoulli numbers turn up in descriptions of the image of the J-homomorphism, algebraic K-theory of the integers, and the classification of exotic spheres.)