Sum of first n nth power

Question

Is there any way to find the value of $\sum_{j=1}^n j^k$ for k fixed.

For instance can we compute the general value of $1^5+2^5\cdots{}+n^5$?

Answer 1

Faulhaber's formula shows that for each $k$ the value you're after is a polynomial of degree $k+1$.

The formula also gives a concrete form of the coefficients of the polynomial (involving Bernoulli numbers), but for small $k$ it is easier just to remember that you're looking for a polynomial of degree $k+1$ and then just find it by computing the answers for $n=0,1,2,\ldots,k+1$ by hand and fitting a polynomial to those.