Where $B_n(x)$ is the Bernoulli polynomial and $B_n$ are the
Bernoulli numbers
Conclude that $$1^m + 2^m + \dots + n^m = \frac{B_{m+1}(n+1) -
B_{m+1}}{m+1}$$ Amongst other things, all you have to do now is
find out what on earth is a Bernoulli polynomial!!