Define Bernoulli polynomials as: $P_0(x)=1$, $P'_n(x)=nP_{n-1}(x)$, $\int_0^1P_n(x)=0$ if $n\geq1$
Need to prove that for $n\geq2$ we have
$\sum_{r=1}^{k-1} r^n= \frac{P_{n+1}(k)-P_{n+1}(0)}{n+1}$
I have already proved that:
$P_n(1)=P_n(0)$ if $n\geq2$ and $P_n(x+1)-P_n(x)=nx^{n-1}$ if $n\geq1$
I tried using induction, but got stuck making the inductive step. Could anybody provide any hints? Thanks!