Further sequences and series

Summation of series

$\sum r=\frac12n(n+1)$, $\sum r^2=\frac16n(n+1)(2n+1)$, $\sum r^3=\frac14n^2(n+1)^2$. Combine them for polynomial sums. When each term is a difference $f(r)-f(r+1)$, almost everything cancels: the method of differences.