I found the expression to the sum of powers long ago and ofcours I think it is true but i don't know for sure, the problem is, it's little though for me to test and try it out. Also i'd like to know how should i prove it.
for k>=0 $$\sum_{m=1}^{n} m^{k+1} =\sum^{k}_{s=0}\sum^{s}_{i=0} {s \choose i}{n+1 \choose s+2}(n-i)^{k} (-1)^{s+i} $$
I revised this because i was looking for additional ways to get the zeta function, this was the formula were i needed it: $\zeta(-s)= 1/(2^s-1)*(2^s\sum_{n=1}^{m/2} n^s +2^s\sum_{n=1}^{(m-1)/2} n^s-\sum_{n=1}^{m} n^s$)
After using both formulas is there a way to simplify it?