# Sum of the series $\sum_i |i|^\alpha$

I am looking for the sum to the following finite series:

$\sum_{i=0}^N |i|^\alpha$

where $0<\alpha<1$.

Thanks.

• actually $\alpha>0$ ...
– yoki
Aug 12 '13 at 14:47
• are you assuming $\alpha$ is rational? Aug 13 '13 at 9:19

Whenever you want to know how to solve such sums, let $S_n = \sum_{i = 0}^n i^\alpha$
Then $$S_{n+1} - S_n = (n+1)^\alpha$$ is the equation you want to solve.
$$S_{n+1} - S_n = (n+1)^\alpha$$