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

Question

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

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

where $0<\alpha<1$.

Thanks.

are you assuming $\alpha$ is rational? – Willie Wong Aug 13 '13 at 09:19 — Willie Wong, Aug 13 '13 at 09:19

score 5 · Accepted Answer · answered Aug 12 '13 at 14:50

I do not think there is any "nice" way to write the answer,

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.

If you type it into wolfram alpha, you will get an expression using Hurwitz and Riemann Zeta functions. Neither can be calculated by hand easily.

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