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

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

where $0<\alpha<1$.


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

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.

$$S_{n+1} - S_n = (n+1)^\alpha$$


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