I want to evaluate the sum $$\sum_{k=1}^\infty \left(\frac{i+1}{\sqrt{2}}\right)^k\cdot k^{-\alpha}$$ where $i=\sqrt{-1}$ and $\alpha\in[\frac{3}{4},1]$ with 8 digits accuracy.If I am willing to expend up to a second of CPU time per calculation, how should I proceed?Suppose I want a really fast scheme for the evaluation and I am willing to expend a lot of work. How should I proceed? Any leads, clues or suggestions would be greatly appreciated.
Evaluate the sum
numerics;performance;fortran;c;special functions