Dirichlet Eta Function - Numerical Algorithms

Numerical Algorithms

Most of the series acceleration techniques developed for alternating series can be profitably applied to the evaluation of the eta function. One particularly simple, yet reasonable method is to apply Euler's transformation of alternating series, to obtain

$\eta(s)=\sum_{n=0}^\infty \frac{1}{2^{n+1}} \sum_{k=0}^n (-1)^{k} {n \choose k} \frac {1}{(k+1)^s}.$

Note that the second, inside summation is a forward difference.

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