Editing Dirichlet eta function (section)

==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 [[Binomial transform#Euler transform|Euler's transformation of alternating series]], to obtain
<math display="block">\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}. </math>

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

=== Borwein's method ===
[[Peter Borwein]] used approximations involving [[Chebyshev polynomials]] to produce a method for efficient evaluation of the eta function.<ref>{{cite book | first=Peter | last=Borwein | author-link=Peter Borwein | url=http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf | chapter=An efficient algorithm for the Riemann zeta function | series=Conference Proceedings, Canadian Mathematical Society | year=2000 | title=Constructive, Experimental, and Nonlinear Analysis | volume=27 | pages=29–34 | isbn=978-0-8218-2167-1 | editor-first=Michel A. | editor-last=Théra | publisher=[[American Mathematical Society]], on behalf of the [[Canadian Mathematical Society]] | location=Providence, RI | access-date=2008-09-20 | archive-date=2011-07-26 | archive-url=https://web.archive.org/web/20110726090927/http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf | url-status=dead }}</ref> If
<math display="block">d_k = n\sum_{\ell=0}^k \frac{(n+\ell-1)!4^\ell}{(n-\ell)!(2\ell)!}</math>
then
<math display="block">\eta(s) = -\frac{1}{d_n} \sum_{k=0}^{n-1}\frac{(-1)^k(d_k-d_n)}{(k+1)^s}+\gamma_n(s),</math>
where for <math>\Re(s) \ge \frac{1}{2} </math> the error term {{math|''γ''<sub>''n''</sub>}} is bounded by
<math display="block">|\gamma_n(s)| \le \frac{3}{(3+\sqrt{8})^n} (1+2|\Im(s)|)\exp\left(\frac{\pi}{2}|\Im(s)|\right).</math>

The factor of <math>3+\sqrt{8}\approx 5.8</math> in the error bound indicates that the Borwein series converges quite rapidly as ''n'' increases.