accelerated convergence | convergence | convergence acceleration

The improvement of the convergence properties of a series, also called convergence acceleration or accelerated convergence, such that a series reaches its limit to within some accuracy with fewer terms than required before. Convergence improvement can be effected by forming a linear combination with a series whose sum is known. Useful sums include sum_(n = 1)^∞ 1/(n(n + 1)) | = | 1 sum_(n = 1)^∞ 1/(n(n + 1)(n + 2)) | = | 1/4 sum_(n = 1)^∞ 1/(n(n + 1)(n + 2)(n + 3)) | = | 1/18 sum_(n = 1)^∞ 1/(n(n + 1)...(n + p)) | = | 1/(p·p!).

Euler transform | lozenge algorithm | Wilf-Zeilberger pair | Wynn's epsilon method