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Method of summation of some slowly convergent series
Authors:Pawe? Wo?ny
Affiliation:Institute of Computer Science, University of Wroc?aw, ul. Joliot-Curie 15, 50-383 Wroc?aw, Poland
Abstract:A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189-371] or the technique recently introduced by ?í?ek et al. [J. ?í?ek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962-968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.
Keywords:Convergence acceleration   Extrapolation   Levin-type method   Nonlinear sequence transformation   Iterative methods   Power series   Orthogonal series   Hypergeometric series   Basic hypergeometric series
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