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Increasing the convergence rate of series. (English) Zbl 0633.65002

A method of accelerating the speed of convergence of a given series is presented and demonstrated by several examples. The main assumption is the existence of a specific asymptotic expansion of the terms in the series. In the examples the method works quite well. The paper does not contain proofs or theoretical backgrounds. The list of references only consists of papers of the author, except for the Handbook of mathematical functions (1967; Zbl 0171.385) by M. Abramowitz and I. A. Stegun. Other publications of similar content and close connection to the topic of the paper are not found under the references, and are not referred to in the text, e.g. J. Wimp [SIAM J. Numer. Anal. 14, 194-204 (1977; Zbl 0355.65003)], A. H. Stroud [Numerical quadrature and solution of ordinary differential equations (1974; Zbl 0298.65018), in particular p. 145-162], C. Brezinski [Numer. Math. 35, 175-187 (1980; Zbl 0444.65001)], the reviewer [Proc. 5th South Afr. Symp. Numerical mathematics, Durban 1979, 1-20 (1979; Zbl 0435.65100)], the reviewer and G. Merz [Praktische Mathematik. II. (1982; Zbl 0476.65001), in particular chapter 12].
Reviewer: G.Meinardus

MSC:

65B10 Numerical summation of series
65B05 Extrapolation to the limit, deferred corrections
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References:

[1] Abramowitz, M.; Stegun, I. A., Mathematical Functions with Formulas, Graphs and Mathematical Tables (1968), Dover: Dover New York · Zbl 0515.33001
[2] Longman, I. M., The summation of power series and Fourier series, J. Comput. Appl. Math., 12&13, 447-457 (1985) · Zbl 0574.65003
[3] Longman, I. M., The summation of series, Appl. Numer. Math., 2, 135-141 (1986) · Zbl 0603.65002
[4] I.M. Longman, The summation of Fourier, Chebyshev and Legendre series, Appl. Math. & Comp.; I.M. Longman, The summation of Fourier, Chebyshev and Legendre series, Appl. Math. & Comp. · Zbl 0638.40001
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