A simple approach to the summation of certain slowly convergent series
Author:
Stanisław Lewanowicz
Journal:
Math. Comp. 63 (1994), 741-745
MSC:
Primary 65B10
DOI:
https://doi.org/10.1090/S0025-5718-1994-1250774-0
MathSciNet review:
1250774
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Abstract | References | Similar Articles | Additional Information
Abstract: Summation of series of the form $\sum \nolimits _{k = 1}^\infty {k^{\nu - 1}}r(k)$ is considered, where $0 \leq \nu \leq 1$ and r is a rational function. By an application of the Euler-Maclaurin summation formula, the problem is reduced to the evaluation of Gauss’ hypergeometric function. Examples are given.
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
- Philip J. Davis, Spirals: from Theodorus to chaos, A K Peters, Ltd., Wellesley, MA, 1993. With contributions by Walter Gautschi and Arieh Iserles. MR 1224447
- Walter Gautschi, A class of slowly convergent series and their summation by Gaussian quadrature, Math. Comp. 57 (1991), no. 195, 309–324. MR 1079017, DOI https://doi.org/10.1090/S0025-5718-1991-1079017-5 I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, New York, 1980.
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Additional Information
Keywords:
Slowly convergent series,
Euler-Maclaurin formula,
Gauss’ hypergeometric function
Article copyright:
© Copyright 1994
American Mathematical Society
