A new summation method

for power series

with rational coefficients

Authors:
Silvia Dassié, Marco Vianello and Renato Zanovello

Journal:
Math. Comp. **69** (2000), 749-756

MSC (1991):
Primary 65B10, 65D20; Secondary 33E20, 41A60

DOI:
https://doi.org/10.1090/S0025-5718-99-01113-8

Published electronically:
May 21, 1999

MathSciNet review:
1651749

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that an asymptotic summation method, recently proposed by the authors, can be conveniently applied to slowly convergent power series whose coefficients are rational functions of the summation index. Several numerical examples are presented.

**1.**P. Baratella and B. Gabutti,*Uniform approximation on unit disk of series arising in plate contact problems*, J. Comput. Appl. Math.**62**(1995), 181-196. MR**96h:65003****2.**S. Dassiè,*Convergence acceleration of power series by means of asymptotic summation and extrapolation*(Italian), Laurea Thesis, University of Padova, 1995 (advisors: M. Vianello and R. Zanovello).**3.**S. Dassiè, Marco Vianello and Renato Zanovello,*Asymptotic summation of power series*, Numer. Math.**80**(1998), 61-73. CMP**98:17****4.**W. Gautschi,*A class of slowly convergent series and their summation by Gaussian quadrature*, Math. Comp.**57**(1991), 309-324. MR**91j:65049****5.**W. Gautschi,*On certain slowly convergent series arising in plate contact problems*, Math. Comp.**57**(1991), 325-338. MR**91j:40002****6.**W. Gautschi,*Orthogonal polynomials: applications and computation*, Acta Numerica 1996, Cambridge University Press, Cambridge, 1996, pp. 45-119. CMP**98:14****7.**P. Henrici,*Applied and computational complex analysis*, Vol. 1, Wiley, New York, 1984. MR**51:8378****8.**S. Lewanowicz,*A simple approach to the summation of certain slowly convergent series*, Math. Comp.**63**(1994), 741-745. MR**95a:65010****9.**G.V. Milovanovi\'{c},*Summation of series and Gaussian quadratures*, Approximation and Computation (R.V.M. Zahar, ed.), ISNM vol.119, Birkhäuser, Basel, 1994, pp. 459-475. MR**96c:65041****10.**G.V. Milovanovi\'{c},*Summation of series and Gaussian quadratures. II*, Numer. Algorithms**10**(1995), 127-136. MR**96f:65031****11.**A. Sidi,*Analysis of convergence of the T-transformation for power series*, Math. Comp.**35**(1980), 833-850. MR**83d:41039****12.**S. Wolfram,*Mathematica: A system for doing mathematics by computer*, 2nd Edition, Addison-Wesley, Reading, 1991.

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Additional Information

**Silvia Dassié**

Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy

**Marco Vianello**

Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy

Email:
marcov@math.unipd.it

**Renato Zanovello**

Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy

Email:
zanov@math.unipd.it

DOI:
https://doi.org/10.1090/S0025-5718-99-01113-8

Keywords:
Slowly convergent power series,
rational coefficients,
asymptotic expansions,
asymptotic summation

Received by editor(s):
April 28, 1998

Published electronically:
May 21, 1999

Additional Notes:
Work supported, in part, by the Italian MURST, and by the GNIM-CNR

Article copyright:
© Copyright 2000
American Mathematical Society