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On reducing to zero the remainder of an asymptotic series

Part of: Approximations and expansions

Published online by Cambridge University Press:  09 April 2009

J. E. Drummond
J. E. Drummond
Affiliation:
Department of Applied Mathematics Australian National University Canberra Australia
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Abstract

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If a weighted Euler transformation is applied to the asymptotic series for ezE1(z) the remainder can be expressed as an integral. Examination of this integral shows that for a transformation of given order the smallest term of the resulting series remains at approximately a constant distance from the start of the series. If, however, there is no restriction on the order of transformation the remainder may be decreased to zero by increasing the number of terms used, but if z is close to the negative real axis the rate of decrease is small. A more general theorem for alternating real series and Taylor's series is also given.

MSC classification

Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 3 , November 1978 , pp. 309 - 316
Copyright
Copyright © Australian Mathematical Society 1978

References

Abramowitz, M. and Stegun, I. A. (1964), Handbook of Mathematical Functions (National Bureau of Standards, Appl. Math. Series), 55.Google Scholar
Drummond, J. E. (1972), “A formula for accelerating the convergence of a general series, Bull. Austral. Math. Soc. 6, 69–74.CrossRefGoogle Scholar
Shanks, D. (1955), “Non-linear transformations of divergent and convergent sequences”, J. Math. Phys. 34, 1–42.CrossRefGoogle Scholar
Sirovich, L. (1971), Techniques of Asymptotic Analysis, App. Math. Sc. 2 (Springer-Verlag).CrossRefGoogle Scholar