Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-26T02:52:47.870Z Has data issue: false hasContentIssue false
  • English
  • Français

On interpreting the sums of asymptotic series of positive terms

Published online by Cambridge University Press:  17 February 2009

J. E. Drummond
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Four different kinds of positive asymptotic series are identified by the limiting ratio of successive terms. When the limiting ratio is 1 the series is unsummable. When the ratio tends rapidly to a constant, whether greater or less than 1, the series is easily summed. When the ratio tends slowly to a constant not equal to 1 the series is compared with a binomial model which is then used to speed the convergence. When the ratio increases linearly, a limiting binomial and an exponential integral model are both used to speed convergence. The two resulting model sums are consistent and in this case are complex numbers. Truncation at the smallest term is found to be unreliable in the second case, invalid in the third case, and the exponential integral is used to produce a significantly improved truncation in the third case. A divergent series from quantum mechanics is also examined.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 1 , July 1982 , pp. 28 - 39
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions (N.B.S. Appl. Math. Series 55, Washington, D. C., 1964).Google Scholar
[2]Aitken, A. C., “On Bernoulli's numerical solution of algebraic equations”, Proc. Roy. Soc. Edinburgh 46 (1925–1926), 289–305.CrossRefGoogle Scholar
[3]Baker, G. A., Essentials of Padé approximants (Academic Press, New York, 1975).Google Scholar
[4]Drummond, J. E., “The anharmonic oscillator; perturbation series for cubic and quartic energy distortion”, J. Phys. A14 (1981), 1651–1661.Google Scholar
[5]Hehenberger, M., McIntosh, H. V. and Brändas, E., “Weyl's theory applied to the Stark effect in the hydrogen atom”, Phys. Rev. A10 (1974), 1494–1506.CrossRefGoogle Scholar
[6]Padé, H., “Sur la représentation approchée d'une fonction par des fractiones rationelles”, Ann. Ecole Norm. Superior Suppl. [3] 9 (1892), 1–93.Google Scholar
[7]Shanks, D., “Non-linear transformations of divergent and convergent sequences”, J. Math. Phys: 34 (1955), 1–42.CrossRefGoogle Scholar
[8]Silverstone, H. J., “Perturbation theory of the Stark effect in hydrogen to arbitrarily high order”, Phys. Rev. A18 (1978), 1853–1864.Google Scholar
[9]Smith, D. A. and Ford, W. F., “Acceleration of linear and logarithmic convergence”, SIAM J. Numer. Anal. 16 (1979), 223–240.CrossRefGoogle Scholar
[10]Broeck, J. Vanden and Schwartz, L. W., “A one parameter family of sequence transformations”, SIAM J. Math. Anal. 10 (1979), 658–666.Google Scholar