Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-15T20:23:27.317Z Has data issue: false hasContentIssue false
  • English
  • Français

A formula for summing divergent series

Published online by Cambridge University Press:  17 February 2009

J. E. Drummond
J. E. Drummond
Affiliation:
Department of Applied Mathematics, School of General Studies, Australian National University, Canberra A.C.T., 2600, Australia.
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.

A formula is given for assigning sums to divergent series where the ratio of adjacent terms varies slowly along the series. This formula consistş of a weighted average of partial sums and is shown to be a general formula which can be easily calculated using a simple recurrence relation. It appears to be more powerful than a repeated Aitken or Shanks e1 process as long as the transformed series remains divergent and it is also compared with the Padé approximants. It is demonstrated on a factorial series, on a nearly geometric divergent series and for the extrapolation of a velocity formula for small amplitudes of motion.

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 2 , December 1975 , pp. 200 - 214
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Aitken, A. C., ‘On Bernoulli's numerical solution of algebraic equations’, Proc. Roy. Soc. Edinburgh 46 (1925–1926),289–305.CrossRefGoogle Scholar
[2]Drummond, J. E., ‘A formula for accelerating the convergence of a general series’, Bull. Austral. Math. Soc. 6 (1972),69–74.CrossRefGoogle Scholar
[3]Lubkin, S., ‘A method of summing infinite seeies’, J. Res. Nat. Bur. Standards 48 (1952),228–254.CrossRefGoogle Scholar
[4]Padé, H., Thesis, Ann. de l'Ecol. Nor. Sup. (3), 9 (1892).CrossRefGoogle Scholar
[5]Shanks, D., ‘Non-linear transformations of divergent and slowly convergent sequences’. J. Maths. and Phys. 34 (1955),1–42.CrossRefGoogle Scholar
[6]Wall, H. S., Analytical Theory of Continued Fractions (Van Nostrand, Princeton, N.J., 1948).Google Scholar
[7]Wynn, P., ‘On a device for computing the em(Sn) transformation’, Math. Tables Aids Comput. 10 (1956),91–96.CrossRefGoogle Scholar