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A remarkable formula for approximating the sum of alternating series

Published online by Cambridge University Press:  01 August 2016

Thomas J. Osler
Thomas J. Osler*
Affiliation:
Mathematics Department, Rowan University, Glassboro, NJ 08028, USAe-mail: osler@rowan.edu
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Extract

In this paper we present a simple formula that can often be used to assist in the approximation of the sum of an alternating series

The ideas presented here were motivated by translating Euler's papers [1] and [2]. Suppose we begin by summing a terms of this series exactly, then the approximation that we will obtain is given by

Type
Articles
Information
The Mathematical Gazette , Volume 93 , Issue 526 , March 2009 , pp. 76 - 82
Copyright
Copyright © The Mathematical Association 2009

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References

1. Euler, L., Methodus universalis serierum convergentium summas quam proxime inveniendi, (A general method for finding approximations to the sums of convergent series (E46)), originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741 pp. 3–9. Opera Omnia: Series 1, 14, pp. 101–107. A Translation with notes byJacob, W. and Osier, T. J., as well as the original paper are available on the web at the Euler Archive, http://www.math.dartmouth.edu/~euler/ Google Scholar
2. Euler, L., Remarques sur un beau rapport entre les series des puissances tant directes que reciproques (Remarks on a beautiful relation between direct as well as reciprocal power series (E254)), Originally published in Memoires de l’academie des sciences de Berlin 17, (1768) pp. 83–106. Opera Omnia: Series 1, 15, pp. 70–90. A Translation with notes by Willis, L. and Osier, T. J. is available on the web at the Euler Archive, http://www.math.dartmouth.edu/~euler/ Google Scholar
3. Knopp, Konrad, Theory and application of infinite series, Dover Publications, New York, 1990. (A translation by Young, R. C. H. of the 4th German edition of 1947.)Google Scholar