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Recycling Stirling's series acceleration technique

Published online by Cambridge University Press:  03 February 2017

Paul Levrie  and
Amrik Singh Nimbran
Paul Levrie
Affiliation:
Faculty of Applied Engineering, University of Antwerp, B-2020 Antwerpen, Belgium Department of Computer Science, KU Leuven, P.O. Box 2402, B-3001 Heverlee, Belgium e-mail: paul.levrie@cs.kuleuven.be
Amrik Singh Nimbran
Affiliation:
B3-304, Palm Grove Heights, Ardee City, Gurgaon, Haryana, India e-mail: amrikn622@gmail.com
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Extract

Infinite series are an important topic in mathematical analysis and convergence is the most crucial concept in the theory of infinite series. The speed at which the sequence of the partial sums of a series approaches its limiting sum has been a subject of investigation for many a mathematician. Euler [1], Kummer [2] and Markoff [3] all developed techniques for accelerating the convergence of slowly converging series. Euler's method only works for alternating series, Kummer's and Markoff's are suitable for general series with a special form.

Gosper [4] illustrates how the rate of convergence of infinite series can be accelerated by a suitable splitting of each term into two parts and then combining the second part of the n th term with the first part of n + 1 th the term and leaving the first part of the first term. Repeated application of this process yields a new series which approaches 0 and the series of the left out first parts (‘orphans’) that converges faster than the original series.

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 550 , March 2017 , pp. 69 - 82
Copyright
Copyright © Mathematical Association 2017 

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