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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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References

1. Euler, L., Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (1755) Part II, chapter I, available at http://eulerarchive.maa.org/page/E212.html Google Scholar
2. Kummer, E., Eine neue Methode, die numerischen Summen langsam convergirender Reihen zu berechnen, J. Reine Angew. Math. 16 (1837) pp. 206–214.Google Scholar
3. Markoff, A., Mémoire sur la transformation des séries peu convergentes en séries très convergentes, Mémoires de L'Académie Impériale des Sciences de St.-Pétersbourg, VIIE Série. Tome XXXVII, No. 9 (1890).Google Scholar
4. Gosper, R., Acceleration of series, Massachusetts Institute of Technology (1974) available at http://dspace.mit.edu/bitstream/handle/1721.1/6088/AIM-304.pdf Google Scholar
5. Stirling, J., Methodus differentialis: sive tractatus de summatione et interpolatione serierum infinitarum, Londini: Typis Gul. Bowyer; impensis G. Strahan (1730).Google Scholar
6. Tweddle, I., James Stirling's Methodus Differentialis: an annotated translation of Stirling's text, Springer-Verlag (2003).Google Scholar
7. Romik, D., Some comments on Euler's series for , Math. Gaz. 86 (July 2002) pp. 281–285.CrossRefGoogle Scholar
8. Schumacher, R., Rapidly convergent summation formulas involving Stirling series, Cornell University Library (2016), available at http://arxiv.org/abs/1602.00336.Google Scholar
9. Nielsen, N., Die Gammafunktion, Chelsea, New York (1965).Google Scholar
10. Schur, I. and Knopp, K., Über die Herleitung der Gleichung , Arch. Math. Phys. 3, 27 (1918) pp. 174–176.Google Scholar
11. Catalan, E., Mémoire sur la transformation des séries, et sur quelques intégrales définies, Mémoires Couronnés et Mémoires des Savants Étrangers 33 (1867) pp. 1–50.Google Scholar
12. Knopp, K., Theory and application of infinite series, (2nd edn.), (4th reprint, 1954) pp. 266–267.Google Scholar
13. Euler, L., Démonstration de la somme de cette suite etc., Journ. lit. d'Allemagne, de Suisse et du Nord, 2, 1 (1743), pp. 115–127, available at: http://eulerarchive.maa.org/page/E212.html Google Scholar
14. Mohammed, M., Infinite families of accelerated series for some classical constants by the Markov-WZ Method, Discrete Math. Theor. Comput. Sci. 7 (2005) pp. 11–24.CrossRefGoogle Scholar
15. Kondratieva, M. and Sadov, S., Markov's transformation of series and the WZ-method, Adv. in Applied Math. 34 (2005), pp. 393–407, available at: http://arXiv.org/abs/math/0405592 CrossRefGoogle Scholar
16. Graham, R., Knuth, D. and Patashnik, O., Concrete mathematics, (2nd edn.), Addison-Wesley, Reading, MA (1994).Google Scholar