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A Regular Summability Method which Sums the Geometric Series to its Proper Value in the Whole Complex Plane

Published online by Cambridge University Press:  20 November 2018

Ludwig Tomm
Ludwig Tomm*
Affiliation:
Univ. of Western Ontario, London, Ontario
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Abstract

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In this paper an explicit regular sequence-to-sequence summability method is presented which sums the geometric series to the value 1/(1-z) in all of ℂ\{1} and to infinity at the point 1. The method also provides compact convergence in ℂ \ [ 1, ∞) and therefore improves well-known results by Le Roy, Lindelöf and Mittag-Leffler.

Keywords

40G99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 2 , 01 June 1983 , pp. 179 - 188
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Roy, Le, Annales de la Faculté des sciences de Toulouse (2), 2 (1900), 317–430 (323).Google Scholar
2. Lindelöf, , Journal de mathématiques (5), 9 (1903), 213–21.Google Scholar
3. Mittag-Leffler, , Atti del IV congressi Internaz. i. 67–85; Rome 1908.Google Scholar
4. Tomm, L., Über die Summierbarkeit der geometrischen Reihe mit regulären Verfahren; author's dissertation; Ulm, West Germany.Google Scholar