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Summability of alternating gap series

Published online by Cambridge University Press:  20 January 2009

J. P. Keating  and
J. B. Reade
J. P. Keating
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK and BRIMS, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK
J. B. Reade
Affiliation:
Department of Mathematics, The University of Manchester, Manchester M13 9PL, UK
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Abstract

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The Abel and Cesàro summabilities of two alternating gap series are investigated. We prove that the series is summable at x = 1 (in both senses), but that is not. In 1907, Hardy obtained essentially the same result for the latter series; our proof is shorter and more elementary: we use the Poisson summation formula to derive an explicit estimate for the size of the oscillations as x → 1_. This represents an example of a general method for determining the Abel summability of similar series.

Keywords

summabilityPoisson summationgap series
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 95 - 101
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

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