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Some relations connecting measures of convergence for series with non-increasing terms

Part of: Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  26 February 2010

R. W. K. Odoni
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Exeter, North Park Road, Exeter. EX4 4QE
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Let an be a non-increasing real sequence such that converges; then clearly an ↓ 0. We shall ignore the trivial case where an = 0 for all large n, and so we assume that an > 0 for all n, from now onwards. In [1] J. B. Wilker introduced certain new sequences associated with the rate of convergence of , and obtained various relations between them, in order to investigate packing problems in convex geometry. Let us define

We write P, Q and T respectively for the inferior limits of pn, qn and tn, and P1, Q1 and T1 for the corresponding superior limits. Further, we put

It is immediately clear from these definitions and our assumptions about an that

the latter since nan → 0 by Olivier's theorem [2, p. 124].

MSC classification

Secondary: 40A05: Convergence and divergence of series and sequences
Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 17 - 24
Copyright
Copyright © University College London 1984

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References

1.Wilker, J. B.. Sizing up a solid packing. Periodica Math. Hungarica, 8 (1977), 117–134.CrossRefGoogle Scholar
2.Knopp, K.. Theory and application of Infinite Series, 2nd edition (Blackie, London & Glasgow, 1951).Google Scholar
3.Bromwich, T. J.. Introduction to the theory of infinite series (Macmillan, London, 1926).Google Scholar
4.Szász, O.. Quasimonotone series. Amer. J. Math., 70 (1948), 203–206.CrossRefGoogle Scholar
5.Hardy, G. H. and Littlewood, J. E.. Contributions to the arithmetic theory of series. Proc. London Math. Soc. (2), 11 (1913), 411–478.CrossRefGoogle Scholar