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On the approach of a series to its Cesàro limit

Published online by Cambridge University Press:  20 January 2009

J. M. Hyslop
J. M. Hyslop
Affiliation:
The University, Glasgow
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The object of this paper is to investigate some properties of series which satisfy conditions of the form

where 0 < ρ ≦ p. denotes, as usual, the n-th Cesàro sum of order p for the series ∑an and the binomial coefficient . It is convenient to state here some properties of and to which we must constantly refer in the sequel.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 4 , September 1938 , pp. 182 - 201
Copyright
Copyright © Edinburgh Mathematical Society 1938

References

REFERENCES

1.Andersen, A. F., Studier over Cesàro's Summabilitetsmetode (Copenhagen, 1921).Google Scholar
2.Bohr, H., Bidrag til de Dirichlet'ske Raekkers Theori (Copenhagen, 1910).Google Scholar
3.Hardy, G. H., Researches in the Theory of Divergent Series and Divergent Integrals, Quart. J. of Math., 35 (1904), 22–66.Google Scholar
4.Hardy, G. H., On certain Oscillating Series, Quart. J. of Math., 38 (1907), 269–288.Google Scholar
5.Hardy, G. H., Generalization of a Theorem in the Theory of Divergent Series, Proc. London Math. Soc. (2), 6(1908), 255–264, and footnote (2) 8 (1909), 277–294.CrossRefGoogle Scholar
6.Hardy, G. H., Theorems relating to the Summability and Convergence of slowly oscillating Series, Proc. London Math. Soc. (2), 8 (1910), 301–320.CrossRefGoogle Scholar
7.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
8.Hyslop, J. M., The Generalization of a Theorem on Borel Summability, Proc. London Math. Soc. (2), 41 (1936), 243–256.CrossRefGoogle Scholar
9.Hyslop, J. M., On the Summability of Series by a Method of Valiron, Proc. Edinburgh Math. Soc. (2), 4 (1936), 218–223.CrossRefGoogle Scholar
10.Valiron, G., Remarques sur la Sommation des Séries Divergentes par les Méthodes de M. Borel, Rendiconti di Palermo, 42 (1917), 267–284.CrossRefGoogle Scholar