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An Application of Abel's Lemma to Double Series1

Published online by Cambridge University Press:  20 January 2009

C. E. Walsh
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Let bmn be a positive function of m and n which decreases steadily with n, so that bmn ≥ bm, n+1 for all values of m and n. Assume also that |am1 + am+2 + … + bmn| < K for all values of m and n, K being finite. Denote bySmn the sum of the first n terms in the first m rows of the double series

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 1 , Issue 4 , July 1929 , pp. 193 - 196
Copyright
Copyright © Edinburgh Mathematical Society 1929

References

1 For various similar results, and for other applications of this Lemma to multiple series, together with extensions of it, see e.g. Hardy:—Proc. London. Math. Soc. 2, 1 (1903), 124–128; 2 (1904), 190; Proc. Cambridge Phil. Soc. 19 (1919), 86, etc. Also, Bromwich, Proc. London Math. Soc. 2, 6 (1907), 58–76 ; and papers and theorems by Hadamard, Ferrar, Proc. London Math. Soc., 29 (1929), and others.