Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-02T15:48:38.321Z Has data issue: false hasContentIssue false
  • English
  • Français

On rearrangements of infinite series

Published online by Cambridge University Press:  18 May 2009

A. P. Robertson
A. P. Robertson
Affiliation:
The Univebsity Glasgow
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.

(i) What is the condition on a given series for every rearrangement to converge?

(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?

The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 3 , Issue 4 , July 1958 , pp. 182 - 193
Copyright
Copyright © Glasgow Mathematical Journal Trust 1958

References

REFERENCES

1.Agnew, R. P., Properties of generalised definitions of limit, Bull. Amer. Math. Soc., 45 (1939), 689–730.CrossRefGoogle Scholar
2.Agnew, R. P., Permutations preserving convergence of series, Proa. Amer. Math. Soc., 6 (1955), 563–564.CrossRefGoogle Scholar
3.Banach, S., Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
4.Bourbaki, N., Topologie geńirale, 2me. ed. (Paris, 1951).Google Scholar
5.Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
6.Levi, F. W., Rearrangement of convergent series, Duke Math. J., 13 (1946), 579–585.CrossRefGoogle Scholar
7.Rado, R., The distributive law for products of infinite series, Quart. J. of Math. (Oxford), 11 (1940), 229–242.Google Scholar
8.Rogers, C. A., Linear transformations which apply to all convergent sequences and series, J. London Math. Soc., 21 (1946), 123–128, 182–185.CrossRefGoogle Scholar