Geodesic
On rearrangements of infinite series
Glasgow mathematical journal, Tome 2 (1958) no. 4, pp. 182-193

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Robertson, A. P. On rearrangements of infinite series. Glasgow mathematical journal, Tome 2 (1958) no. 4, pp. 182-193. doi: 10.1017/S2040618500033694
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[1] 1.Agnew, R. P., Properties of generalised definitions of limit, Bull. Amer. Math. Soc., 45 (1939), 689–730. Google Scholar | DOI

[2] 2.Agnew, R. P., Permutations preserving convergence of series, Proa. Amer. Math. Soc., 6 (1955), 563–564. Google Scholar | DOI

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[7] 7.Rado, R., The distributive law for products of infinite series, Quart. J. of Math. (Oxford), 11 (1940), 229–242. Google Scholar

[8] 8.Rogers, C. A., Linear transformations which apply to all convergent sequences and series, J. London Math. Soc., 21 (1946), 123–128, 182–185. Google Scholar | DOI

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