On rearrangements of infinite series

Abstract

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).