On convergence sets of conditionally convergent series

Abstract

For a given convergent series we consider the set of permutations of ℤ+ which leave the series convergent. This is called the convergence set of the series. We characterize these sets, describe the relationship of convergence sets of different series, and investigate the possibility of reconstructing the series given its convergence set. In particular, we give a surprising extension of a result of Agnew, [2]: we show that the permutations preserving the summability of all conditionally convergent series also preserve their sums, see Theorem 3.2. We also prove in Theorem 3.7 that the convergence sets of two conditionally convergent series are either equal or not comparable when ordered by set inclusion.