On the set of limit points of conditionally convergent series
Volume 237 / 2017
Abstract
Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $\tau $ be a permutation of the natural numbers. We study the set $\operatorname {LIM}(\sum_{n=1}^\infty x_{\tau (n)})$ of all limit points of the sequence $(\sum_{n=1}^p x_{\tau (n)})_{p=1}^\infty $ of partial sums of the rearranged series $\sum_{n=1}^\infty x_{\tau (n)}$. We give a full characterization of such limit sets in finite-dimensional spaces. Namely, every such limit set in $\mathbb R^m$ is either compact and connected, or closed with all connected components unbounded. On the other hand, each set of one of these types is the limit set of some rearranged conditionally convergent series. Moreover, this characterization does not hold in infinite-dimensional spaces.
We show that if $\sum_{n=1}^\infty x_n$ has the Rearrangement Property and $A$ is a closed subset of the closure of the sum range of $\sum_{n=1}^\infty x_n$ and it is $\varepsilon $-chainable for every $\varepsilon \gt 0$, then there is a permutation $\tau $ such that $A=\operatorname {LIM}(\sum_{n=1}^\infty x_{\tau (n)})$.
