On the $c_0$-equivalence and permutations of series

TL;DR

This paper investigates how permutations of a convergent series can produce subsequences of partial sums that are $c_0$-equivalent to any given sequence, linking to the classical Riemann series theorem.

Contribution

It proves the existence of permutations that align subsequences of series partial sums with arbitrary sequences, extending classical results on series rearrangements.

Findings

Existence of permutations matching subsequences to arbitrary sequences

Connection established with Riemann series theorem

Results applicable to series with conditionally convergent subsequences

Abstract

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given arbitrary sequence $(b_n)$ of real numbers there exists a permutation $\sigma\colon \N \to \N$ such that $\sigma(n) = n$ for every $n \notin A$ and $(b_n)$ is $c_0$-equivalent to a subsequence of the sequence of partial sums of the series $\sum\limits_{n=1}^\infty a_{\sigma(n)}$. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.