On the $c_0$-equivalence and permutations of series

Abstract

Assume that a convergent series of real numbers $\sum _{n=1}^{\infty }{a}_n$ has the property that there exists a set $A\subseteq \mathbb{N}$ such that the series $\sum _{n\in A}{a}_n$ is conditionally convergent. We prove that for a given arbitrary sequence $(b_n)$ of real numbers there exists a permutation $\sigma :\mathbb{N}\to \mathbb{N}$ 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 $\sum _{n=1}^{\infty }{a}_{\sigma (n)}$. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.