On the $c_0$-equivalence and permutations of series

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$hasthepropertythatthereexistsaset$A\subseteq \N$ 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 $\sigma \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$ 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 ${\sum }_{n=1}^{\mathrm{\infty }}{a}_{\sigma (n)}$. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.