  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.
