On the c_0 -equivalence and permutations of series

ume that a convergent series of real numbers ∑ _n=1^∞ a_n has the property that there exists a set A⊆ℕ such that the series ∑ _n ∈ A a_n is conditionally convergent. We prove that for a given arbitrary sequence (b_n) of real numbers there exists a permutation σ :ℕ→ℕ such that σ (n) = n for every n ∉ A and (b_n) is c_0 -equivalent to a subsequence of the sequence of partial sums of the series ∑ _n=1^∞ a_σ (n) . Moreover, we discuss a connection between our main result with the classical Riemann series theorem.