On the c0-equivalence and permutations of series

Assume 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.