On the $c_0$-equivalence and permutations of series

Abstract

Assume that a convergent series of real numbers ∞ n=1 an has the property that there exists a set A ⊆ N such that the series n∈A an is conditionally convergent. We prove that for a given arbitrary sequence (bn) of real numbers there exists a permutation σ : N → N such that σ(n) = n for every n / ∈ A and (bn) is c 0 -equivalent to a subsequence of the sequence of partial sums of the seriesMoreover, we discuss a connection between our main result with the classical Riemann series theorem.2010 Mathematics Subject Classification. 40A05, 40A35. Key words and phrases. c 0 -equivalence, center of distances, potentially convergent series, von Neumann's theorem, Riemann rearrangement theorem, hypernumber.