Subseries and signed series

Abstract

For any positive decreasing to zero sequence a_n such that Ʃa_n diverges we consider the related series Ʃk_na_n and Ʃj_na_n. Here, k_n and j_n are real sequences such that Ʃk_nє{0,1} and j_nє{-1,1}. We study their convergence and characterize it in terms of the density of 1's in the sequences k_n and j_n. We extend our results to series m_na_n, with Ʃm_nє{-1,0,1} and apply them to study some associated random series.