Convergence of positive series and ideal convergence

Let ℐ ⊆ 2 N be an admissible ideal, we say that a sequence ( 𝑥 𝑛 ) of real numbers ℐ− converges to a number 𝐿 , and write ℐ − lim 𝑥 𝑛 = 𝐿 , if for each 𝜀 > 0 the set 𝐴 𝜀 = { 𝑛 : | 𝑥 𝑛 − 𝐿 | ≥ 𝜀 } belongs to the ideal ℐ . In this paper we discuss the relation ship between convergence of positive series and the convergence properties of the summand sequence. Concretely, we study the ideals ℐ having the following property as well: