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Let p; q be given positive numbers and a; b non-negative ones. In this paper we study and characterize the class S(a; b; p; q) of all admissible ideals I ⊄ 2N with the following property... for all sequences (an) of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic. [ABSTRACT FROM AUTHOR]

Let I ⊆ 2N be an admissible ideal, we say that a sequence (xn) of real numbers I-converges to a number L, and write I - lim xn = L, if for each ε > 0 the set Aε = {n: |xn - L| ≥ n} 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: ... where 0 < α ≤ 1 ≤ β ≤ 1/α are real numbers and (an), (bn) are sequences of positive real numbers. We characterize T(α, β, an, bn) the class of all such admissible ideals I. This accomplishment generalized and extended results from the papers [4, 7, 12, 16], where it is referred that the monotonicity condition of the summand sequence in so-called Olivier's Theorem (see [13]) can be dropped if the convergence of the sequence (nan) is weakend. In this paper we will study I-convergence mainly in the case when I stands for I, Ic(q), I≤q, respectively. [ABSTRACT FROM AUTHOR]