On Almost Increasing Sequences for Generalized Absolute Summability

Abstract

In this paper, we establish a summability factor theorem for summability |A, δ|k as defined in (2) where A is a lower triangular matrix with non-negative entries satisfying certain conditions. This paper is an extension of the main result of [3] using definition (2) below. Let A be a lower triangular matrix, {sn} a sequence. Then An := ∑v=0nanvsv. A series ∑a n is said to be summable |A|k, k ≥ 1 if ∑n=1∞ nk-1|An - A n-1|k < ∞. (1) and it is said to be summable |A, δ|k, k ≥ 1 and δ ≥ 0 if (see,[1]) ∑n=1∞ nδk+k-1|An - An-1|k < ∞. © ELEMENT.