A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability

Quite recently, Savaş 1 obtained sufficient conditions for ∑ anλn to be summable |A, δ|k, k ≥ 1, 0 ≤ δ < 1/k. The purpose of this paper is to obtain the corresponding result for quasif-increasing sequence. Our result includes and moderates the conditions of his theorem with the special case μ 0. A sequence {λn} is said to be of bounded variation bv if ∑ n |Δλn| < ∞. Let bv0 bv ∩ c0, where c0 denotes the set of all null sequences. The concept of absolute summability of order k ≥ 1 was defined by Flett 2 as follows. Let ∑ an denote a series with partial sums {sn}, andA a lower triangular matrix. Then ∑ an is said to be absolutelyA-summable of order k ≥ 1,written thatan is summable |A|k, k ≥ 1, if