Abstract
Mazhar (1971) gave the characterization for the series∑anϵnto be summable|N,pn|whenever∑anis summable|C,α|k,α≥0,k≥1. Here we prove two theorems, the first concerns the sufficient conditions and the second the necessary conditions satisfied by{ϵn}in order to have∑anϵnsummable|N¯,pn|kwhenever∑anis summable|C,α|k,k≥1.
Highlights
Let an be a given infinite series with partial sums sn
Let σnδ and rnδ denote the nth Cesàro mean of order δ (δ > −1) of the sequences {sn} and {nan}, respectively
Consider the inclusion map i : A → B defined by i(x) = x, i is continuous which follows as A and B are BK-spaces
Summary
Let an be a given infinite series with partial sums sn. Let σnδ and rnδ denote the nth Cesàro mean of order δ (δ > −1) of the sequences {sn} and {nan}, respectively. The series an is said to be summable |C, δ|k, k ≥ 1 if nk−1 σnδ − σnδ−1 k < ∞, (1) A series an is said to be summable |N, pn|k, k ≥ 1, (see [1]), if
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