Generalization of Some Absolute Summability Methods

Mazhar (1971) gave the characterization for the series ∑ an n to be summable |N,pn|whenever ∑ an is 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 { n } in order to have ∑ an n summable |N̄,pn|k whenever ∑ an is summable |C,α|k, k≥ 1. 2000 Mathematics Subject Classification. Primary 40G. Let ∑ an be a given infinite series with partial sums sn. Let σδ n and rδ n 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 ∞ ∑ n=1 nk−1 ∣∣σδ n −σδ n−1∣∣k <∞, (1)