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Let α≥0 and β>— 1. The main result gives necessary and sufficient conditions for the sequence (εn) in order that the sequence (εnUn) will be absolutely summable by the Cesàro method Cβ for each sequence (Un) which is bounded or summable by the method Cα
Another theorem is proven when Cα and Cβ are replaced by triangular methods A = (ank) and B=(bnk) satisfying 
, where (ξnk) = (ank)-1.
