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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 (εn U n) 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.
