When given an infinite series Pan with the partial sums (sn) and a normal matrix A = (anv), i. e. a lower triangular matrix of non-zero diagonal entries, A defines the sequence-to-sequence transformation, mapping the sequence s = (sn) to As = (An(s)), In this case, by | A |k summability of this infinite series we mean the convergence of the series Pn k?1 | ?An(s) | k , by Tanovic-Miller in [5], where k ? 1 and ?An(s) = An(s) ? An?1(s). Let (pn) be a sequence of positive numbers such that Pn = Pn v=0 pv ? ? as n ? ?, (P?i = p?i = 0, i ? 1). Sulaiman in [4] defined | A, pn |k summability of the series. Specifically, when anv = pv Pn , | A, pn |k summability is equivalent to | N , pn |k summability which was introduced by Bor in [1]. Bor in [2] proved the sufficient conditions for | N , pn |k summability of the series Pan?n, later Mazhar in [3] also proved under weaker conditions by using an almost increasing sequence. The object of this paper is to show that these two results can be generalized to a wide class of summability methods.