A summability factor theorem by using an almost increasing sequence
AuthorÖğdük, H. Nedret
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CitationÖğdük, H. N. (2009). A summability factor theorem by using an almost increasing sequence. Maltepe Üniversitesi. s. 180.
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 , 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  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 . Bor in  proved the sufficient conditions for | N , pn |k summability of the series Panλn, later Mazhar in  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.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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