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A. C. Aitken absolutely convergent an(x bound of f(x bounded function CHAPTER convergence the series convergent or properly convergent series cosh D.Sc defined definition denote double series E E amn E\an Eamn Ean and Ebn Ean is convergent Ebn converges Ebn is properly example finite limit following theorem follows at once follows from Theorem function f(x given Hence infinite product infinite series infinitely oscillating integer interval lim f(x log log lower bound lower limits monotonic decreasing function MULTIPLICATION OF SERIES non-negative terms obtain oscillate finitely positive integral values positive number properly divergent Prove Raabe's test radius of convergence real number repeated series result follows sequence series 00 series converges series E amn series Ean series Ean(x series of non-negative series whose terms set of values sinh Suppose tend to zero tends to infinity Theorem 18 Theorem 39 uniformly convergent upper and lower upper bound vergent