## An Asymptotic Method in the Theory of Series |

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accumulation point analogous arguments asymptotic formula attained Banach space behaviour best possible constant breakdown Carleman's inequality compact set completes the proof conclude this Section consider continuous function convergent series convex function defined denote derive differential equation discussion due to Copson eigenvalue eindige equivalent exists a number fact finite sections fixed following lemma give Hardy's inequality holda holds implies increasing function induction inequality due interval iteration problem iteration process Lagrange multiplier latter lim inf log log log NX maximum mean value theorem Moreover N.G. de Bruijn n+i n n n+n n n=i n notation parameter process breaks proof of lemma proof of theorem properties recurrence relation Remark replaced result rf((log satisfies sequence solutions of A.1 strict inequality substitutions theorem 1.2 whereas wordt write Xqnq zero zn(X zn+1