An equivalent reformulation of summability by weighted mean methods (Q1375092)

A theorem of G. H. Hardy (1921) showed that a series \(\sum^\infty_0a_k\) is Cesàro (\(C,1)\) summable to a finite number \(L\) if and only if the series \(\sum^\infty_0b_n\), where \(b_n:= \sum^\infty_{k=n} a_k/(k+1)\), converges to the same value \(L\). It is the object here to give a similar theorem in which \((C,1)\)-summability is replaced by a general weighted mean, though the method of proof is quite different from Hardy's. Thus given a fixed sequence \(p:= (p_k)\) of positive numbers with \(P_n: =p_0+ \cdots +p_n\) \((n=0,1, \dots)\), the \((M,p)\) [or \((\overline N,p)]\) transform of a series \(\sum^\infty_0 a_k\), with partial sums \((s_n)\), is defined by \(\sigma_n: =P^{-1}_n \sum^n_0 p_ks_k\) \((n=0,1, \dots)\), with the related series \(\sum^\infty_0 b_n\) defined by \(b_n: =p_n \sum^\infty_{k=n} a_k/P_k\) \((n=0,1,\dots)\). It is then shown that if \(P_n\to \infty\), \(p_n/P_n \to 0\), and if certain summations involving \((p_n)\) and \((P_n)\) are bounded then \(\sum a_k\) is \((M,p)\) summable to a finite number \(L\) if and only if \(\sum b_n\) converges to \(L\).