Let \(P_n=\sum_{y=0}^np_y\) as being required to define the standard power series method, where \(s=(s_k)\) is transformed into \(t_s (x)=\sum_{k=0}^\infty p_ks_kx^k/p(x)\), \(p(x)=\sum_{k=0}^\infty p_kx^k\) \((0<x<1)\). K. Ishiguro [Proc. Japan Acad. 40, 807–812 (1964; Zbl 0125.30903)] proved (*) \(t_s(x)\to \sigma\), \(x\uparrow 1\), to infer \(s_n\to\sigma\) under the Tauberian condition \(s_n-s_{n-1}=o (p_n/P_n)\), assumptions provided that include (**) \(P_n=O(p (x_n))\), where \(x_n=1-1/n\). In line with this result and employing the same Tauberian condition, discrete power series methods have been considered by B. Watson [Analysis, München 22, No. 4, 361–365 (2002; Zbl 1028.40003)] and the present authors. As concernes Watson’s work, some \(x_n=1-1/ \lambda_n \uparrow 1\) took the place of \(x\) in (*) and of the \(x_n\) in (**). The paper under review displays \(x\) in (*), (**) as being replaced by appropriate functions of \(x_n=1-1/\lambda_n\). [Improper notation such as “\(\lim f_s(z_n) =s\)”; “\(f\) and \(h\) are any given functions”; “\(f_s\)”\((\equiv t_s)\) vs. “\(f_n\)” \((\equiv P_n)\).]