\textit{H. Bor} and \textit{H. Seyhan} [Indian J. Pure Appl. Math. 30, No. 10, 1041-1046 (1999; Zbl 0942.40001)] proved a theorem on \(|\overline N,p_n, \delta|_k\), \(k\geq 1\) and \(0\leq \delta<1/k\) summability factors of infinite series. In this paper the authors have improved the previous results under suitable conditions by using an almost increasing sequence. A sequence \(\{b_n\}\) is said to be almost increasing if there exists a positive increasing sequence \(\{c_n\}\) and two positive constants \(A\) and \(B\) such that \(Ac_n\leq b_n\leq Bc_n\). The present result for \(|\overline N,p_n, \delta|\) holds for \(|c,1,\delta |_k\) and \(|\overline N,1/n+1, \delta|_k\).