Let (1) \(\sum^{\infty}_{n=1}a_ n\) be an infinite series with partial sums \(s_ n\) \((n=0,1,...)\). Denote by \(u^{\alpha}_ n\) the nth Cesàro means of order \(\alpha >-1\) of the sequence \(\{s_ n\}\). The series (1) is said to be \(| C,\alpha;\gamma |_ k\) summable (k\(\geq 1\), \(\gamma\geq 0)\) if \(\sum^{\infty}_{n=1}n^{\gamma k+k- 1}| u^{\alpha}_ k-u^{\alpha}_{k-1}|^ k<+\infty.\) Let \(p_ k\) be a sequence of positive real numbers with \(P_ n=\sum^{n}_{k=1}p_ k\to +\infty\) (n\(\to \infty)\). Put \(T_ n=(1/P_ n)\sum^{n}_{k=0}p_ ks_ k\) \((n=0,1,...)\). The series (1) is said to be \(| \bar N,p_ n;\gamma |_ k\) summable (k\(\geq 1\), \(\gamma\geq 0)\) if \[ \sum^{\infty}_{n=1}(P_ n/p_ n)^{\gamma k+k-1}| T_ n-T_{n-1}|^ k<+\infty. \] The author shows that if \(1-\gamma k>0\) then from the \(| \bar N,p_ n;\gamma |_ k\) summability of (1) the \(| C,1,\gamma |_ k\) summability of this series follows.