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Let(   p  n   ){(p_{n})}    and(   q  n   ){(q_{n})}    be any two non-negative real sequences, withR  n   :=    ∑   k  =  0   np  k   ⁢   q   n  -  k≠  0{R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0}    (n  ∈  ℕ{n\in\mathbb{N}}   ). Let∑   k  =  0   ∞    a  k{\sum_{k=0}^{\infty}a_{k}}    be a series of real or complex numbers with partial sums(   s  n   ){(s_{n})}   , and sett  n   p  ,  q    :=    1   R  n    ⁢    ∑   k  =  0   np  k   ⁢   q   n  -  k    ⁢   s  k{t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}}    forn  ∈  ℕ{n\in\mathbb{N}}   . In this paper, we present the necessary and sufficient conditions under which the existence of the limitlim   n  →  ∞    ⁡   s  n    =  L{\lim_{n\to\infty}{s_{n}}=L}    follows from that oflim   n  →  ∞    ⁡   t  n   p  ,  q=  L{\lim_{n\to\infty}t_{n}^{p,q}=L}   . These conditions are one-sided or two-sided if(   s  n   ){(s_{n})}    is a sequence of real or complex numbers, respectively.

A Tauberian theorem for the generalized Nörlund summability method