An Alternative Proof of a Tauberian Theorem for the Weighted Mean Method of Summability

Let (pn) be a sequence of nonnegative numbers such that p(0) > 0 and We give an alternative proof of a Tauberian theorem stating that the existence of the limit is a sequence of complex numbers, these Tauberian conditions are two-sided. Significance Statement: If a sequence converges, then its weighted means converge to the same number. But, the converse of this implication is not true in general and its partial converse might be valid. This manuscript presents an alternative proof of a well-known Tauberian theorem stating that convergence of a slowly decreasing sequence (in case of sequences of real numbers) or a slowly oscillating sequence (in case of sequences of complex numbers) follows from its weighted mean summability. Corollaries of the main results in this manuscript consist of well-known Tauberian theorems for Cesaro and logarithmic summability methods.