We prove that the Weakly-Vanishing Mean Oscillation (W-VMO) property of a sequence or a series is a necessary and sufficient condition under which the convergence (C0) follows from the Abel summability (A0) to the same limit. Hence, this result shows the Tauberian converse, with the largest possible space of sequences, of the Abel (1826) theorem on power series for which (A0) ⇒(C0). The inversion of the Cesàro summability (C1) ⇒ (C0) is also addressed within the same unified setting and solved with the necessary and sufficient W-VMO Tauberian condition.