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 $(C_{0})$ follows from the Abel summability $(A_{0})$ 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 $(A_{0})\Rightarrow (C_{0})$ . The inversion of the Cesàro summability $(C_{1})\Rightarrow (C_{0})$ is also addressed within the same unified setting and solved with the necessary and sufficient W-VMO Tauberian condition.
