The classical Adomian decomposition method frequently used to solve linear and nonlinear algebraic or integro-differential equations of ordinary and partial type is revisited. Rewriting the technique in an elegant form, a parameter so-called as the convergence control parameter, is embedded into the method to control the convergence and the rate of convergence of the method. Besides the constant level curves for determining suitable values, an effective approach for obtaining the best possible convergence control parameter is later devised based on the squared residual error of the studied problem. The optimum Adomian decomposition method is proved to converge to the true solution where the classical Adomian decomposition method fails to converge. When both methods are convergent, the present algorithm is observed to accelerate the rate of convergence. Moreover, the restricted domain of convergent physical solution obtained by the classical Adomian method is shown to be greatly extended to a finer interval by the optimum Adomian decomposition method. The justification of the new scheme is made clear on several mathematical/physical examples selected from the open literature. Finally, an example is provided to demonstrate the better accuracy of the optimum Adomian decomposition method over the recently popular homotopy analysis method.