There exist in the literature several related formulations of regular perturbation theory (cf. [l] and [2]). In deciding which formalism to apply to a specific problem we must take into account the radius of convergence of the various methods, and, if possible, the ease of application. In this note we shall develop a formalism, closely related to Fredholm’s resolvent, which has an infinite radius of convergence. The present formulation shares with the formalism derived directly from the Fredholm resolvent [l] the disadvantage of being rather cumbersome in its application. Consequently, we shall go on to develop an asymptotic approximation to the result. By this means we shall obtain a relatively simple procedure for evaluating the result of a formulation which is unrestricted as regards radius of convergence. Finally, we shall discuss the close relation between the present theory and Feenberg’s [3] formulation of regular perturbation theory.