On the convergence of planar diagram expansions

Renormalizable quantum field theories whose perturbation expansions are described by planar Feynman diagrams only, such as SU() gauge theory, are considered in 4 dimensional Euclidean space. For studying asymptotic properties of the perturbation series one might wish to isolate first all those planar diagrams that do not contain any ultraviolet divergent subgraphs. In this paper it is proved that this infinite set of diagrams, when summed, converges within a finite radius of convergence for the coupling constant.