In

Roma 8/8/1988

This is a concise discussion of a proof of n! bounds on the renormalized series for phi-4 scalar theoriy in 4 dimensions. The analysis is via the renormalizations group and the beta function. As a corollary one obtains the convergence of the planar theory in its expansion in terms of the running coupling constants and its Borel summability inb terms of the renormalized copulings.