Ultraviolet divergences in1Nexpansions of asymptotically free theories

We calculate the ultraviolet divergences of the nonlinear $\sigma$ and Gross-Neveu models in the $\frac{1}{N}$ expansion near two dimensions ($d=2-\epsilon$). Beyond the leading order the theories develop logarithmic as well as pole singularities in $\epsilon$. Identical divergences occur when the renormalization constants calculated in perturbation theory are expanded in powers of $\frac{1}{N}$ rather than in powers of the coupling constants. The necessary infinite renormalizations of the $\frac{1}{N}$ expansions are completely determined by the two-loop $\beta$ functions and one-loop anomalous dimensions of perturbation theory. These results can be extended to non-Abelian gauge theories in four dimensions which admit $\frac{1}{N}$ expansions.