Truncation of Schwinger-Dyson equations and the $\frac{1}{N}$ expansion in the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model

We use the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model as a nonperturbative example to study the effect of truncating the Schwinger-Dyson sequence of equations. We show that truncation of this hierarchy leads in the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model to solutions for the propagators and vertex functions correct to a definite power of $\frac{1}{N}$. The procedure can be used to generate a systematic expansion in $\frac{1}{N}$ for an arbitrary Green's function of the model.