Large-order shifted 1/N expansions through the asymptotic iteration method

The perturbation technique within the framework of the asymptotic iteration method is used to obtain large-order shifted 1/N expansions, where N is the number of spatial dimensions. This method is contrary to the usual Rayleigh-Schr"{o}dinger perturbation theory, no matrix elements need to be calculated. The method is applied to the Schr"{o}dinger equation and the non-polynomial potential $V(r)=r^2+\frac{b{r}^2}{(1+c{r}^2)}$ in three dimensions is discussed as an illustrative example.