  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 $V(r)=r^2+\frac{b r^2}{(1+cr^2)}$ in three dimensions
is discussed as an illustrative example.
