Rayleigh–Rice perturbation theory is compared with the more rigorously derived perturbation theory based on the extinction theorem (or extended boundary condition) for the case of Dirichlet boundary conditions. Numerical calculations for a sinusoidal surface profile show that the two theories give identical results when carried out to high order. This is true even when the two series do not seem to converge numerically. Numerical convergence of the two series is found in several cases, including some cases in which the Rayleigh hypothesis is invalid. For surfaces of arbitrary shape, the two perturbation series appear superficially to differ at third and higher order. It is shown, however, that a reciprocity transformation of one series makes it identical to the other through fifth order. Finally, it is shown that reciprocity holds in both perturbation theories, establishing analytically the equivalence of the two theories at least through fifth order. These results are surprising given the limited regime ...