Closed‐form expressions are derived for the Rayleigh–Rice and extinction‐theorem perturbation theories for the case of Dirichlet boundary conditions. It is shown that the two theories are related through the reciprocity transformation to all orders. It follows that a proof of invariance of either of the theories with respect to the reciprocity transformation would imply the formal equivalence of the two theories. Using the reciprocity relationship between the two theories, a recursion relation is derived for annth‐order term of the extinction‐theorem perturbation theory (similar to the one for the Rayleigh–Rice perturbation theory). In spite of considerable effort, no general proof of invariance of the terms of the two perturbation series with respect to the reciprocity transformation was obtained. Two possible approaches to such a proof are suggested. While the question of convergence of the two series is not examined in this work, it is possible that the formulas derived here will prove useful in such a study.