An ab initio method is introduced, called the maximum radius of convergence (MAXR(c)) perturbation theory, that exploits the added degrees of freedom permitted with flexible energy denominator perturbation theory [J. Chem. Phys. 109, 7725 (1998)] by defining the energy-denominator factors of a Rayleigh-Schrodinger perturbative expansion to be (approximately) optimal. This method can yield rapid convergence as long as there is no quasidegeneracies in first order between the reference-space state and one of the orthogonal-space states. (C) 2000 American Institute of Physics. [S0021-9606(00)30314-2].