Convergence features of the Rayleigh-Schrödinger perturbation theory (PT) strongly depend on the partitioning applied. We investigate the large order behavior of the Møller-Plesset and Epstein Nesbet partitionings in comparison with a less known partitioning obtained by level shift parameters minimizing the norm of operator Q^W^, with W^ being the perturbation operator while Q standing for the reduced resolvent of the zero order Hamiltonian H^0. Numerical results, presented for molecular systems for the first time, indicate that it is possible to find level shift parameters in this way which convert divergent perturbation expansions to convergent ones in some cases. Besides numerical calculations of high-order PT terms, convergence radii of the corresponding perturbation expansions are also estimated using quadratic Padé approximants.