A class of perturbation problems is considered, in which the Rayleigh-Schrodinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:φ2m:g(x))2, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex4 anharmonic oscillator are given.