We discuss a method for systematically correcting results obtained using variational expressions in scattering theory. The approach taken is to compute a sequence of Pade approximants of the form [N/N] for the error in an initial variational estimate obtained using a basis-set expansion. The relationship between the Pade-approximant approach and the iterative Schwinger method for correcting variational estimates is also examined. We discuss a large class of general variational expressions to which the Pade-approximant approach can be applied. The variational expressions considered include those for the wave function, for photoionization transition matrix elements, as well as for scattering matrix (K-matrix) elements. We have applied this approach to the 5σ photoionization of CO using the frozen-core Hartree-Fock and fixed-nuclei approximations. We find that the Pade-approximant method converges rapidly and reliably. Both total photoionization cross sections and photoelectron angular distributions from threshold to 40 eV are presented and compared to previous experimental and theoretical results. We find major quantitative discrepancies between the present results for the total cross section and previous theoretical results.