Scheme dependence and the limit of QCD perturbation series

A proof is presented that the much-discussed fastest-apparent-convergence (FAC) and principle-of-minimal-sensitivity (PMS) procedures for selecting the renormalization scheme will inevitably result in a zero limit for the perturbation series, if a limit exists. An alternative procedure, based on fixing the scheme-dependent $\beta$-function coefficients to be zero and optimizing the renormalization point, is suggested. This is shown to result in a finite limit closely related to the Borel sum, which is conjectured to be the maximum obtainable for any sequence of schemes leading to a limit of the series.