In what schemes can QCD perturbation series converge?

We extend our earlier results and are able to prove that the PMS (principle of minimal sensitivity) and FAC (fastest apparent convergence) procedures for selecting the renormalization scheme will yield a finite limit, if and only if the perturbation series is convergent in the schemes with all scheme-dependent $\beta$-function coefficients equal to zero ("zero schemes"). In this case our modified PMS procedure ($\stackrel{\mathrm{̃}}{\mathrm{P}}$ $\stackrel{\mathrm{̃}}{\mathrm{M}}$ $\stackrel{\mathrm{̃}}{\mathrm{S}}$) will yield the same finite limit as PMS and FAC. In the more likely and interesting case of a divergent although Borel-summable series in these "zero schemes", $\stackrel{\mathrm{̃}}{\mathrm{P}}$ $\stackrel{\mathrm{̃}}{\mathrm{M}}$ $\stackrel{\mathrm{̃}}{\mathrm{S}}$ serves to Borel sum the series order by order. An explicit expression for the scheme invariants ${\rho }_i$ is derived and some new results concerning them are discussed.