  Using renormalization-group methods, differential equations can be obtained
for the all-orders summation of leading and subsequent non-leading logarithmic
corrections to QCD perturbative series for a number of processes and
correlation functions. For a QCD perturbative series known to four orders, such
as the e+ e- annihilation cross-section, explicit solutions to these equations
are obtained for the summation to all orders in alpha_s of the leading set and
the subsequent two non-leading sets of logarithms. Such summations are shown
for a number of processes to lead to a substantial reduction in sensitivity to
the renormalization scale parameter. Surprisingly, such summations are also
shown to lower the infrared singularity within the perturbative expression for
the e+ e- annihilation cross-section to coincide with the Landau pole of the
naive one-loop running QCD couplant.
