  For any perturbative series that is known to $k$-subleading orders of
perturbation theory, we utilise the process-appropriate renormalization-group
(RG) equation in order to obtain all-orders summation of series terms
proportional to $\alpha^n \log^{n-k}(\mu^2)$ with $k = {0,1,2,3}$,
corresponding to the summation to all orders of the leading and
subsequent-three-subleading logarithmic contributions to the full perturbative
series. These methods are applied to the perturbative series for semileptonic
$b$-decays in both MS-bar and pole-mass schemes, and they result in RG-summed
series for the decay rates which exhibit greatly reduced sensitivity to the
renormalization scale $\mu$. Such summation via RG-methods of all logarithms
accessible from known series terms is also applied to perturbative QCD series
for vector- and scalar-current correlation functions, the perturbative static
potential function, the (single-doublet standard-model) Higgs decay amplitude
into two gluons, as well as the Higgs-mediated high-energy cross-section for
$W^+W^-\to ZZ$ scattering. The resulting RG-summed expressions are also found
to be much less sensitive to the renormalization scale than the original series
for these processes.
