  We propose a renormalon-inspired resummation of QCD perturbation theory based
on approximating the renormalization scheme (RS) invariant effective charge
beta-function coefficients by the portion containing the highest power of
$b$=$\frac{1}{6}(11N$--$2N_{f})$, for SU($N$) QCD with $N_{f}$ quark flavours.
This can be accomplished using exact large-$N_{f}$ all-orders results. The
resulting resummation is RS-invariant and the exact next-to-leading order (NLO)
and next-to-NLO (NNLO) coefficients in any RS are included. This improves on a
previously employed naive resummation of the leading-$b$ piece of the
perturbative coefficients which is RS-dependent, making its comparison with
fixed-order perturbative results ambiguous. The RS-invariant resummation is
used to assess the reliability of fixed-order perturbation theory for the
$e^{+}e^{-}$ $R$-ratio, the analogous $\tau$-lepton decay ratio $R_{\tau}$, and
Deep Inelastic Scattering (DIS) sum rules, by comparing it with the exact NNLO
results in the effective charge RS. For the $R$-ratio and $R_{\tau}$, where
large-order perturbative behaviour is dominated by a leading ultra-violet
renormalon singularity, the comparison indicates fixed-order perturbation
theory to be very reliable. For DIS sum rules, which have a leading infra-red
renormalon singularity, the performance is rather poor. In this way we estimate
that at LEP/SLD energies ideal data on the $R$-ratio could determine
$\alpha_{s}(M_{Z})$ to three-significant figures, and for the $R_{\tau}$ we
estimate a theoretical uncertainty $\delta\alpha_{s}(m_{\tau})\simeq0.008$
corresponding to $\delta\alpha_{s}(M_{Z})\simeq0.001$. This encouragingly small
uncertainty is much less than has recently been deduced from comparison with
the ambiguous naive resummation.
