  We examine the large-order behaviour of a recently proposed
renormalization-group-improved expansion of the Adler function in perturbative
QCD, which sums in an analytically closed form the leading logarithms
accessible from renormalization-group invariance. The expansion is first
written as aneffective series in powers of the one-loop coupling, and its
leading singularities in the Borel plane are shown to be identical to those of
the standard "contour-improved" expansion. Applying the technique of conformal
mappings for the analytic continuation in the Borel plane, we define a class of
improved expansions, which implement both the renormalization-group invariance
and the knowledge about the large-order behaviour of the series. Detailed
numerical studies of specific models for the Adler function indicate that the
new expansions have remarkable convergence properties up to high orders. Using
these expansions for the determination of the strong coupling from the the
hadronic width of the $\tau$ lepton we obtain, with a conservative estimate of
the uncertainty due to the nonperturbative corrections, $\alpha_s(M_\tau^2)=
0.3189^{+ 0.0173}_{-0.0151}$, which translates to $\alpha_s(M_Z^2)=
0.1184^{+0.0021}_{-0.0018}$.
