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We examine the large-order behaviour of a recently proposedrenormalization-group-improved expansion of the Adler function in perturbativeQCD, which sums in an analytically closed form the leading logarithmsaccessible from renormalization-group invariance. The expansion is firstwritten as aneffective series in powers of the one-loop coupling, and itsleading singularities in the Borel plane are shown to be identical to those ofthe standard "contour-improved" expansion. Applying the technique of conformalmappings for the analytic continuation in the Borel plane, we define a class ofimproved expansions, which implement both the renormalization-group invarianceand the knowledge about the large-order behaviour of the series. Detailednumerical studies of specific models for the Adler function indicate that thenew expansions have remarkable convergence properties up to high orders. Usingthese expansions for the determination of the strong coupling from the thehadronic width of the $\tau$ lepton we obtain, with a conservative estimate ofthe uncertainty due to the nonperturbative corrections, $\alpha_s(M_\tau^2)=0.3189^{+ 0.0173}_{-0.0151}$, which translates to $\alpha_s(M_Z^2)=0.1184^{+0.0021}_{-0.0018}$.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane