  We consider a new class of perturbation expansions, which incorporate in a
systematic way the available information about the divergent character of the
perturbation series in QCD. The new expansion functions, which replace the
powers of the coupling, are defined by the analytic continuation in the Borel
plane, achieved through an optimal conformal mapping. We consider several
possibilities of implementing the known behaviour of the Borel transform near
the leading singularities in the complex plane and using the corresponding
expansions. These expansions have the same asymptotic convergence rate, but
differ at low orders. We show that the new expansions allow a precise
determination of the strong coupling $\alpha_s$ from the hadronic decays of the
$\tau$ lepton.
