  Perturbation expansions appear to be divergent series in many physically
interesting situations, including in quantum field theories like quantum
electrodynamics (QED) and quantum chromodynamics (QCD), where the perturbative
coefficients exhibit a factorial growth at large orders. While this feature has
little impact on physical predictions in QED, it can have nontrivial
consequences in applications of perturbative QCD at moderate energies. In
particular, it affects the theoretical error in the extraction of the strong
coupling $\alpha_s$ from hadronic $\tau$ decays, despite progress of
perturbative calculations available at present to four loops. We discuss a new
type of perturbative expansion for QCD correlators, which uses instead of the
standard powers of the coupling a new set of expansion functions. These
functions are defined by means of an optimal conformal mapping of the Borel
complex plane, which implements the known features of the high-order divergence
in terms of the lowest Borel-plane singularities. The properties of the
expansion functions resemble those of the expanded correlators, by exhibiting
in particular the singular behaviour of the correlators at $\alpha_s=0$. We
prove the good convergence properties of the new expansions on mathematical
models that simulate the physical polarization function for light quarks and
its derivative (the Adler function), in various prescriptions of
renormalization-group summation.
