  The difference between fixed-order (FO) and contour-improved (CI)
formulations of QCD perturbation theory limits the precision of the strong
coupling determined from the hadronic decay of the $\tau$ lepton. Recently,
several attempts to understand the mathematical origin of the difference and to
solve it by subtracting the dominant infrared renormalon divergence have been
made. Motivated by these studies, we review in this paper an improved
perturbative QCD expansion, defined some time ago, which also exploits the
renormalons by means of a suitable conformal mapping of the Borel plane. In
particular, we revisit the convergence of the new expansion, by completing the
proof presented in a previous paper and showing that the domain of convergence
is larger than stated before. We also check the validity of the convergence
conditions for the Adler function and the CI and FO expansions of the $\tau$
hadronic spectral function moments, and compare the approach based on conformal
mapping with recent solutions to the CIPT-FOPT discrepancy proposed in the
literature.
