  Recently, it was demonstrated that the discrepancy between the fixed-order
(FOPT) and contour-improved (CIPT) perturbative expansions for $\tau$-lepton
decay hadronic spectral function moments, which had been affecting the
precision of $\alpha_s$ determinations for many years, is related to the CIPT
expansion being inconsistent with the standard formulation of the operator
product expansion (OPE). Even though the problem can be alleviated
phenomenologically for the most part by employing a renormalon-free scheme for
the gluon-condensate matrix element, the principal inconsistency of CIPT
remains. The CIPT expansion is special because it is not a power expansion, but
represents an asymptotic expansion in a sequence of functions of the strong
coupling. In this article we provide a closer look at the mathematical aspects
of the asymptotic sequence of the functions the CIPT method is based on, and we
expose the origin of the CIPT inconsistency as well as the reasons for its
apparent good convergence at low orders. Our results are of general interest,
and may in particular provide a useful tool to check for the consistency of
expansion methods that are similar to CIPT.
