  We show that the Borel representations of tau hadronic spectral function
moments based on contour-improved perturbation theory (CIPT) in general differ
from those obtained within fixed-order perturbation theory (FOPT) in the
presence of IR renormalons in the underlying Adler function. The Borel sums
obtained from both types of Borel representations in general differ as well,
and the apparently conflicting behavior of the FOPT and CIPT spectral function
moment series at intermediate orders, which has been subject to many studies in
the past literature, can be understood quantitatively using concrete Borel
function models. The difference between the CIPT and FOPT Borel sums, which we
call the "asymptotic separation", can be computed analytically for any Borel
function model and is proportional to inverse exponential terms in the strong
coupling. Even though moments can be designed where the asymptotic separation
is strongly suppressed, it is as a matter of principle unavoidable. If the
Borel function of the Euclidean Adler function has a sizeable gluon condensate
renormalon cut, the asymptotic separation can explain the observed disparity of
the CIPT and FOPT spectral function moments at the 5-loop level. The existence
of the asymptotic separation implies that the power corrections in the operator
product expansion for the spectral function moments in the CIPT expansion
approach do not have the commonly assumed analytic standard form.
