  We consider a Borel sum definition of all-orders perturbation theory for
Minkowskian QCD observables such as the R(e+e-) ratio, and show that both this
perturbative component and the additional non-perturbative Operator Product
Expansion (OPE) component can remain separately well-defined for all values of
energy sqrt(s), with the perturbative component dominating as s-->infinity, and
with both components contributing as s-->0. In the infrared s-->0 limit the
perturbative correction to the parton model result for R(e+e-) has an
all-orders perturbation theory component which smoothly freezes to the value
R(0)=2/b, where b=(33-2Nf)/6 is the first QCD beta-function coefficient with Nf
flavours of massless quarks. For freezing one requires Nf<9. The freezing
behaviour is manifested by the ``contour-improved'' or ``Analytic Perturbation
Theory'' (APT), in which an infinite subset of analytical continuation terms
are resummed to all-orders. We show that for the Euclidean Adler-D function,
D(Q^2), the perturbative component remains defined into the infrared provided
that ALL the renormalon singularities care taken into account, but no analogue
of the APT reorganisation of perturbation theory is possible. We perform
phenomenological comparisons of suitably smeared low-energy data for the
R(e+e-) ratio, with the perturbative freezing perdictions, and find good
agreement.
