  We reconsider in some detail a construction allowing (Borel) convergence of
an alternative perturbative expansion, for specific physical quantities of
asymptotically free models. The usual perturbative expansions (with an explicit
mass dependence) are transmuted into expansions in 1/F, where $F \sim 1/g(m)$
for $m \gg \Lambda$ while $F \sim (m/\Lambda)^\alpha$ for $m \lsim \Lambda$,
$\Lambda$ being the basic scale and $\alpha$ given by renormalization group
coefficients. (Borel) convergence holds in a range of $F$ which corresponds to
reach unambiguously the strong coupling infrared regime near $m\to 0$, which
can define certain "non-perturbative" quantities, such as the mass gap, from a
resummation of this alternative expansion. Convergence properties can be
further improved, when combined with $\delta$ expansion (variationally improved
perturbation) methods. We illustrate these results by re-evaluating, from
purely perturbative informations, the O(N) Gross-Neveu model mass gap, known
for arbitrary $N$ from exact S matrix results. Comparing different levels of
approximations that can be defined within our framework, we find reasonable
agreement with the exact result.
