  Recently-generated long strong-coupling series for the two-point Green's
functions of asymptotically free ${\rm O}(N)$ lattice $\sigma$ models are
analyzed, focusing on the evaluation of dimensionless renormalization-group
invariant ratios of physical quantities and applying resummation techniques to
series in the inverse temperature $\beta$ and in the energy $E$. Square,
triangular, and honeycomb lattices are considered, as a test of universality
and in order to estimate systematic errors. Large-$N$ solutions are carefully
studied in order to establish benchmarks for series coefficients and
resummations. Scaling and universality are verified. All invariant ratios
related to the large-distance properties of the two-point functions vary
monotonically with $N$, departing from their large-$N$ values only by a few per
mille even down to $N=3$.
