  By using the results of a high-statistics (O(10^7) measurements) Monte Carlo
simulation we test several predictions of perturbation theory on the O(n)
non-linear sigma-model in 2 dimensions. We study the O(3) and O(8) models on
large enough lattices to have a good control on finite-size effects. The
magnetic susceptibility and three different definitions of the correlation
length are measured. We check our results with large-n expansions as well as
with standard formulae for asymptotic freedom up to 4 loops in the standard and
effective schemes.
  For this purpose the weak coupling expansions of the energy up to 4 loops for
the standard action and up to 3 loops for the Symanzik action are calculated.
For the O(3) model we have used two different effective schemes and checked
that they lead to compatible results. A great improvement in the results is
obtained by using the effective scheme based on the energy at 3 and 4 loops. We
find that the O(8) model follows very nicely (within few per mille) the
perturbative predictions. For the O(3) model an acceptable agreement (within
few per cent) is found.
