  The ground state energy of integrable asymptotically free theories can be
conjecturally computed by using the Bethe ansatz, once the theory has been
coupled to an external potential through a conserved charge. This leads to a
precise prediction for the perturbative expansion of the energy. We provide a
non-trivial test of this prediction in the non-linear sigma model and its
supersymmetric extension, by calculating analytically the associated Feynman
diagrams at next-to-leading order in the $1/N$ expansion, and at all loops. By
investigating the large order behaviour of the diagrams, we locate the position
of the renormalons of the theory and we obtain an analytic expression for the
large $N$ trans-series associated to each. As a spin-off of our calculation, we
provide a direct derivation of the beta function of these theories, at
next-to-leading order in the $1/N$ expansion.
