  In theories with renormalons the perturbative series is factorially divergent
even after restricting to a given order in $1/N$, making the $1/N$ expansion a
natural testing ground for the theory of resurgence. We study in detail the
interplay between resurgent properties and the $1/N$ expansion in various
integrable field theories with renormalons. We focus on the free energy in the
presence of a chemical potential coupled to a conserved charge, which can be
computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples,
like the first $1/N$ correction to the free energy in the non-linear sigma
model, the terms in the $1/N$ expansion can be fully decoded in terms of a
resurgent trans-series in the coupling constant. In the principal chiral field
we find a new, explicit solution for the large $N$ free energy which can be
written as the median resummation of a trans-series with infinitely many,
analytically computable IR renormalon corrections. However, in other examples,
like the Gross-Neveu model, each term in the $1/N$ expansion includes
non-perturbative corrections which can not be predicted by a resurgent analysis
of the corresponding perturbative series. We also study the properties of the
series in $1/N$. In the Gross-Neveu model, where this is convergent, we
analytically continue the series beyond its radius of convergence and show how
the continuation matches with known dualities with sine-Gordon theories.
