  The generating function of the cumulants in random matrix models, as well as
the cumulants themselves, can be expanded as asymptotic (divergent) series
indexed by maps. While at fixed genus the sums over maps converge, the sums
over genera do not. In this paper we obtain alternative expansions both for the
generating function and for the cumulants that cure this problem. We provide
explicit and convergent expansions for the cumulants, for the remainders of
their perturbative expansion (in the size of the maps) and for the remainders
of their topological expansion (in the genus of the maps). We show that any
cumulant is an analytic function inside a cardioid domain in the complex plane
and we prove that any cumulant is Borel summable at the origin.
