Should we worry about renormalons in the epsilon-expansion?

Turning the divergent epsilon-expansion into a numerically sensible algorithm, relies on the knowledge of the behaviour of the large order contributions. Two different pictures are known to compete there. The first one was based on Lipatov's instantons, which is known to deal with the multiplicity of Feynman diagrams which grows factorially at high orders. However this was challenged by 't Hooft's renormalons who pointed out that renormalization could yield a similar growth through one single diagram. We study here a well-known model, the $O(N)$ model, in the large $N$ limit. The reason for returning to this familiar model, is that it deals with diagrams known to give renormalon effects.Through an explicit analytic result, we find no sign of a non-analyticity of perturbation theory due to these renormalons.