Asymptotic behaviour of the derivative expansion in the ERG

We show that the derivative expansion of the exact (functional) renormalization group is a divergent series in any dimension, both for an exponential cutoff and more general smooth cutoffs. We prove this by showing that within massless $\lambda\varphi^4$ perturbation theory, such divergences arise first at two loops. From several lines of theoretical argument and by analysing infinite classes of two- and three-loop contributions, we conclude that the derivative expansion is an asymptotic series that initially converges towards the exact result before divergent behaviour takes over.