Can asymptotic series resolve the problems of inflation?

We discuss a cosmological scenario in which inflation is driven by a potential which is motivated by an effective Lagrangian approach, to gravity. We exploit the recent arguments that an effective Lagrangian L-eff which, by definition, contains operators of arbitrary dimensionality, is in general not a convergent but rather an asymptotic series with factorially growing coefficients. The potential we use is obtained by resumming this series. It provides a natural realization of the inflationary scenario. By caking into account a cosmological infrared cutoff, we propose a mechanism which simultaneously provides a graceful exit from inflation and relaxes to zero the contribution to the cosmological constant which stems from the inflation Lagrangian.