A method for deriving accurate analytic approximations for Markovian open quantum systems was recently introduced by Lucas and Hornberger [Phys. Rev. Lett. 110 240401 (2013)]. Here, we present a detailed derivation of the underlying nonperturbative jump expansion, which involves an adaptive resummation to ensure optimal convergence. Applying this to a set of exemplary master equations, we find that the resummation typically leads to convergence within the lowest two to five orders. Besides facilitating analytic approximations, the optimal jump expansion thus provides a numerical scheme for the efficient simulation of open quantum systems.