  In quantum mechanics and quantum field theory perturbation theory generically
requires the inclusion of extra contributions non-perturbative in the coupling,
such as instantons, to reproduce exact results. We show how full
non-perturbative results can be encoded in a suitable modified perturbative
series in a class of quantum mechanical problems. We illustrate this explicitly
in examples which are known to contain non-perturbative effects, such as the
(supersymmetric) double-well potential, the pure anharmonic oscillator, and the
perturbative expansion around a false vacuum.
