  Given a truncated perturbation expansion of a physical quantity, one can,
under certain circumstances, obtain lower or upper bounds (or both) to the sum
of the full perturbation series by using the Borel transform and a variational
conformal map. The method is illustrated by applying it to various mathematical
toy-models for which exact results are known. One of these models is used to
exemplify how non-perturbative contributions supplement the sum of a
Borel-nonsummable series to give the final exact and unambiguous result.
Finally, the method is applied to some physical problems. In particular, some
speculations are made on the phase of quantum electrodynamics at super-high
temperatures from a study of its perturbative free-energy density.
