Bounds On The Sum Of A Divergent Series

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.