  We show how to convert divergent series, which typically occur in many
applications in physics, into rapidly convergent inverse factorial series. This
can be interpreted physically as a novel resummation of perturbative series.
Being convergent, these new series allow rigorous extrapolation from an
asymptotic region with a large parameter, to the opposite region where the
parameter is small. We illustrate the method with various physical examples,
and discuss how these convergent series relate to standard methods such as
Borel summation, and also how they incorporate the physical Stokes phenomenon.
We comment on the relation of these results to Dyson's physical argument for
the divergence of perturbation theory. This approach also leads naturally to a
wide class of relations between bosonic and fermionic partition functions, and
Klein-Gordon and Dirac determinants.
