  We give an explicit demonstration that the derivative expansion of the QED
effective action is a divergent but Borel summable asymptotic series, for a
particular inhomogeneous background magnetic field. A duality transformation
B\to iE gives a non-Borel-summable perturbative series for a time dependent
background electric field, and Borel dispersion relations yield the
non-perturbative imaginary part of the effective action, which determines the
pair production probability. Resummations of leading Borel approximations
exponentiate to give perturbative corrections to the exponents in the
non-perturbative pair production rates. Comparison with a WKB analysis suggests
that these divergence properties are general features of derivative expansions
and effective actions.
