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We give an explicit demonstration that the derivative expansion of the QEDeffective action is a divergent but Borel summable asymptotic series, for aparticular inhomogeneous background magnetic field. A duality transformationB\to iE gives a non-Borel-summable perturbative series for a time dependentbackground electric field, and Borel dispersion relations yield thenon-perturbative imaginary part of the effective action, which determines thepair production probability. Resummations of leading Borel approximationsexponentiate to give perturbative corrections to the exponents in thenon-perturbative pair production rates. Comparison with a WKB analysis suggeststhat these divergence properties are general features of derivative expansionsand effective actions.Comment: 18 pp, Revtex, 2 fig

Borel summation of the derivative expansion and effective actions