[en] Starting from the path-integral representation for the electron propagator without fermion loops in QED, we analytically investigate the strong-coupling behavior in an arbitrary background electromagnetic field through a series expansion in powers of 1/e. Contrary to the perturbation theory expansion in e the new series only contains positive powers of the derivative operator p. Due to infrared singularities in the path integral the series does not exist beyond the lowest orders, although one can build a systematic expansion in powers of p (not 1/e) which can be calculated up to any order. To handle infinities we regularize using a Pauli-Villars approach. The introduction of fermion loops would not correspond to higher orders in 1/e, so a priori our results are only pertinent to the sector of QED we have chosen. 17 refs., 1 fig