  We continue an effort to obtain information on the QED perturbation series at
high loop orders, and particularly on the issue of large cancellations inside
gauge invariant classes of graphs, using the example of the l - loop N - photon
amplitudes in the limit of large photons numbers and low photon energies. As
was previously shown, high-order information on these amplitudes can be
obtained from a nonperturbative formula, due to Affleck et al., for the
imaginary part of the QED effective lagrangian in a constant field. The
procedure uses Borel analysis and leads, under some plausible assumptions, to a
number of nontrivial predictions already at the three-loop level. Their direct
verification would require a calculation of this `Euler-Heisenberg lagrangian'
at three-loops, which seems presently out of reach. Motivated by previous work
by Dunne and Krasnansky on Euler-Heisenberg lagrangians in various dimensions,
in the present work we initiate a new line of attack on this problem by
deriving and proving the analogous predictions in the simpler setting of 1+1
dimensional QED. In the first part of this series, we obtain a generalization
of the formula of Affleck et al. to this case, and show that, for both Scalar
and Spinor QED, it correctly predicts the leading asymptotic behaviour of the
weak field expansion coefficients of the two loop Euler-Heisenberg lagrangians.
