  In recent years, the Euler-Heisenberg Lagrangian has been shown to be a
useful tool for the analysis of the asymptotic growth of the N-photon
amplitudes at large N. Moreover, certain results and conjectures on its
imaginary part allow one, using Borel analysis, to make predictions for those
amplitudes at large loop orders. Extending work by G.V. Dunne and one of the
authors to the three-loop level, but in the simpler context of 1+1 dimensional
QED, we calculate the corresponding Euler-Heisenberg Lagrangian, analyse its
weak field expansion, and study the congruence with predictions obtained from
worldline instantons. We discuss the relevance of these issues for Cvitanovic's
conjecture.
