  We review what is presently known about higher loop corrections to the
Euler-Heisenberg Lagrangian and its Scalar QED analogue. The use of those
corrections as a tool for the study of the properties of the QED perturbation
series is outlined. As a further step in a long-term effort to prove or
disprove the convergence of the N photon amplitudes in the quenched
approximation, we present a parameter integral representation of the three-loop
Euler-Heisenberg Lagrangian in 1+1 dimensional QED, obtained in the worldline
formalism.
