  The Landau-Lifshitz equation is obtained from the Lorentz-Abraham-Dirac
equation through `reduction of order'. It is the first in a divergent series of
approximations that, after resummation, eliminate runaway solutions. Using
Borel plane and transseries analysis we explain why this is, and show that a
non-perturbative formulation of reduction of order can retain runaway
solutions. We also apply transseries analysis to solutions of the
Lorentz-Abraham-Dirac equation, essentially treating them as expansions in both
time and a coupling. Our results illustrate some aspects of such expansions
under changes of variables and limits.
