  The Landau-Lifshitz equation is considered as an approximation of the
Abraham-Lorentz-Dirac equation. It is derived from the Abraham-Lorentz-Dirac
equation by treating radiation reaction terms as a perturbation. However, while
the Abraham-Lorentz-Dirac equation has pathological solutions of
pre-acceleration and runaway, the Landau-Lifshitz equation and its finite
higher order extensions are free of these problems. So it seems mysterious that
the property of solutions of these two equations is so different. In this paper
we show that the problems of pre-acceleration and runaway appear when one
consider a series of all-order perturbation which we call it the
Landau-Lifshitz series. We show that the Landau-Lifshitz series diverges in
general. Hence a resummation is necessary to obtain a well-defined solution
from the Landau-Lifshitz series. This resummation leads the pre-accelerating
and the runaway solutions. The analysis is focusing on the non-relativistic
case, but we can extend the results obtained here to relativistic case at least
in one dimension.
