  We study the accuracy of the post-Newtonian (PN) approximation and its formal
region of validity, by investigating its optimal asymptotic expansion for the
quasi-circular, adiabatic inspiral of a point particle into a Schwarzschild
black hole. By comparing the PN expansion of the energy flux to numerical
calculations in the perturbative Teukolsky formalism, we show that (i) the
inclusion of higher multipoles is necessary to establish the accuracy of
high-order PN terms, and (ii) the region of validity of PN theory is largest at
relative O(1/c^6) (3PN order). The latter result suggests that the series
diverges beyond 3PN order, at least in the extreme mass-ratio limit, probably
due to the appearance of logarithmic terms in the energy flux. The study
presented here is a first formal attempt to determine the region of validity of
the PN approximation using asymptotic analysis. Therefore, it should serve as a
template to perform similar studies on other systems, such as comparable-mass
quasi-circular inspirals computed by high-accuracy numerical relativistic
simulations.
