Gravitational waves from binary systems in circular orbits: Convergence of a dressed multipole truncation

The gravitational radiation originating from a compact binary system in circular orbit is usually expressed as an infinite sum over radiative multipole moments. In a slow-motion approximation, each multipole moment is then expressed as a post-Newtonian expansion in powers of v/c, the ratio of the orbital velocity to the speed of light. The bare multipole truncation of the radiation consists in keeping only the leading-order term in the post-Newtonian expansion of each moment, but summing over all the multipole moments. In the case of binary systems with small mass ratios, the bare multipole series was shown in a previous paper to converge for all values v/c < 2/e, where e is the base of natural logarithms. In this paper, we extend the analysis to a dressed multipole truncation of the radiation, in which the leading-order moments are corrected with terms of relative order (v/c)^2 and (v/c)^3. We find that the dressed multipole series converges also for all values v/c < 2/e, and that it coincides (within 1%) with the numerically ``exact'' results for v/c < 0.2.