Gravitational waves from binary systems in circular orbits: Does the   post-Newtonian expansion converge?
  Gravitational radiation can be expressed in terms of an infinite series of radiative, symmetric trace-free (STF) multipole moments which can be connected to the behavior of the source. We consider a truncated model for gravitational radiation from binary systems in which each STF mass and current moment of order l is given by the lowest-order, Newtonian-like l-pole moment of the orbiting masses; we neglect post-Newtonian corrections to each STF moment. Specializing to orbits which are circular (apart from the radiation-induced inspiral), we find an explicit infinite series for the energy flux in powers of $v/c$, where v is the orbital velocity. We show that the series converges for all values $v/c < 2/e$ when one mass is much smaller than the other, and $v/c < 4/e$ for equal masses,where e is the base of natural logarithms. These values include all physically relevant values for compact binary inspiral. This convergence cannot indicate whether or not the full series (obtained from the exact moments) will converge. But if the full series does not converge, our analysis shows that this failure to converge does not originate from summing over the Newtonian part of the multipole moments. 
L.E. Simone, S.W. Leonard, E. Poisson, and C.M. Will