In the very well written article by C. Will, On the Unreasonable Effectiveness of the post-Newtonian Approximation in Gravitational Physics, he states:
The one question that remains open is the nature of the post-Newtonian sequence; we still do not know if it converges, diverges or is asymptotic. Despite this, it has proven to be remarkably effective.
What is the current status of the convergence of the post-Newtonian sequence? Are there indications that it is possibly asymptotic?
The question about PN sequence is raiser mathematical one, nor physical or philosophical. It has a good answer in the paper "Newtonian and post-Newtonian approximations are asymptotic to general relativity" by T. Futamase and Bernard, F. Schutz, Phys. Rev. D 28, 2363 – Published 15 November 1983. The Abstract to this paper is clearly stated that the PN approximations "are genuine asymptotic approximations to general relativity". Also this question discussed in many papers such as Blanchet, L., “Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries”, Living Rev. Relativity, 9, lrr-2006-4, (2006). URL (cited on 3 August 2006): http://www.livingreviews.org/lrr-2006-4. 2.2, 3
"The Post-Newtonian Approximation for Relativistic Compact Binaries", by Toshifumi Futamase&Yousuke Itoh, Living Reviews in Relativity, 12 Mar 2007, 10(1):2
"Accuracy of the post-Newtonian approximation: Optimal asymptotic expansion for quasicircular, extreme-mass ratio inspirals" by Nicolás Yunes and Emanuele Berti, Phys. Rev. D 77, 124006 – Published 5 June 2008; Erratum Phys. Rev. D 83, 109901 (2011).
