  The problem of a compact binary system whose components move on circular
orbits is addressed using two different approximation techniques in general
relativity. The post-Newtonian (PN) approximation involves an expansion in
powers of v/c<<1, and is most appropriate for small orbital velocities v. The
perturbative self-force (SF) analysis requires an extreme mass ratio m1/m2<<1
for the components of the binary. A particular coordinate-invariant observable
is determined as a function of the orbital frequency of the system using these
two different approximations. The post-Newtonian calculation is pushed up to
the third post-Newtonian (3PN) order. It involves the metric generated by two
point particles and evaluated at the location of one of the particles. We
regularize the divergent self-field of the particle by means of dimensional
regularization. We show that the poles proportional to 1/(d-3) appearing in
dimensional regularization at the 3PN order cancel out from the final gauge
invariant observable. The 3PN analytical result, through first order in the
mass ratio, and the numerical SF calculation are found to agree well. The
consistency of this cross cultural comparison confirms the soundness of both
approximations in describing compact binary systems. In particular, it provides
an independent test of the very different regularization procedures invoked in
the two approximation schemes.
