  Recent numerical and analytic computations based on the self-force (SF)
formalism in general relativity showed that half-integral post-Newtonian (PN)
terms, i.e. terms involving odd powers of 1/c, arise in the redshift factor of
small mass-ratio black-hole binaries on exact circular orbits. Although those
contributions might seem puzzling at first sight for conservative systems that
are invariant under time-reversal, they are in fact associated with the
so-called non-linear tail-of-tail effect. We shall describe here how the
next-to-next-to-leading order contributions beyond the first half-integral
5.5PN conservative effect (i.e. up to order 7.5PN included) have been obtained
by means of the standard PN formalism applied to binary systems of point-like
objects. The resulting redshift factor in the small mass-ratio limit fully
agrees with that of the SF approach.
