We compute the energy flux of the gravitational waves radiated by a particle of mass μ in circular orbits around a rotating black hole of mass M up to the 11th post-Newtonian order (11PN), i.e. v22 beyond the leading Newtonian approximationwhere v is the orbital velocity of the particle [L. Blanchet, Living Rev. Relativity 5, 3 (2002)]. By comparing the PN results for the energy flux with high-precision numerical results in black hole perturbation theory, we find the region
of validity in the PN approximation becomes larger with increasing PN order. If one requires the relative error of the energy flux in the PN approximation to be less than 10−5, the energy flux at 11PN (4PN) can be used for v  0.33 (v  0.13). The region of validity can be further
extended to v  0.4 if one applies a resummation method to the energy flux at 11PN. We then compare the orbital phase during a two-year inspiral from the PN results with the high-precision
numerical results. We find that, for late (early) inspirals when q ≤ 0.3 (q ≤ 0.9), where q is the dimensionless spin parameter of the black hole, the difference in the phase is less than 1 (10−4) rad and hence these inspirals may be detected in the data analysis for space detectors such as eLISA/New Gravitational wave Observatory by the PN templates. We also compute the energy flux radiated into the event horizon for a particle in circular orbits around a non-rotating black
hole at 22.5PN, i.e. v45 beyond the leading Newtonian approximation, which is comparable to the PN order derived in our previous work for the energy flux to infinity at 22PN.