  In this work we examine the Cauchy convergence of both post-Newtonian
(T-approximant) and re-summed post-Newtonian (P-approximant) templates for the
case of a test-mass orbiting a Kerr black hole along a circular equatorial
orbit. The Cauchy criterion demands that the inner product between the $n$ and
$n+1$ order approximation approaches unity, as we increase the order of
approximation. In previous works, it has been shown that we achieve greater
fitting factors and better parameter estimation using the P-approximant
templates for both Schwarzschild and Kerr black holes. In this work, we show
that the P-approximant templates also display a faster Cauchy convergence
making them a superior template to the standard post-Newtonian templates.
