  Near-future, space-based, radio- and gravitational-wave interferometry
missions will enable us to rigorously test whether the Kerr solution of general
relativity accurately describes astrophysical black holes, or if it requires
some kind of modification. At the same time, recent work has greatly improved
our understanding of theories of gravity that modify the Einstein-Hilbert
action with terms quadratic in the curvature, allowing us to calculate black
hole solutions to (essentially) arbitrary order in a slow-rotation expansion.
Observational constraints of such quadratic gravity theories require the
calculation of observables that are robust against the expansion order of the
black hole solution used.
  We carry out such a study here and determine the accuracy with respect to
expansion order of ten observables associated with the spacetime outside a
rotating black hole in two quadratic theories of gravity,
dynamical-Chern-Simons and scalar-Gauss-Bonnet gravity. We find that for all
but the most rapidly rotating black holes, only about the first eight terms in
the spin expansion are necessary to achieve an accuracy that is better than the
statistical uncertainties of current and future missions.
