  In this note we have compared two different perturbation techniques that
could be used to generate solutions of Einstein's equations in presence of
negative cosmological constant. One of these two methods is derivative
expansion and the other is an expansion in inverse powers of dimension. Both
the techniques generate space-time with a singularity shielded by a dynamical
event horizon. We have shown that in the appropriate regime of parameter space
and with appropriate choice of coordinates, the metrics and corresponding
horizon dynamics, generated by these two different techniques, are exactly
equal to the order the solutions are known both sides. This work is essentially
extension of \cite{prevwork} where the authors have shown the equivalence of
the two techniques up to the first non-trivial order.
