  We address the problem of analyzing the radius of convergence of perturbative
expansion of non-equilibrium steady states of Lindblad driven spin chains. A
simple formal approach is developed for systematically computing the
perturbative expansion of small driven systems. We consider the paradigmatic
model of an open $XXZ$ spin 1/2 chain with boundary supported ultralocal
Lindblad dissipators and treat two different perturbative cases: (i) expansion
in system-bath coupling parameter and (ii) expansion in driving (bias)
parameter. In the first case (i) we find that the radius of convergence quickly
shrinks with increasing the system size, while in the second case (ii) we find
that the convergence radius is always larger than $1$, and in particular it
approaches $1$ from above as we change the anisotropy from easy plane ($XY$) to
easy axis (Ising) regime.
