  We consider a class of unbounded quasiperiodic Schr\"odinger-type operators
on $\ell^2(\mathbb Z^d)$ with monotone potentials (akin to the Maryland model)
and show that the Rayleigh--Schr\"odinger perturbation series for these
operators converges in the regime of small kinetic energies, uniformly in the
spectrum. As a consequence, we obtain a new proof of Anderson localization in a
more general than before class of such operators, with explicit convergent
series expansions for eigenvalues and eigenvectors. This result can be
restricted to an energy window if the potential is only locally monotone and
one-to-one. A modification of this approach also allows the potential to be
non-strictly monotone and have a flat segment, under additional restrictions on
the frequency.
