  We establish the existence of two weak coupling regime effective dynamics for
an open quantum system of repeated interactions (vanishing strength and
individual interaction duration, respectively). This generalizes known results
in that the von Neumann algebras describing the system and the chain element
may not be of finite type. Then (but now assuming that the small system is of
finite type), we prove that both effective dynamics capture the long-term
behaviour of the system: existence of a unique asymptotic state for them
implies the same property for the respective exact dynamics--provided that the
perturbation parameter is sufficiently small. The zero-th order term in a power
series expansion in the perturbation parameter of such an asymptotic state is
given by the asymptotic state of the effective dynamics. We conclude by working
out the case in which the small system and the chain element are spins.
