  We prove a new convergence condition for the activity expansion of
correlation functions in equilibrium statistical mechanics with possibly
negative pair potentials. For non-negative pair potentials, the criterion is an
if and only if condition. The condition is formulated with a sign-flipped
Kirkwood-Salsburg operator and known conditions such as Koteck${\'y}$-Preiss
and Fern${\'a}$ndez-Procacci are easily recovered. In addition, we deduce new
sufficient convergence conditions for hard-core systems in $\mathbb R^d$ and
$\mathbb Z^d$ as well as for abstract polymer systems. The latter improves on
the Fern${\'a}$ndez-Procacci criterion.
