  We prove a novel inversion theorem for functionals given as power series in
infinite-dimensional spaces and apply it to the inversion of the
density-activity relation for inhomogeneous systems. This provides a rigorous
framework to prove convergence for density functionals for inhomogeneous
systems with applications in classical density function theory, liquid
crystals, molecules with various shapes or other internal degrees of freedom.
The key technical tool is the representation of the inverse via a fixed point
equation and a combinatorial identity for trees, which allows us to obtain
convergence estimates in situations where Banach inversion fails. Moreover, the
new method for the inversion gives for the (homogeneous) hard sphere gas a
significantly improved radius of convergence for the virial expansion improving
the first and up to now best result by Lebowitz and Penrose (1964).
