  We study $N$-electron atoms with nuclear charge $Z$. It is well known that,
in the cationic ($Z > N$) high-$Z$ region, the atom behaves as a weakly
interacting system. The anionic ($Z < N$) regime, on the other hand, is
characterized by an instability threshold at $Z_c \lesssim N-1$, below which
the atom spontaneously emits an electron. We construct a regularized
perturbative series (RPS) for the ionization potential of ions in an
isoelectronic sequence that exactly reproduces both, the large $Z$ and the $Z$
near $Z_c$ limits. The large-$Z$ expansion coefficients are analytically
computed from perturbation theory, whereas the slope of the energy curve at
$Z=N-1$ is computed from a kind of zero-range forces theory that uses as input
the electron affinity and the covalent radius of the neutral atom with $N-1$
electrons. Relativistic effects, at the level of first-order perturbation
theory, are considered. Our RPS formula is to be used in order to check the
consistency of the ionization potential values for atomic ions contained in the
NIST database.
