  In the standard framework of self-consistent many-body perturbation theory,
the skeleton series for the self-energy is truncated at a finite order $N$ and
plugged into the Dyson equation, which is then solved for the propagator $G_N$.
For two simple examples of fermionic models -- the Hubbard atom at half filling
and its zero space-time dimensional simplified version -- we find that $G_N$
converges when $N\to\infty$ to a limit $G_\infty\,$, which coincides with the
exact physical propagator $G_{\rm exact}\ \,$ at small enough coupling, while
$G_\infty \neq G_{\rm exact}\ \,$ at strong coupling. We also demonstrate that
it is possible to discriminate between these two regimes thanks to a criterion
which does not require the knowledge of $G_{\rm exact}\ \,$, as proposed in
[Rossi et al., PRB 93, 161102(R) (2016)].
