  In the framework of the KIDS generalized energy density functional (EDF), the
nuclear equation of state (EoS) is expressed as an expansion in powers of the
Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an
optimal number of converging terms was obtained in specific cases of fits to
empirical data and pseudodata, the degree of convergence remains to be examined
not only for homogeneous matter but also for finite nuclei. One goal of the
present work is to validate the minimal and optimal number of EoS parameters
required for the description of homogeneous nuclear matter over a wide range of
densities relevant for astrophysical applications. The major goal is to examine
the validity of the adopted expansion scheme for an accurate description of
finite nuclei. To this end we vary the values of the high-order derivatives of
the EoS, namely the skewness of the energy of symmetric nuclear matter and the
kurtosis of the symmetry energy, at saturation and examine the relative
importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For
given sets of EoS parameters determined in this way, we define equivalent
Skyrme-type functionals and examine the convergence in the description of
finite nuclei focusing on the masses and charge radii of closed-shell nuclei.
The EoS of symmetric nuclear matter is found to be efficiently parameterized
with only 3 parameters and the symmetry energy (or the energy of pure neutron
matter) with 4 parameters when the EoS is expanded in the power series of the
Fermi momentum. Higher-order EoS parameters do not produce any improvement, in
practice, in the description of nuclear ground-state energies and charge radii,
which means that they cannot be constrained by bulk properties of nuclei.
