The correlation energy of finite systems of electrons in a neutralizin
g spherical positively charged background (jellium) is investigated wi
thin various approximations. The correlation energy is defined with re
spect to the Hartree-Fock energy (with exact treatment of exchange). O
n one hand, the exact second-order energy contribution is calculated f
rom many-body perturbation theory. On the other hand, a quasi-boson ap
proximation to harmonic order only allows one to estimate the RPA grou
nd-state correlation energy and to emphasize the relative importance o
f Coulomb-interaction screening corrections (ring diagrams). One finds
that the second-order contribution accounts for most of the correlati
on energy. Comparisons with other approaches, such as density-function
al theory and CI calculations, confirm this assertion.