  The operators of localized spins within a magnetic material commute at
different sites of its lattice and anticommute on the same site, so they are
neither fermionic nor bosonic operators. Thus, to construct diagrammatic
many-body perturbation theory, the spin operators are usually mapped to the
bosonic ones with Holstein-Primakoff (HP) transformation being the most widely
used in magnonics and spintronics literature. However, to make calculations
tractable, the square root of operators in the HP transformation is expanded
into a Taylor series truncated to some low order. This poses a question on the
range of validity of truncated HP transformation when describing nonequilibrium
dynamics of localized spins interacting with each other or with conduction
electron spins. Here we apply exact diagonalization techniques to Hamiltonian
of fermions (i.e., electrons) interacting with HP bosons vs. Hamiltonian of
fermions interacting with the original localized spin operators in order to
compare their many-body states and one-particle equilibrium or nonequilibrium
Green functions. The Hamiltonian of fermions interacting with HP bosons gives
incorrect ground state and electronic spectral function, unless large number of
terms are retained in truncated HP transformation. Furthermore, tracking
nonequilibrium dynamics of localized spins over longer time intervals requires
progressively larger number of terms in truncated HP transformation. Finally,
we show that recently proposed [M. Vogl et al., Phys. Rev. Research 2, 043243
(2020); J. K\"{o}nig et al., SciPost Phys. 10, 007 (2021)] resummed HP
transformation resolves the trouble with truncated HP transformation, while
allowing us to derive an exact (manifestly Hermitian) Hamiltonian consisting of
finite and fixed number of boson-boson and electron-boson interacting terms.
