  We have implemented three approaches to describe the thermodynamic properties
of ferrimagnetic ($S=5/2, s=2$) spin chains. The application of cumulant
expansion has been generalized to the ferrimagnetic chain in the presence of an
external magnetic field. Using cumulants, we have obtained the field dependent
effective Hamiltonian in terms of the classical variables up to the second
order of quantum corrections. Thermodynamic functions, the internal energy, the
specific heat and the magnetic susceptibility are obtained from the effective
Hamiltonian. We have also examined the modified spin wave theory to derive the
same physical properties. Finally, we have studied our model using quantum
Monte Carlo simulation to obtain accurate results. The comparison of the above
results and also the high temperature series expansion shows that cumulant
expansion gives good results for moderate and high temperature regions while
the modified spin wave theory is good for low temperatures. Moreover, the
convergence regions of the cumulant expansion and the modified spin wave theory
overlap each other which propose these two as a set of complement methods to
get the thermodynamic properties of spin models.
