Limiting free energy per particle for Ising Model by approximating its functional integral

There have been a lot of methods aimed at studying the limiting free energy per particle (LFEPP) for 3-dimensional (3D) Ising model in absence of an external magnetic field. These methods are elegant, but most of them are complicated and often require specialized knowledge and special skills. Here we approximate the LFEPP for Ising model from its functional integral using classic mathematical-physical methods. The resulting LFEPPs for 1-dimensional (1D) to 3D Ising model have similar structures and forms. We then verify that these LFEPPs are correct for two limiting cases of the 1D and 2-dimensional (2D) models, as well as for the critical inverse temperature $z_c$ of the 2D model. Based on these verifications, we derive naturally the LFEPP and the $z_c$ ($\approx 0.21\sim 0.22$) for the 3D model. Furthermore, we suggest similar LFEPPs for 1D-3D Ising models with an external magnetic field, although they are too complicated.