The equation of state for a collection of ideal bosons in both the low-density and high-density regions is found using the method of cluster expansion with a new generating function. The importance of the radius of convergence in the cluster expansion and its connection to the Bose–Einstein condensation phenomenon are studied. The radius of convergence of the partition function is calculated and the values of critical density, fugacity and other thermodynamic properties at condensation are obtained using Mayer’s convergence method.

Bose–Einstein condensations (BEC) for an ideal Bose gas and an imperfect Bose gas are presented using cluster expansion method by using a new generating function obtained by Ushcats. The saturation density is calculated from the known values of virial coefficients for both ideal and uniform hard-sphere imperfect Bose gas. The values of saturation densities are found for some experimentally observed Bose–Einstein condensates and the fractional shift in the saturation densities are also calculated using this method, which are found to be positive.