For a given N and τ, this equation can be solved for τ and then a series solution for z can be found by the method of inversion of series, either in powers of τ or as an asymptotic expansion in inverse powers of τ. From these expansions, we can find the behavior of the gas near T = 0 and in the Maxwell–Boltzmann as T approaches infinity. In particular, we are interested in the limit as N approaches infinity, which can be easily determined from these expansions.