limit
In a manner completely analogous to what was done for the fermion case, the
low density limit can be treated by perturbation theory. Note that
if is not close to 1, then the divergent terms, which
have a 
prefactor accompanying them, will vanish in the thermodynamic
limit. Thus, for the proceeding analysis, these terms can be neglected.
As before, we assume the fugacity can be expanded as
Then the equation for the density becomes
By equating like powers of 
on both sides, the coefficients
can be determined as they were for the fermion gas.
Working to first order in gives
and the equation of state is
which is just the classical ideal gas equation. To second order, however, we find
and the second order equation of state becomes
The second virial coefficient can be read off and is given by
Interestingly, in contrast to the fermionic system, the pressure is actually decreased from its classical value as a result of bosonic spin statistics. Thus, it appears that there is an ``effective attraction'' between the particles. This fact is not entirely unexpected, given that any number of bosons can occupy the same quantum state.