For the bosonic ideal gas, one must solve the equations
in order to obtain the equation of state. Examination of these equations,
however, shows an immediate problem: The term 
is
divergent both for the pressure and the average particle number.
These terms need to be treated carefully, and so we split them
off from the rest of the sum, giving:
where 
means that the term is excluded. With these
divergent terms split off, the thermodynamic limit can be taken and the
remaining sums converted to integrals as was done in the fermion case.
Thus, for the pressure, we find
where the change of variables
has been made. Using the expansion
the pressure equation becomes
and by a similar procedure, the average particle number becomes
In this equation, the term that has been split off represents the
average occupation of the ground ( ) energy state:
Since 
must be greater than or equal to 0, it can be
seen that there are restrictions on the allowed values of 
.
Firstly, since 
, must be a positive number.
However, in order that the average occupation of the ground state be
positive,
from which it follows that
The fact that as 
causes to diverge
will have interesting consequences to be discussed below. However,
let us first consider the low density limit with 
.