  The expansion of the partition function for large coordination number $Z$ is
a long standing method and has formerly been used to describe the Ising model
at finite temperatures. We extend this approach and study the interacting Bose
gas at finite temperatures. An analytical expression for the free energy is
derived which is valid for weakly interacting and strongly interacting bosons.
The transition line which separates the superfluid phase from Mott
insulating/normal gas phase is shown for fillings $\langle\hat n\rangle=1$ and
$\langle\hat n\rangle=2$. For unit filling, our findings agree qualitatively
with Quantum Monte-Carlo results. Contrary to the well-known mean-field result,
the shift of the critical temperature in the weakly interacting regime is
apparent.
