A simple way of approximating the canonical partition functions in statistical mechanics

We propose a simple pedagogical way of introducing the Euler-MacLaurin summation formula in an undergraduate course on statistical mechanics. We put forward two alternative routes: the first one is the simplest and yields the first two terms of the expansion. The second one is somewhat more elaborate and takes into account all the correction terms. We apply both to the calculation of the simplest one-particle canonical partition functions for the translational, vibrational and rotational degrees of freedom.