In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth function. Verifying the conditions of the test shows why the series converges uniformly on closed intervals that do not contain any multiples of 2[pi]. We investigate the problem both graphically and numerically before proceeding to an analytical solution. (Contains 1 table and 3 figures.)