The asymptotic rates of convergence of thermodynamic properties with respect to the number of Fourier coefficients, k(max), included in Fourier path integral calculations are derived. The convergence rates are developed both with and without partial averaging for operators diagonal in coordinate representation and for the energy. Properties in the primitive Fourier method are shown to converge asymptotically as 1/k(max) whereas the asymptotic convergence rate is shown to be 1/k(max)(2) when partial averaging is included. Properties are shown to converge at the same rate whether full partial averaging or gradient partial averaging is used. The importance of using the proper operator to optimize convergence rates in partial averaging calculations is emphasized.