Any truncated-path-integral partition function of a nonrelativistic quantum system in thermodynamic equilibriumone obtained by means of the Feynman path-integral-procedure using a finite number of such integralsis known to have a value not less than that of the exact one corresponding to it. A rigorous asymptotic lower bound obtained for the relative disparity in their valuesthe difference in their values divided by that of the exact partition function confirms asymptotic positive-definiteness of the original upper bound. Values determined directly for a linear harmonic oscillator agree asymptotically with values of they bound.