Feynman path integral methods based on the Trotter approximation represent paths by a set of P discrete points. We prove that the M-point partition function is an upper bound of the P-point one if M is a divisor of P. Also for this case, we show that, during calculations using P-point paths, it is possible - at negligible additional cost -to obtain M-point estimators of the partition function that, for each individual path, converge monotonically. This permits accurate extrapolation to infinite P, which greatly improves the accuracy of calculations of thermodynamic quantities. (C) 2003 Published by Elsevier B.V.