  We present and discuss a detailed derivation of a new analytical method that
systematically improves the convergence of path integrals of a generic $N$-fold
discretized theory. We develop an explicit procedure for calculating a set of
effective actions $S^{(p)}$, for $p=1,2,3,...$ which have the property that
they lead to the same continuum amplitudes as the starting action, but that
converge to that continuum limit ever faster. Discretized amplitudes calculated
using the $p$ level effective action differ from the continuum limit by a term
of order $1/N^p$. We obtain explicit expressions for the effective actions for
levels $p\le 9$. We end by analyzing the speedup of Monte Carlo simulations of
two different models: an anharmonic oscillator with quartic coupling and a
particle in a modified P\"oschl-Teller potential.
