  We present a new analytical method that systematically improves the
convergence of path integrals of a generic $N$-fold discretized theory. Using
it we calculate the effective actions $S^{(p)}$ for $p\le 9$ which lead to the
same continuum amplitudes as the starting action, but that converge to that
continuum limit as $1/N^p$. We checked this derived speedup in convergence by
performing Monte Carlo simulations on several different models.
