Abstract
Combinatorial relations can be used to convert the non-relativistic time-sliced Feynman path integral into perturbation expansions. These methods reveal that when the time interval is sliced into N increments, each order of perturbation theory sustains an error O(1/ square root N). In this way the author provides exact path integral results for the following potentials: delta-function comb, finite well, tunnelling barrier, and a generalized exponential cusp. For the tunnelling barrier it is seen how the celebrated (-1) reflection factor arises in the limit of infinite barrier height. The one-dimensional Coulomb problem is solved as a limiting case of the exponential cusp. In addition, for power potentials the author indicates how this path integral approach yields sometimes divergent, nevertheless asymptotic perturbation expansions.