  I provide a straightforward proof that a simple harmonic oscillator perturbed
by an (almost) arbitrary positive interaction has a perturbative expansion for
any finite-time Euclidian transition amplitude which obeys the following
result: the difference of the sum of the first N terms of the series and the
exact result is bounded in absolute value by the next term in the series. The
proof makes no assumptions about either the strength of the interactions or the
convergence of the perturbation series. I then argue that the result
generalizes immediately to a much broader class of quantum mechanical systems,
including bare perturbation theory in quantum field theory. The case of
renormalized perturbation theory is more subtle and remains open, as does the
generalization to energy levels and connected S-matrix elements.
