Convergence of the optimized δ expansion for the connected vacuum amplitude: Zero dimensions

Recent proofs of the convergence of the linear {δ} expansion in zero and one dimension have been limited to the analogue of the vacuum generating functional in field theory. In zero dimensions it was shown that with an appropriate, N-dependent, choice of an optimizing parameter {λ}, which is an important feature of the method, the sequence of approximants ZN tends to Z with an error proportional to e^-cN. In the present paper we establish the convergence of the linear {δ} expansion for the connected vacuum function W=lnZ. We show that with the same choice of {λ} the corresponding sequence WN tends to W with an error proportional to e^-c0ex0exN. The rate of convergence of the latter sequence is governed by the positions of the zeros of ZN.