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 ${\mathit{Z}}_{\mathit{N}}$ tends to Z with an error proportional to ${\mathit{e}}^{\mathrm{-}\mathit{c}\mathit{N}}$. 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 ${\mathit{W}}_{\mathit{N}}$ tends to W with an error proportional to ${\mathit{e}}^{\mathrm{-}\mathit{c}\mathrm{\surd }\mathit{N}}$. The rate of convergence of the latter sequence is governed by the positions of the zeros of ${\mathit{Z}}_{\mathit{N}}$.