  Recent proofs of the convergence of the linear delta expansion in zero and in
one dimensions 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 $\l$, which is an
important feature of the method, the sequence of approximants $Z_N$ tends to
$Z$ with an error proportional to ${\rm e}^{-cN}$. In the present paper we
establish the convergence of the linear delta expansion for the connected
vacuum function $W=\ln Z$. We show that with the same choice of $\l$ the
corresponding sequence $W_N$ tends to $W$ with an error proportional to ${\rm
e}^{-c\sqrt N}$. The rate of convergence of the latter sequence is governed by
the positions of the zeros of $Z_N$.
