  We present a method for evaluating divergent non-Borel-summable series by an
analytic continuation of variational perturbation theory. We demonstrate the
power of the method by an application to the exactly known partition function
of the anharmonic oscillator in zero spacetime dimensions. In one spacetime
dimension we derive the imaginary part of the ground state energy of the
anharmonic oscillator for {\em all negative values of the coupling constant
$g$, including the nonanalytic tunneling regime at small-$g$. As a highlight of
the theory we retrieve from the divergent perturbation expansion the action of
the critical bubble and the contribution of the higher loop fluctuations around
the bubble.
