  We present a method for extracting tunnelling amplitudes from perturbation
expansions which are always divergent and not Borel-summable. We show that they
can be evaluated by an analytic continuation of variational perturbation
theory. The power of the method is illustrated by calculating the imaginary
parts of the partition function of the anharmonic oscillator in zero spacetime
dimensions and of the ground state energy of the anharmonic oscillator for all
negative values of the coupling constant $g$ and show that they are in
excellent agreement with the exactly known values. As a highlight of the theory
we recover from the divergent perturbation expansion of the tunnelling
amplitude the action of the instanton and the effects of higher loop
fluctuations around it.
