Finite path integral limits work in cases where the perturbative series is not Borel summable

The perturbative expansion in powers of the coupling of observables in quantum field theory and quantum mechanics is known to yield an asymptotic series. If the original physical system is well-behaved and a finite observable is expected, this can often be calculated via a Borel resummation of the asymptotic series. However, there are cases where a system is well-behaved and the series is not Borel summable. This typically occurs when the physical system has a non-trivial vacuum structure. It has recently been shown that if the perturbative series is carried out under finite path integral limits, one can obtain a convergent series that yields observables even at strong coupling. This was recently used to obtain the energy at strong coupling for the anharmonic oscillator. This is a Borel summable case so the question is whether finite path integral limits work when the series is not Borel summable. To begin answering this question we consider a simple non-Borel summable case: the series stemming from a basic integral where the function has a double-well shape and hence two minima. The integral has an exact analytical expression that the series can be compared to. Under finite integral limits that run from $-L$ to $L$, where $L$ is finite, positive and real, we develop two perturbative series in powers of the coupling: one by expanding the integral about the local maximum at the origin and the other by expanding it about one of the minima. In both cases, we obtain an absolutely convergent series and the series sums to the exact analytical expression of the original integral in the infinite $L$ limit. It is significant that a perturbative expansion about one of the minima reproduces the exact analytical expression because this implies that it captures the full effect of both minima.