Convergence proof for optimized $\delta$ expansion: Anharmonic oscillator

A recent proof of the convergence of the optimized $\delta$ expansion for one-dimensional non-Gaussian integrals is extended to the finite-temperature partition function of the quantum anharmonic oscillator. The convergence is exponentially fast, with the remainder falling as $e^{-c{N}^{\frac{2}{3}}}$ at order $N$ in the expansion, independently of the size of the coupling or the sign of the mass term. In particular, the approach gives a convergent resummation procedure for the double-well (non-Borel-summable) case.