The mean energy for the anharmonic oscillator in the strong-coupling regime

We consider a single anharmonic oscillator with frequency ω and coupling constant λ, respectively, in the strong-coupling regime. We are assuming that the system is in thermal equilibrium with a reservoir at temperature β-1. Using the strong-coupling perturbative expansion, we obtain the mean energy for the oscillator in the regime λ⪢ω, up to order 1/λ.