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For many nonlinear physical systems, approximate solutions are pursued byconventional perturbation theory in powers of the non-linear terms.Unfortunately, this often produces divergent asymptotic series, collectivelydismissed by Abel as "an invention of the devil." An alternative method, theself-consistent expansion (SCE), has been introduced by Schwartz and Edwards.Its basic idea is a rescaling of the zeroth-order system around which thesolution is expanded, to achieve optimal results. While low-order SCEs havebeen remarkably successful in describing the dynamics of non-equilibriummany-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergenceproperties have not been elucidated before. To address this issue we apply thistechnique to the canonical partition function of the classical harmonicoscillator with a quartic $gx^{4}$ anharmonicity, for which perturbationtheory's divergence is well-known. We obtain the $N$th order SCE for thepartition function, which is rigorously found to converge exponentially fast in$N$, and uniformly in $g\ge0$. We use our results to elucidate the relationbetween the SCE and the class of approaches based on the so-called"order-dependent mapping." Moreover, we put the SCE to test against othermethods that improve upon perturbation theory (Borel resummation,hyperasymptotics, Pad\'e approximants, and the Lanczos $\tau$-method), and findthat it compares favorably with all of them for small $g$ and dominates overthem for large $g$. The SCE is shown to successfully capture the correctpartition function for the double-well potential case, where no perturbativeexpansion exists. Our treatment is generalized to the case of many oscillators,as well as to any nonlinearity of the form $g|x|^{q}$ with $q\ge0$ and complex$g$. These results allow us to treat the Airy function, and to see thefingerprints of Stokes lines in the SCE.

A Deal with the Devil: From Divergent Perturbation Theory to an Exponentially-Convergent Self-Consistent Expansion