For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory (PT) in powers of the nonlinear term. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "the invention of the devil.'" Here we analyze an alternative method, called the Self-Consistent Expansion (SCE), due to Schwartz and Edwards. Its basic idea is an optimal and dynamical rescaling of the zeroth system around which the solution is expanded. It can be seen as a procedure to systematically improve upon the variational approximation. While low-order SCE calculations have been remarkably successful in describing the dynamics of nonequilibirium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this, we study the canonical partition function of a classical harmonic oscillator with a quartic g*x^4 anharmonicity, for which PT's divergence is well-known. We find the explicit Nth-order SCE of the partition function, and derive analytical error bounds. We prove that SCE converges exponentially fast in N and uniformly in g>0, regardless of the coupling strength. Numerically, SCE outperforms other asymptotic techniques, especially in the strong-coupling g>>1 regime. SCE is shown to converge even in the double-well potential case, where no PT exists. Our results are generalized to a g*|x|^q perturbation with any real q>0 and complex g. This allows us to treat the Airy function as a representative WKB application, and observe the manifestation of Stokes lines in the SCE formulation. We also propose extensions of the method to the many-oscillators and quantum cases.