  Conventional weak-coupling Rayleigh-Schr\"odinger perturbation theory suffers
from problems that arise from resonant coupling of successive orders in the
perturbation series. Multiple-scale analysis, a powerful and sophisticated
perturbative method that quantitatively analyzes characteristic physical
behaviors occurring on various length or time scales, avoids such problems by
implicitly performing an infinite resummation of the conventional perturbation
series. Multiple-scale perturbation theory provides a good description of the
classical anharmonic oscillator. Here, it is extended to study (1) the
Heisenberg operator equations of motion and (2) the Schr\"odinger equation for
the quantum anharmonic oscillator. In the former case, it leads to a system of
coupled operator differential equations, which is solved exactly. The solution
provides an operator mass renormalization of the theory. In the latter case,
multiple-scale analysis elucidates the connection between weak-coupling
perturbative and semiclassical nonperturbative aspects of the wave function.
