We study the behaviour of truncated Rayleigh-Schrödinger series for the low-lying eigenvalues of the one-dimensional, time-independent Schrödinger equation, in the semiclassical limit ℏ→0. Under certain hypotheses on the potential V(x), we prove that for any given small ℏ>0 there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and exact eigenvalue is smaller than exp (-C/ℏ) for some positive constant C. We also prove the analogous results concerning the eigenfunctions.