Let \(K(a_ n(x)/b_ n(x))\) be a continued fraction, where \(a_ n(x)\) and \(b_ n(x)\) are polynomials with nonnegative coefficients, in a real variable x. Let the continued fraction correspond at \(x=0\) to a formal power series in x or at \(x=\infty\) to a formal power series in \(x^{- 1}\). Conditions are given which insure that the corresponding series are asymptotic expansions of the functions to which the odd and even parts of the continued fraction converge as \(x\to 0+\) or as \(x\to +\infty\).