A numerical method is presented for rapidly calculating the energy eigenvalues of one-dimensional Schrödinger equations. It is applicable to systems for which the potential is either analytic or has no pole of order greater than two. The method is based on a power-series expansion of the wave function at large distances. With the use of high-speed computing machines the large number of terms required in the power series can be computed easily. The method is illustrated by obtaining energy eigenvalues for a number of one-dimensional systems with potentials of the type V=kx2n/2n. It is also applicable to a variety of systems of physical interest. As an example, an exact energy eigenvalue for a rotating Morse oscillator has been calculated. This is compared with that obtained from Pekeris' approximate analytical solution.

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