  Based on the Green's function (GF) equation-of-motion formalism, we develop a
method to expand the double time Green's function into Taylor series of the
parameter $\lambda$ in the Hamiltonian $H=H_0 + \lambda H_1$. Here $H_0$ is the
exactly solvable part and $H_1$ is regarded as the perturbation. To restore the
analytical structure of GF, we use the continued fraction to do resummation for
the obtained series. The problem of zero-temperature divergence is identified
and remedied by the self-consistent series expansion. To demonstrate the
implementation of this method, we carry out the weak- as well as the
strong-coupling expansion for the Anderson impurity model to order $\lambda^2$.
Improved result for the local density of states is obtained by self-consistent
second-order strong-coupling expansion and continued fraction resummation.
