Higher-degree moment tensor representation of seismic sources is obtained for a horizontally layered homogeneous medium. A Taylor series expansion of Green’s function around a reference source position and time is made, and this enables us to approximate the seismic radiation in both the regional and teleseismic distances through a sequence of terms representing increasingly detailed aspects of the source behavior. Source coefficients and orientation factors of the first- and second-degree moment tensors are obtained. The representation is applied to a unilateral rupture Haskell fault model, and the synthetic seismograms of different models calculated by the higher-degree moment tensors are compared with the theoretical solutions for a propagating source. Our results show that, the representation of higher- degree moment tensors up to degree 2 can describe the response of a moving source well enough, and it’s possible to use the moment series as a tool for calculating seismograms from finite and propagating faults in the forward sense. The computation takes much less time than the method of summing point sources over the fault surface. The information yielded by the higher-degree moments may solve problems such as the fault-plane ambiguity and the space–time evolution of the rupture propagation of an earthquake.