The energy eigenvalues of bound states of an electron in the general exponential cosine screened Coulomb potential are obtained using the shifted 1/N expansion method. The energies for the states from 1s to 8k are calculated from six to eight significant figures. The energy eigenvalues for the 1s, 2s - 2p, 3s - 3d, and 4s - 4f states are also presented as a function of the screening parameter lambda. Results are compared with the ones obtained by other workers. The agreement reduces roughly for large lambda. It is also observed that the convergence of the expansion series increases remarkably as l increases.