EM Gaussian beams are the most celebrated and used kind of laser beams. Their description beyond paraxial regimes has a long and venerable history, culminating may be with the building of a scheme of approximations which can be named the Davis scheme of approximations whose convergence has been considered as granted. Strange as it may be, a paper by Wang and Webb demonstrated that, actually, the Davis scheme is divergent. This quite unexpected result has been dramatically overlooked. This is the motivation for the present paper which reviews diverging and converging schemes of approximations to describe fundamental EM Gaussian beams. One of the new results obtained in the present framework is that a scheme of approximations known as the improved standard scheme, introduced more than two decades ago, is diverging as well. These divergences are the result of the behavior of asymptotic series similar to the ones encountered in quantum electrodynamics.