A dominated convergence theorem for Eisenstein series

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp tau = 0. As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane