The mathematical foundation of the methods using addition theorems to evaluate multicenter integrals over Slater-type orbitals is actually well understood. However, many numerical aspects of such approaches still require further investigations. In the framework of these methods, multicenter integrals are generally represented by infinite series which under certain circumstances are very slowly convergent. Accordingly, the determination of the convergence type of such series is of great importance since it allows one to choose adequately the convergence accelerator to be used in the summation procedure. In this work, the convergence of the two-range addition theorem proposed by Barnett and Coulson [Philos. Trans. R. Sec. London, Ser. A 243, 221 (1951)] is analyzed. The results obtained from this study are then applied to study the convergence of three-center nuclear integrals, and most importantly, to discuss the choice of the convergence accelerator to be used in the summation procedure.