The magnetic anisotropy of epitaxial Co_N/Cu(111),1 ≤ N ≤ 7, films is investigated in terms of the relativistic spin-polarized screened Korringa-Kohn-Rostoker method by taking into account uniform relaxations of the Co interlayer distance between -4% and +3% with respect to the Cu parent lattice. While the spin-orbit coupling induced (band energy) part of the magnetic anisotropy is found to favor a perpendicular magnetization for N ≤ 2, because of the dominating contribution of the magnetic dipole-dipole interaction to the magnetic anisotropy energy, an in-plane magnetization is energetically preferred for essentially all relaxations and layer thicknesses. Only for N = 2, 3 the anisotropy between an in-plane and a perpendicular orientation of the magnetization is not significantly different. The theoretical results are in good agreement with recent experiments based on pulsed layer deposition.