The polarizations of prompt J/psi and psi(2S) mesons are measured in proton-proton collisions at root s = 7 TeV, using a dimuon data sample collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 4.9 fb(-1). The prompt J/psi and psi(2S) polarization parameters lambda, lambda, and lambda, as well as the frame-invariant quantity (lambda) over bar. are measured from the dimuon decay angular distributions in three different polarization frames. The J/psi results are obtained in the transverse momentum range 14 < P-T < 70 GeV, in the rapidity intervals |y| < 0.6 and 0.6 < |y| < 1.2. The corresponding psi(2S) results cover 14 < PT < 50 GeV and include a third rapidity bin, 1.2 < |y| < 1.5. No evidence of large polarizations is seen in these kinematic regions, which extend much beyond those previously explored. (C) 2013 CERN. Published by Elsevier B.V. All rights reserved.