Let (u(n)) be a sequence of real numbers. In this paper, we give new Tauberian conditions imposed on the general control modulo of the oscillatory behavior of integer order m >= 1 of a sequence (u(n)), under which, convergence follows from the Abel summability. These are generalizations of some of the well-known classical Tauberian conditions. (C) 2011 Elsevier Ltd. All rights reserved.