The tangent bundle TkM of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k. For a Banach manifold M and a natural number k first we determine a smooth manifold structure on TkM which also offers a fiber bundle structure for (πk, TkM,M). Then we introduce a particular lift of linear connections on M to geometrize TkM as a vector bundle over M. More precisely based on this lifted nonlinear connection we prove that TkM admits a vector bundle structure over M if and only if M is endowed with a linear connection. Also, using the projective limit techniques, we declare a generalized Fr´echet vector bundle structure for T∞M over M.
