Regularity of the powers of an ideal

For a homogeneous ideal I of S = k[X-o,...,X-o] we compare the Castelnuovo-Mumford regularities of the powers of I to the regularity reg I of I itself. We show that if dim S/I = 0 and I is generated in degrees at most d then reg I-k less than or equal to (k-1)d + reg I, and that if dim S/I less than or equal to 1 then reg I-k less than or equal to k reg I. To do this, we use results on generic initial ideals.