Consider a renewal process [N(t), t>0]. For fixed t > 0 and each n ≥ 1, let yn,1, …, Yn,n be independent exponentials each having mean t/n, independent of the renewal process. Ross [2] developed a recursion for the sequence of approximations mn = EN(Yn,1 + … + Yn,n) that converges to m(t)if the renewal function m(·) = EN(·) is continuous at t > 0. In this note, we derive an upper bound on the rate of convergence of this sequence under mild conditions on m near t. Tightness of this bound is discussed in terms of regularity conditions on m.