Let A be a densely defined closed operator, and {A(n)}, {B(n)} be two sequences of bounded operators on a Grothendieck space X with the Dunford-Pettis property such that {A(n) - I} is uniformly power bounded, B(n)A subset-of AB(n) = I - A(n), A(n)A subset-of AA(n), \\AA(n)X\\ --> 0 for x is-an-element-of X and \\A(n)*A*x*\\ --> 0 for x* is-an-element-of D(A*). If {A(n)} converges strongly on X, then both {A(n)} and {B(n)\R(A)} converge uniformly. Implications of this result in the cases of discrete semigroups, n-times integrated semigroups and cosine operator functions are then described.
