Abstract: Let be an array of random variables and be an array of constants. A new concept of integrability (call residually -integrability) for an array of random variables with respect to an array of constants is introduced, which is weaker than other related notions of integrability, such as -integrability, -integrability. Under this assumption of integrability and appropriate conditions on the array of weights, we investigate strong convergence and mean convergence for weighted sums of dependent random variables. Some related results in literature are extended and improved.