If you add a convergent sequence {A_k} to a divergent sequence {B_k} will the sum converge or diverge?

Conan has a good explanation. For a rigorous proof that the sum diverges, recall that sums and differences of two convergent series is convergent. Suppose {A_k} converges and {A_k + B_k} converges. Then the difference {A_k + B_k} - {A_k} = {B_k} converges. Thus, is {B_k} diverges and {A_k} converges, it must be that the sum diverges.