$\displaystyle A-B=N$, where A, B, and N diverge, but A does not equal N, and B does not equal N.
Is it possible to prove that $\displaystyle N/B$ diverges/ converges?
Please provide a clearer description of your question. The only plausible interpretation I can put on this would make it a series or sequence question wich would be belong in either the calculus or discrete forums and not in number theory.