An=Bn-Cn Bn and Cn are both diverging Does An converge??
Definitely not!
Consider $\displaystyle B_n=n^2 \text{ and} C_n=5n^4$
Clearly the difference diverges as well. The difference can sometimes lead to a converging series but you need to use convergence tests to verify. An example of such a case is a telescoping series, where almost all the terms cancel out and the remaining terms sum to a finite value.