The comparison test provides a way to use the convergence of a series we know to help us determine the convergence of a new series. Suppose we have two series
where 0 ≤ an < bn. Then if B converges, so does A. Also, if A diverges, then so does B. So if we suspect that a series A converges, we can try to find a similar series B where the terms are all bigger than the terms of A and where B is known to converge, thus proving that A converges.
Conversely, if we have a series B that we suspect diverges, we can try to find a similar series A where the terms are all smaller than the terms of B and where A is known to diverge, thus proving that B diverges.