From the alternating series test, you know that if and if decreases monotonically to zero, then converges. However, it is not enough to have having a limit of zero, you also need decreasing, as the following example shows.
Take your favorite convergent series with positive terms, say , and take your favorite divergent positive term series whose terms go to zero, say . Now ``shuffle'' these together to form the following series:
This alternates, the terms go to zero, but the terms are not decreasing monotonely to zero. This series also diverges. The divergent harmonic series overpowers to force the sum off to . (This can be made rigorous by looking at partial sums).