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).