The alternating series test is used when the terms of the underlying sequence alternate. Suppose we have a series
where the an alternate positive and negative. If an+1 < an (i.e., the terms get smaller) and if
then the series converges.
If a series Σ | an | converges then the series Σ an converges and is said to converge absolutely. If Σ | an | diverges but Σ an converges, then Σ an is said to converge conditionally.
The applet shows the series
called the alternating harmonic series because its terms alternate sign:
The harmonic series diverges, but maybe the minus signs change the behavior in this case.
The alternating series test requires that the an alternate sign,
get smaller and approach zero as n approaches infinity, which is true in this case.
So this series does converge and is said to converge conditionally.
While the harmonic series diverges, the alternating harmonic series has
enough negative numbers in it to counterbalance the growth, resulting in convergence.