The alternating series test (AST) says, briefly, that if
- $a_k>0$
- $a_k \geq a_{k+1}$, and
- $a_k \to 0$ as $k \to \infty$
then $\sum_k (-1)^k a_k$ converges.
This seems to be a one-way test (that is, if an alternating series fails the test, we don't know that it diverges).
This document says so explicitly. It then gives an example of an alternating series which fails the AST. That doesn't prove the series is divergent (but it turns out to be divergent, anyway, by $n$th term test).
Is there a convergent, alternating series that fails the AST?
Of course, if the series fails condition 3, then it also fails the $n$th term test, and must diverge. And if it fails the first condition, then it's not strictly alternating, anyway.
So it must be a series that is not getting smaller (condition 2), but still converges.
Best Answer
Yes, there are such series. Consider, for example, a sequence such as
$$1, 2, \frac 1 2, 1, \frac 1 4, \frac 1 2, \frac 1 8, \frac 1 4, \dots$$
The series
$$1 - 2 + \frac 1 2 - 1 + \frac 1 4 - \frac 1 2 + \frac 1 8 - \frac 1 4 + \dots$$
is alternating and (absolutely) convergent, but it clearly fails to be monotonically decreasing.