The alternating series test is a sufficient condition for the convergence of a numerical series. I am searching for a counterexample for its inverse: i.e. a series (alternating, of course) which converges, but for which the hypothesis of the theorem are false.
In particular, if one writes the series as $\sum (-1)^n a_n$, then $a_n$ should not be monotonically decreasing (since it must be infinitesimal, for the series to converge).
On
If you want a conditionally convergent series in which the signs alternate, but we do not have monotonicity, look at $$\frac{1}{2}-1+\frac{1}{4}-\frac{1}{3}+\frac{1}{6}-\frac{1}{5}+\frac{1}{8}-\frac{1}{7}+\cdots.$$ It is not hard to show that this converges to the same number as its more familiar sister.
Put: $$ b_n = \begin{cases} n^{-2} &: n \text{ odd} \\ 2^{-n} &: n \text{ even} \end{cases} $$
$b_n$ is not monotonically decreasing. Still, $\sum (-1)^n b_n$ converges.