The following alternating series fails the alternating series test with flying colors
$$\sum_{n=1}^{\infty} (-1)^n\dfrac{\cos^2\left(n\right)}{n^2}.$$
This series is however is absolutely convergent, which is easy to show.
In addition, you can take many absolutely convergent series which do pass the test and create one which does not:
Given $\ \sum\limits_{n=1}^{\infty} (-1)^n b_n\ $ where $\sum\limits_{n=1}^{\infty} b_n$ converges, and $b_n < b_{n-1}$ for all $n$. Make a new series by defining $a_{2n-1} = b_{2n}$ and $a_{2n}=b_{2n-1}$. Then $\{a_n\}$ is not decreasing, as for all $n$, $a_{2n-1}=b_{2n}<b_{2n-1}=a_{2n}$. However, because the original series is absolutely convergent
$$\sum\limits_{n=1}^{\infty} (-1)^n b_n = \sum\limits_{n=1}^{\infty} (-1)^n a_n$$