Monotonicity in alternating Series
Consider the sequence $$a_n = \begin{cases}\dfrac2{n+1} & \text{if }n \text{ is odd}\\ \dfrac4{n^2} & \text{if } n \text{ is even}\end{cases}$$ Now consider the series $$S = \sum_{n=1}^{\infty}(-1)^{n-1} a_n = \dfrac11 - \dfrac1{1^2} + \dfrac12 - \dfrac1{2^2} + \dfrac13 - \dfrac1{3^2} \pm \cdots = \sum_{n=1}^{\infty} \left(\dfrac1n - \dfrac1{n^2} \right) \tag{$\spadesuit$}$$ The divergent series $(\spadesuit)$ is an alternating series and the individual terms tend to zero as $n \to \infty$, but not in a monotonic way.