For the convergence of an alternating series, the sequence $\{p_n\}$ needs to be a non-negative, monotonically decreasing sequence with a limit of zero.
However, I'm having difficulty thinking of an example where the absence of monotonicity is an issue, i.e.,:
A non-negative sequence with limit zero whose alternating series diverges.
I'm sure there's a rather simple example, but I can't seem to pin one down.
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.
