Monotonicity in Alternating Series Test

Alternating series test states that if {$x_n$} is a decreasing sequence converging to $0$, then $\sum_{n=1}^{\infty}(-1)^{n+1}x_n$ converges. Monotonicity is important because otherwise examples such the one here can be constructed, where $\lim_{x \rightarrow \infty}x_n = 0$ but the series diverges. I wanted to know how to construct these examples. Does this have to do with rearrangement? In the link the person gives an example, but I am not sure what thinking to have when constructing these examples.

• You want the sum of positive elements to diverge to infinity faster then the sum of negative elements or to have that sum of negative elements doesn't converge. – kingW3 Apr 1 '18 at 23:03