Example of Leibniz alternating series criterion for convergence?

Question

As I understand, the two conditions for an alternating series to converge using the Leibniz criterion are:

1. Absolute value of sequence terms is monotome decreasing and

2. Limit of the terms of the sequence to infinity is 0.

I initially thought that the second condition must necessarily imply the first condition, so I was wondering why both conditions were stated; however I now appreciate that these two conditions are seperate, although I cannot think of an example that will help me to wrap my head around this.

Does anyone have any examples of an alternating series for which the limit of the sequence formed by the terms is zero but the terms are not monotome decreasing and therefore the series fails the Leibniz criterion and diverges?

Answer 1

$$ \sum_{n=2}^\infty\frac{(-1)^n}{n^p+(-1)^n} $$ diverges if $0<p\le1/2$ and converges if $p>1/2$.