Is an alternating sequence never divergent?

Question

My question:

lets say the sequence $\{a_n\}_{n \in \mathbb{N}}$ is alternating and not against $0$, to avoid the Leibniz criterion.

Is that said sequence always divergent then?

Answer 1

Do you mean:

Suppose $a_n \geq 0$, and $a_n \not \rightarrow 0$. Is is true that the series $\Sigma _{n=1}^\infty (-1)^n a_n$ diverges?

Then the answer is that the series can't possibly converge to a real number, because the terms of a convergent series always converge to zero. No conclusion can be made about if it diverges to $\pm \infty$ or just doesn't converge at all.