What are the values of $\alpha \in \mathbb{R}$ for which the series $\sum_1^{\infty} \frac{(-1)^n}{n^{\alpha}} $ converges

Question

What are the values of $\alpha \in \mathbb{R} $ for which the series $$ \sum_1^{\infty} \frac{(-1)^n}{n^{\alpha}} $$ converges?

I've used Liebnitz Test and proved that for all $\alpha \gt 0$ the series converges. Now for $\alpha \lt 0 \ $ I can sense that the sequence
$$\frac{(-1)^n}{n^{\alpha}}$$ becomes oscillatory but how do I formally prove that in this case the series diverges?

Answer 1

If $\alpha\leqslant0$, then$$\left|\frac{(-1)^n}{n^\alpha}\right|=n^{-\alpha}\geqslant1,$$since $-\alpha\geqslant0$. Therefore, you don't have $\lim_{n\to\infty}\frac{(-1)^n}{n^\alpha}=0$, and so your series diverges.