Absolute, conditional or no convergence

$p>1$. $\sum\frac1{n^p}$ converges, so the series converges absolutely.

$0<p\le1$. $\sum\frac1{n^p}$ diverges, so there is no absolute convergence. But $\frac1{n^p}$ is decreasing with limit $0$, so by the alternating series test the series converges. It is conditionally convergent.

$p\le0$. The terms $\frac{(-1)^n}{n^p}=(-1)^nn^{|p|}$ do not tend to $0$ (their absolute value is $\ge1$), so the series diverges.

Summary.
- $p>1$: absolutely convergent,
- $0<p\le1$: conditionally convergent,
- $p\le0$: divergent.

Remark. For $p=1$ the sum is $-\ln2$: a conditional convergence that is deeply sensitive to the order of the terms.