The series $\sum n^{-p}$ converges exactly beyond the threshold $p=1$.Suggest a slogan
For \(p\in\RR\), the series \(\sum_{n\ge1} n^{-p}\) converges if and only if \(p>1\).
If \(p\le0\), then \(n^{-p}\ge1\) for all \(n\), so the terms do not tend to \(0\) and the series diverges. If \(p>0\), the function \(x\mapsto x^{-p}\) is positive and decreasing on \([1,\infty)\), so by the integral test the series converges if and only if \(\int_1^\infty x^{-p}\dx\) is finite, which holds if and only if \(p>1\).