$\sum k^{-p}$ and $\int_\varepsilon^\infty x^{-p}\,dx$ converge exactly for $p>1$ and $\int_0^1 x^{-p}\,dx$ exactly for $p<1$; in $\RR^n$ the threshold for $\abs{x}^{-p}$ is $p=n$.Suggest a slogan
Let \(p\in\RR\), let \(\varepsilon>0\), let \(n\geq1\), and let \(B \coloneqq \theset{x\in \RR^n \suchthat \abs{x} \leq 1}\) be the closed unit ball for the Euclidean norm. Then \[ \begin{aligned} \sum_{k=1}^\infty \frac 1 {k^p} < \infty &\iff p>1, \\ \int_\varepsilon^\infty \frac {dx} {x^p} < \infty &\iff p>1, \\ \int_0^1 \frac {dx} {x^p} < \infty &\iff p<1, \\ \int_B \frac{dx}{\abs{x}^p} < \infty &\iff p < n, \\ \int_{\RR^n\setminus B} \frac{dx}{\abs{x}^p} < \infty &\iff p > n . \end{aligned} \]