I am having trouble with the following question.
For which values of $n, p$ is the integral $\int_{\mathbb{R}^n}\frac{1}{|x|^p}$ convergent? I need to derive a relationship between $n$ and $p$, i.e. $p>n$.
The only thing I seem to have proved is that for n=1 this is always divergent, however I believe this is wrong in higher dimensions. How shall I figure it out? Thank you very much.
You can use the polar integration formula to get $$\int_{\mathbb R^n} \frac{dx}{\lvert x \rvert ^p} dx = C_n \int_0^\infty \frac{r^{n-1}}{r^p} \, dr = C_n\int_0^\infty r^{n-1-p} dr.$$ For convergence near $r=0$ we need $n-1-p > -1$ and for convergence as $r \to \infty$ we need $n-1-p < -1$ so indeed, there are no values of $p,n$ that work. In order to be able to find such $p,n$ you need to either cut out a neighborhood of $0$ or restrict to a bounded domain.