Lebesgue integration of 1/{|x|^{\alpha}}

Question

I would like to know for which values of $\alpha$, the following statement holds: $$ \int_{\mathbb{R}^n}\dfrac{1}{\Vert x\Vert^{\alpha}}<\infty. $$ Thanks!

Answer 1

For no $\alpha$! Either integral diverges on $\{\|x\| \leq 1\}$ or the integral diverges on $\{\|x\| > 1\}$