Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists?
I find it hard when it comes to generalize this statement in $R^n$. I've been able to do something for $n=2$ and $n=3$, always stating $r^2=|x|$, and working with polar or spherical coordinates, but it's imposible to keep going, obiously. Any ideas?
Using $n$-dimensional polar coordinates we have:
$$\int_{B}\frac{1}{\lvert x\rvert^{\alpha}}dx=\int_{\mathbb{R}^{n}}\frac{1}{\lvert x\rvert^{\alpha}}\chi_{B}(x)dx=\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}\frac{1}{r^{\alpha}}\chi_{B}(r)r^{n-1}d\mu dr=\bar{\omega}\int_{0}^{1}r^{n-\alpha-1}dr$$
where $\bar{\omega}$ denotes the volume of the $n$-dimensional unit ball. Notice that the integral is finite provided that $n-\alpha-1>-1$ which occurs iff $n>\alpha$.