Can I always choose an $N$ such that $\int_{|x|>R}\frac{1}{|x|^{m+N}}dx$ converges?

Question

Given $R>0$ and $m\geq 0$ can I choose an $N$ such that the integral in $\mathbb{R}^n$

$$\int_{|x|>R}\frac{1}{|x|^{m+N}}dx$$

converges? And how can I see this? I am essentially looking for an analogy of $p$-integrals in $\mathbb{R}^n$, ie for what powers does the above integral converge?

Answer 1

By polar coordinates $$\int_{|x|>R}\frac{1}{|x|^{m+N}}dx =c_n \int_R^\infty \frac{r^{n-1}}{r^{m+N}}dr =c_n \int_R^\infty \frac{1}{r^{m+N-n+1}}dr$$

which converges if and only if $\color{red}{m+N-n>0}$