Problem on $L^p$ space where $f(x)=x^{-1/n} \in L^p([1,\infty))$ is true or false

Question

Prove or Disprove : $f(x)=x^{-1/n} \in L^p([1,\infty))$ and find $\|f\|_p $ if it exists .

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I think if $\int_1^{\infty} \left(f(x)\right)^p\, dx <\infty $ then $f\in L^p([1,\infty]).$ Now $\int_1^{\infty} \left(x^{-1/n}\right)^p\, dx=\int_1^{\infty}x^{-p/n}\, dx $ Now what is $p/n > or < 1 $ here , where $1<p<\infty $

Answer 1

You think right. So if $p>n$ the integral converges, otherwise it diverges. In case it converges, it is easy to compute the integral. It equals $\frac{n}{p-n}$, so the norm is that number raised to the power $1/p$.