Show that the Cauchy distribution, whose density is $f\left(x\right)=\frac{1}{\left[\pi\left(1+x^{2}\right)\right]}$ does not possess finite moments.

Question

This from Hoel's Statistical Book: Show that the Cauchy distribution, whose density is $f\left(x\right)=\frac{1}{\left[\pi\left(1+x^{2}\right)\right]}$ does not possess finite moments.

I've been reading the recommended section for this problem, but I have no clue of how to solve it.

Answer 1

$\int_R\frac{|x^n|}{1+x^2}dx$ does not exist for $n\ge 1$, since integrand ~$x^{n-2}$ as $x\to \infty$.