TITLE: how can I show this integral diverges?
QUESTION [5 upvotes]: I want to show $E(T_a)=\infty$
$$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$
to show this I need to show this integral diverges.
I know gamma function that
$$\Gamma (\alpha)=\int_0^\infty x^{\alpha -1}e^{-x}dx$$
By benefit from gamma function, how can I show this integral diverges ?
Thank you for helping

REPLY [3 votes]: Since $e^{-a^2/x} \to 1$ at $\infty,$ we have $e^{-a^2/x} > 1/2$ for large $x.$ It follows that the integrand is on the order of $1/\sqrt x$  for large $x.$ Since $\int_b^\infty (1/\sqrt x)\,dx = \infty$ for all $b>0,$ the integral diverges.