Moments of Pareto($\alpha$)

Question

I'm trying to show that a random variable $X\sim\text{Pareto}(\alpha)$ for some $\alpha>1$, i.e. $$P(X>x)=x^{-\alpha},\quad x>1$$ has infinit moment generating function $$\mathbb{E}[e^{hX}]=\infty$$ for all $h>0$. How come?

Thank you in advance.

Answer 1

The density function is$$f_X(x)=\begin{cases}\frac\alpha{x^{\alpha+1}},&x>1\\0,&\text{otherwise}\end{cases}$$

Let $t=\lceil\alpha \rceil+1\ge\alpha+1$. Then $t\in\Bbb Z_{\ge0}$, thus for $x>1,h>0$,$$e^{hx}=1+hx+\frac{h^2x^2}{2!}+...\ge\frac{h^tx^t}{t!}$$This gives

$$\begin{align*}E[e^{hX}]&=\alpha\int_1^\infty\frac{e^{hx}}{x^{\alpha+1}}dx\\&\ge\frac{h^t\alpha}{t!} \int_1^\infty\frac{x^t}{x^{\alpha+1}}dx\\&=\frac{h^t\alpha}{t!(t-\alpha)}[x^{t-\alpha}]_1^\infty\to\infty\end{align*}$$