Moment generating function of $f(x)=\frac{1-\cos(x)}{\pi x^2}$ does not exist

Question

How to prove that $\int_{\mathbb{R}}\exp(tx)\frac{1-\cos(x)}{x^2}dx$ is not finite for any fixed $t$?

Thank you!

Answer 1

Let $t>0$. Since $1-\cos x\ge0$ we have for any $k\in\mathbb{N}$ we have $$ \int_{k\pi}^{(k+1)\pi}\exp(t\,x)\frac{1-\cos x}{x^2}\,dx\ge\frac{e^{k\pi t}}{(k+1)^2\pi^2}\int_{k\pi}^{(k+1)\pi}(1-\cos x)\,dx=\frac{e^{k\pi t}}{(k+1)^2\pi} $$ and $$ \sum_{k=1}^\infty\frac{e^{k\pi t}}{(k+1)^2}=+\infty. $$