For a real-valued r.v. $\xi$ we suppose and existence of its moment generating function $m(t) = \mathsf E\mathrm e^{t\xi}$ for all $t\in (-h,h)$ where $h>0$. I wonder how many moments $M_n = \mathsf E\xi^n$ are finite.
I know that using the differentiation under the Lebesgue integral, local existence of $m(t)$ implies an existence of $M_1 = \mathsf E\xi$. On the other hand for the 2nd moment it seems that the finiteness of $M_2$ is equivalent to the statement that $m''(0)$ exists, hence there should be examples when $m(t)$ in the neighborhood of $0$ while $M_2 = \infty$.
For an example when $M_2$ does exist without an existence of $m(t)$ for $t>0$ we can consider a r.v. with a density $ f(x) = \frac{2}{\pi(1+x^2)^2}. $
Could you provide such an example when $m(t)$ exists in the neighborhood of $t=0$ but $M_2 = \infty$? Maybe you can also refer me to the literature since this question is not covered in my book on the probability.