Example of a general random variable with finite mean but infinite variance

One answer is Pareto distribution with parameters $\alpha , x_0$ which are both positive. The distribution is given by:

$$f_X(x)= \begin{cases} \alpha\,\frac{x_0^\alpha}{x^{\alpha+1}} & x \ge x_0, \\ 0 & x < x_0. \end{cases}$$

Note that $E[X] = \infty$ for $\alpha \leq 1$ and is finite elsewhere.

The variance is not finite for $\alpha \in [1,2) $

Hence it satisfies your question for $(1,2)$

In general,$E[X^n]= \infty \ \ ;n\geq \alpha$.

EDIT: Clarified the answer as suggested.

An example is a random variable $X$ having a student-t distribution with $\nu = 2$ degrees of freedom

Its mean is $E[X] = 0$ for $\nu > 1$, but its second moment $E[X^2] = Var[X] = \infty$ for $1 < \nu \le 2$

---

Edit: Oh wait, finite positive. Well, $X+1$, I guess.