sequence of random variables converge to $X$ in probability with all expectation zero, but $\mathbb{E}[X] = \infty$.

I'm currently working on the following problem:Construct a sequence of random variables $X_n$ and $X$ such that $X_n \overset{\mathbb{P}}{\rightarrow} X$ and $\mathbb{E}[X_n] = 0$ for all $n$ but $\mathbb{E}[X] = \infty$.

I can imagine random variables with infinite expectation like $\mathbb{P}[X] = \frac{1}{n(n+1)}$ for all $n \in \mathbb{N}$ but struggle to see how to construct a sequence that converges to $X$ while meeting the criteria.