Suppose $X_i$ are a sequence of random variables such that
$$X_i \xrightarrow{d} X \sim \mathrm{Gamma}(k,\theta),$$
where $d$ denotes convergence in distribution. Does necessarily
$$\mathbb{E}[X_i^m] \xrightarrow{n \to \infty} \mathbb{E}[X^m],$$
where $m \in \mathbb{N}$?
Suppose $X\sim\text{Gamma}(k, \theta)$. Also, consider another random variable $U\sim\text{Unif}(0,1)$ that is independent of $X$. Define $X_n$ as: \begin{eqnarray*} X_n = \begin{cases} X & \text{when } \frac{1}{n} < U < 1 \\ 2n & \text{when } 0 < U \leq \frac{1}{n} \end{cases}\end{eqnarray*}
Notice that $$X_n \xrightarrow{d} X \sim \mathrm{Gamma}(k,\theta)$$
but $\mathbb{E}(X_n) = \left(1-\frac{1}{n}\right)\mathbb{E}(X) + \left(\frac{1}{n}\right)2n \rightarrow \mathbb{E}(X) + 2$.