When does convergence in $L^p$ imply convergence of the p-th moment? Suppose $X_n$ is a sequence of random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$.
When does convergence in $L^p$,i.e.
$$\mathbb{E}[\vert X_n - X \vert ^p]\rightarrow 0,$$
imply convergence of the p-th moment, i.e.
$$\mathbb{E}[X_n^p] \rightarrow \mathbb{E}[X^p]$$
 A: For the begining we will prove the following lemma

Lemma. Let $p\in\mathbb{N}$ and $X,Y\in L_p(\Omega,\mathcal{F},\mathbb{P})$, then
  $$
\Vert X^p-Y^p\Vert_1\leq \Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\Vert X\Vert_p^k\Vert Y\Vert_p^{p-1-k}\tag{1}
$$

Proof. For $p=1$ ineqality $(1)$ obviously holds. Let $p\geq 2$, then
$$
\begin{align}
\Vert X^p-Y^p\Vert_1
&=\left\Vert(X-Y)\sum\limits_{k=0}^{p-1}X^k Y^{p-1-k}\right\Vert_1
 \leq\Vert X-Y\Vert_p\left\Vert \sum\limits_{k=0}^{p-1}X^k Y^{p-1-k}\right\Vert_q\\
&\leq\Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\left\Vert X^k Y^{p-1-k}\right\Vert_q\\
&=   \Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\left\Vert |X|^{kq} |Y|^{(p-1-k)q}\right\Vert_1^{1/q}\\
\end{align}
$$
We claim that 
$$
\left\Vert |X|^{kq} |Y|^{(p-1-k)q}\right\Vert_1\leq\Vert X\Vert_p^{kq}\Vert Y\Vert_p^{p-kq}\tag{2}
$$ 
Since for any $Z\in L_p(\Omega,\mathcal{F},\mathbb{P})$ we have $\Vert |Z|^{(p-1)q}\Vert_1=\Vert Z\Vert_p^p$, then $(2)$ holds for $k=0$ and $k=p-1$. If $0<k<p-1$, then
consider $r=p/kq>1$ and its adjoint $s=r/(r-1)$. By Holder inequality
$$
\left\Vert |X|^{kq} |Y|^{(p-1-k)q}\right\Vert_1
\leq\Vert |X|^{kq}\Vert_r^{1/r}\Vert|Y|^{(p-1-k)q}\Vert_s^{1/s}
=\Vert |X|^{kqr}\Vert_1^{1/r}\Vert|Y|^{(p-1-k)qs}\Vert_1^{1/s}\\
=\Vert |X|^{p}\Vert_1^{1/r}\Vert|Y|^{p}\Vert_1^{1/s}
=\Vert X\Vert_p^{p/r}\Vert Y\Vert_p^{p/s}
=\Vert X\Vert_p^{kq}\Vert Y\Vert_p^{p-kq}
$$
Thus $(2)$ holds for all $k$. Hence
$$
\begin{align}
\Vert X^p-Y^p\Vert_1
&\leq \Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\left\Vert |X|^{kq} |Y|^{(p-1-k)q}\right\Vert_1^{1/q}\\
&\leq \Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\left(\Vert X\Vert_p^{kq}\Vert Y\Vert_p^{p-kq}\right)^{1/q}\\
&\leq \Vert X-Y\Vert_p\sum\limits_{k=0}^{p-1}\Vert X\Vert_p^{k}\Vert Y\Vert_p^{p-1-k}\\
\end{align}
$$
Now we return the main question.
By assumption we have a sequence $\{X_n:n\in\mathbb{N}\}\subset L_p(\Omega,\mathcal{F},\mathbb{P})$ which converges to $X\in L_p(\Omega,\mathcal{F},\mathbb{P})$, i.e. 
$$
\lim\limits_{n\to\infty}\Vert X_n-X\Vert_p=0\tag{3}
$$
As the consequence there exist $C>0$ such that 
$$
\Vert X_n\Vert_p\leq C\qquad \Vert X\Vert_p\leq C\tag{4}
$$ 
Then from previous lemma and $(4)$ it follows
$$
|\mathbb{E}[X_n^p]-\mathbb{E}[X^p]|
\leq\mathbb{E}[|X_n^p-X^p|]
=\Vert X_n^p-X^p\Vert_1\\
\leq\Vert X_n-X\Vert_p\sum\limits_{k=0}^{p-1}\Vert X_n\Vert_p^{k}\Vert X\Vert_p^{p-1-k}
\leq pC^{p-1}\Vert X_n-X\Vert_p
$$
Now from $(3)$ it follows that
$$
\lim\limits_{n\to\infty}\mathbb{E}[X_n^p]=\mathbb{E}[X^p]
$$
A: Okay I just realized a way to simplify this argument a lot by using a less commonly used version of dominated convergence, recall (Royden Theorem 19, "General Lebesgue Dominated Convergence Theorem"):

Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$. Suppose there is a sequence $\{g_n\}$ of nonnegative measurable functions on $E$ that converges pointwise on $E$ to $g$ and dominates $\{f_n\}$ on $E$ in the sense that $|f_n| \leq g_n$ on $E$ for all $n$. If $\lim \int_E g_n = \int_E g < \infty$, then $\lim \int_E f_n = \int_E f$.

Take $f_n = X_n^p$, $g_n = |X_n|^p$, $g=|X|^p$. We have $E[|X_n|^p] \to E[|X|^p]<\infty$ by triangle inequality. Thus the theorem tells us $E[X_n^p] \to E[X^p]$. And this argument works for all $p \in [1,\infty)$.
