If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so that for any continuous bounded function $f,$ we have $\mathbb{E}f\left(\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right)\to\mathbb{E}f(W)$ where $W\sim\mathcal{N}(0,1).$ Now, $|\cdot|$ is not a bounded function, so it is not necessarily true that

$$\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|\to\mathbb{E}|W|.$$

My question is whether the above is true for this specific distribution of $Z_i.$ If not, what does $\mathbb{E}\left|\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right|$ converge to?