Let $(\Omega, \mathcal{A}, P)$ be a probability space. Let $(g_n)_{n=1}^\infty$ be i.i.d. And let $g_n:\Omega \to \mathbb{R}$ be a random variable that follows the standard normal distribution.
It is clear that $E(\sum_{n=1}^\infty|g_n|^2) = \infty$. However, I wonder whether $E(\sum_{n=1}^\infty |g_n|) = \infty$ or it is finite. If $E(\sum_{n=1}^\infty |g_n|) < \infty$, then I want to know the threshold of $p \in (1, 2)$ suth that $E(\sum_{n=1}^\infty |g_n|^p) < \infty$.