Standard Normal MGF and Even Moments

Let $X \sim N(0,1)$. 1. Show that the moment generating function is $M_X(t) = e^{t^2/2}$, and use it to derive a recursion for the even moments $E[X^{2n}]$ in terms of $E[X^{2n-2}]$. 2. Compute $E[X^2]$, $E[X^4]$, and $E[X^6]$ explicitly. 3. In a quantitative research context, explain how you would use these moment calculations to sanity-check simulated samples that are purportedly standard normal (e.g., from a random number generator), using hypothesis tests.