Find $E(X^{2k-1})$ for $k = 1, 2, 3, 4, ...$
Find $E(X^{2k})$ for $k = 1, 2, 3, 4, ...$
I used the fact that: $e^x = 1+x+ \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} +....$ to get:
$e^{\frac{t^2}{2}} = 1+\frac{t^2}{2}+ \frac{{(\frac{t^2}{2})^2}}{2!} + \frac{{(\frac{t^2}{2})^3}}{3!} + \frac{{(\frac{t^2}{2})^4}}{4!} +....$. I don't know how I use this information to solve for the expected values
Hint
$$\mathbb E[X^k]=\left.\frac{\mathrm d ^k}{\mathrm d t^k}\right|_{t=0}M(t).$$
So, if $$M(t)=\sum_{n=0}^\infty a_nx^n,$$ then $$\mathbb E[X^k]=k!a_k.$$