I am studying the topic of MGF and have come across the result that odd moments of $N(0,1)$ are $0,$ yet I cannot see why.
Given $\mu_k$ the $k$-th moment and from the fact that:
\begin{equation} M(t)=e^{t^2/2} = \sum_{k=0}^{\infty}\frac{t^{2k}}{2^k k!} \end{equation} They claim that
Odd powers of $t$ are all $0$
Therefore $\mu_k = 0$ and $\mu_{2k} = \frac{(2k)!}{2^k k!}$
I cannot see why that is true.
Thanks a lot for any hint.