I have this problem from Proakis book: Let $X$ denote a Gaussian random variable with mean equal to zero and variance equal to $\sigma^2$ . Show that: If $n=2k+1 \rightarrow E[X^n] = 0$ else $E[X^n] = 1\times3\times5\times...\times(n-1)\sigma^n$.
The Solution, start defining $I_n= \int_{-\infty}^{\infty} x^n e^{-x^2/2\sigma^2} dx$ and say: $\frac{\partial^2 I_n}{\partial x^2} = n(n-1)I_{n-2}-\dfrac{2n+1}{\sigma^2}I_n +\dfrac{1}{\sigma^4}I_{n+2}=0 \rightarrow I_{n+2}=\sigma^2(2n+1)I_n-\sigma^4n(n-1)I_{n-2}$
But I don´t understand because the second derivate is 0. The book said $E[x^n]=0$ when $n$ is even, but this doesn't have sense. The remaining of the solution I understand.