Let $\mathbf X = (X_1, . . . , X_n)$ consist of independent and identically Normal $N(0, θ)$ random variables, with mean $0$ and variance $θ > 0.$
The density of $f_{\theta}(\mathbf x)$ is equal to $$\frac{1}{(2\pi \theta)^{\frac{n}{2}}} e^{\frac{1}{2 \theta} \sum^n_{i=1}x^2_i}$$
So, I have to show that the MLE for $\theta$ is $\frac{1}{n}\sum^n_{i=1}X^2_i$.
I have showed that the derivative of the log-likelihood function is equal to $0$ when $\theta$ is evaluated at $\frac{1}{n}\sum^n_{i=1}X^2_i$. Then I showed that it is the maximum by showing the second derivative is less than $0$ . Then by computing the limit of the likelihood function, when $\theta \rightarrow \infty$ I showed that it's $0$. Then only thing left to do is to compute the limit when $\theta \rightarrow 0$. But when I compute it, for some reason it diverges to $\infty$. Am I doing it wrong?