CRLB for estimating $\theta$ of $\sim\text{Geo}(\theta)$

I'd like to ask if the below computation of the information number for the CRLB is correct:

Consider $x_1, x_2$ as iid $\sim Geo(\theta)$

Since $x_1, x_2$ ae iid and the geometric distribution is part of the exponential family, the information number is

$-nE_\theta(\frac{d^2}{d\theta^2}lnf(x|\theta))=-2E_\theta[\frac{d^2}{d\theta^2}ln(\theta(1-\theta)^{x-1})]=\frac{\theta^2(1-\theta)}{2}$

Assuming the Geometric paramaterization you're using with $E[X] = \frac{1}{\theta}$, this simplifies to $- \frac{2}{\theta^2(\theta-1)}$.