I have an iterative equation defined by $\theta_k$ $$ 2\theta_{k+1} = -\theta_k^2 + \sqrt{ \theta_k^4 + 4\theta_k^2} $$ and $\theta_0 = 1$.
How do I show that then $\theta_k(\theta_{k-1}^{-1} - 1)$ asymptotically equals $1-3/k + O(1/k^2)$
I know $\theta_k$ tends to zero and so $\theta_k(\theta_{k-1}^{-1} - 1)$ which is
$$ \dfrac{\sqrt{ \theta_k^2 + 4} - \theta_k}{2} - \dfrac{-\theta_k^2 + \sqrt{ \theta_k^4 + 4\theta_k^2}}{2} $$
tends to 1.
Then I'm not too sure how to calculate the $\dfrac{1}{k}$ coefficient. It seems those similar questions are well studied so have clever tricks that work but not sure it translates to this case easily.
