Limit of an inverse sequence

Let $(u_n)$ be a sequence defined by $$u_1 = \frac{1}{2}, \qquad u_{n+1} = \frac{\sqrt{u_n^2 + 4u_n} + u_n}{2} \quad (n \in \mathbb{N}^*).$$ Let $v_n = \displaystyle\sum_{k=1}^{n} \frac{1}{u_k^2}$ for all $n \in \mathbb{N}^*$. Prove that the sequence $(v_n)$ has a finite limit and find that limit.