I tried to write down the proof for this series. Getting the value is easy assuming convergence. But I couldn't formally prove that it converges using monotone threorem. The series is increasing for some values of $z$ and decreases for some values of $z$, but it is always monotone. How to find proceed now since we don't know if it increases or decreases? Please provide a formal solution.
Q.Let $a>0$ and let $z_1>0$. Define $z_{n+1}=\sqrt{a+z_n} $for n$\in > N$. Show that the limit $(z_n)$ converges and find the limit (where a is just $a$ positive constant).