# Monotone series increases and decreases depending on first term. What is the limit?

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).

• Did you mean $a$ as the parameter in the question, not $z$? – herb steinberg Feb 6 at 0:57
• Please demonstrate some effort. What have you tried? If you assume convergence, can you obtain the value it converges to? Now if $z_1$ is bigger or smaller than that value, it will determine the direction we go. – Michael Feb 6 at 1:03

## 1 Answer

Hint. Don't use monotonicity, go back to the definition.

• Find the limit $$L$$, assuming it exists.
• Check that $$L>1$$.
• Use basic algebra to show that $$|z_{n+1}-L|<\frac1L\,|z_n-L|$$.
• Explain how this solves the problem.