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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).

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    $\begingroup$ Did you mean $a$ as the parameter in the question, not $z$? $\endgroup$ – herb steinberg Feb 6 '19 at 0:57
  • $\begingroup$ 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. $\endgroup$ – Michael Feb 6 '19 at 1:03
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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.
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