0
$\begingroup$

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).

$\endgroup$
  • 3
    $\begingroup$ Did you mean $a$ as the parameter in the question, not $z$? $\endgroup$ – herb steinberg Feb 6 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 at 1:03
2
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.