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October 25th, 2018, 09:00 PM  #1 
Newbie Joined: Oct 2018 From: Indonesia Posts: 4 Thanks: 0 Math Focus: Nonlinear Waves and Differential Equations  Monotone Convergence Theorem and Cauchy Convergence Criterion
Hello. I would be grateful for anyone who can solve this problem for me : Show that the sequence $\left(z_n\right)$ given by $z_{n+1}=\sqrt{a+z_n}$ for $a>0, z_1>0$ converges by using Monotone Convergence Theorem (or Cauchy Convergence Criterion)! 
October 25th, 2018, 10:14 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 502 Thanks: 280 Math Focus: Dynamical systems, analytic function theory, numerics 
Here is a hint: Apply the mean value theorem to successive terms \[z_{n+1}  z_{n} = \sqrt{a + z_n}  \sqrt{a + z_{n1}} \leq \frac{ z_n  z_{n+1}}{2\sqrt{a}} \] 
October 25th, 2018, 10:15 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra 
$$\left.\begin{aligned}z_{n+1} &= \sqrt{a+z_n} \\ z_1 &\gt 0 \end{aligned} \right\} \implies z_{n} \gt 0 \; \forall n \ge 1$$ This means that we can compare the magnitude of $z_{n+1}^2$ and $z_n^2$ as proxies for the magnitude of $z_{n+1}$ and $z_n$. \begin{align} z_{n+1}^2 &= a+z_n \\ z_{n+1}^2  z_n^2 &= a + z_n  z_n^2 \end{align} The quadratic expression on the right factorises as $\big(b  z_n\big)\big((b1) + z_n\big)$ where $b(b1)=a$ which has a unique solution with $b \gt 1$ because $a \gt 0$. Thus we have three cases:
Now, we compare the magnitude of $z_{n+1}^2$ and $b^2$ as proxies for $z_{n+1}$ and $b$. \begin{align}z_{n+1}^2  b^2 &= a + z_n  b^2 \\ &= b(b1) + z_n b^2 \\ &= z_n b\end{align} Here,
Therefore:
Thus:
Last edited by v8archie; October 25th, 2018 at 10:18 PM. 
October 25th, 2018, 11:06 PM  #4 
Newbie Joined: Oct 2018 From: Indonesia Posts: 4 Thanks: 0 Math Focus: Nonlinear Waves and Differential Equations 
Thanks for such a great hint, SDK. I don't think I could use the hint for now, but of course it would be useful in my next course of advanced real analysis.

October 25th, 2018, 11:09 PM  #5 
Newbie Joined: Oct 2018 From: Indonesia Posts: 4 Thanks: 0 Math Focus: Nonlinear Waves and Differential Equations 
Thank you very much, v8archie. This is what I am looking for.


Tags 
cauchy, convergence, criterion, monotone, theorem 
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