Let p>0 and [tex] x = \sqrt{p+\sqrt{p+\sqrt{p+ \cdots }}}[/tex] , where all the square roots are positive. Design a fixed point iteration [tex]x_{n+1} = F (x_{n})[/tex] with some F which has x as a fixed point. We prove that the fixed point iteration converges for all choices of initial guesses greater than -p+1/4.(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]x_{n+1}=F(x_{n})= \sqrt{p+x_{n}}[/tex] so x is a fixed point for F since F(x)=x.

Now let [tex]g(x)=\sqrt{p+x}[/tex].

We have [tex] g'(x)=\frac{1}{2 \sqrt{p+x} } [/tex]

I can see that for [tex] x > -p + 1/4 [/tex], we have that g'(x) <1.

From there I am not sure how to proceed to obtain convergence for [tex] x_{0} > -p +\frac{1}{4} [/tex] .

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# Homework Help: Fixed point iteration

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