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Convergence of sqrt(2+sqrt(sn)) = s_n+1

  1. Nov 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Show convergence of [itex] s_{n+1}= \sqrt{2+\sqrt{s_n}}[/itex] where [itex]s_1 = \sqrt{2}[/itex]

    and that [itex] s_n<2 [/itex] for all n=1,2,3...
    2. Relevant equations

    Let {p_n}be a sequence in metrice space X. {p_n} converges to p iff every neighborhood of p contains p_n for all but a finite number of n.


    3. The attempt at a solution

    I'm only assuming that's the relevant property to know...

    s_n+1 >=s_n so increasing.

    [itex] s_{n+1} > \sqrt{2} [/itex]

    so [itex]\frac{1}{s_{n+1}} <\frac{1}{\sqrt{2}}[/itex]

    but 1/s_n+1 is positive so

    [itex]0< \frac{1}{s_{n+1}} <\frac{1}{\sqrt{2}}[/itex] so it's bounded.

    Since it's bounded and increasing, the sequence is convergent.
     
    Last edited: Nov 8, 2011
  2. jcsd
  3. Nov 8, 2011 #2

    SammyS

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    Is this a typo? " where [itex]s_n = \sqrt{2}[/itex] "

    Did you mean to write: [itex]s_1 = \sqrt{2}[/itex] instead ?
     
  4. Nov 8, 2011 #3
    yes you are correct. I fixed the typo
     
  5. Nov 8, 2011 #4

    Ray Vickson

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    You could also analyze this as the dynamical system x_{n+1} = f(x_n), where f(x) = sqrt(2 + sqrt(x)), using the technique of "cobweb plots"; see, eg.,
    http://www.math.montana.edu/frankw/ccp/modeling/discrete/cobweb/learn.htm [Broken] or http://en.wikipedia.org/wiki/Cobweb_plot .

    RGV
     
    Last edited by a moderator: May 5, 2017
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