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

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Homework Statement



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...

Homework 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.


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.
 
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  • #2
SammyS
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Homework Statement



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

and that [itex] s_n<2 [/itex] for all n=1,2,3...
Is this a typo? " where [itex]s_n = \sqrt{2}[/itex] "

Did you mean to write: [itex]s_1 = \sqrt{2}[/itex] instead ?
 
  • #3
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yes you are correct. I fixed the typo
 
  • #4
Ray Vickson
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Homework Statement



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...

Homework 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.


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.
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
 
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