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

  • Thread starter zfolwick
  • Start date
  • Tags
    Convergence
In summary, the sequence s_n converges to a limit because it is bounded and increasing. This can be analyzed using the dynamical system x_{n+1} = f(x_n), where f(x) = sqrt(2 + sqrt(x)).
  • #1
zfolwick
36
0

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.
 
Last edited:
Physics news on Phys.org
  • #2
zfolwick said:

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
yes you are correct. I fixed the typo
 
  • #4
zfolwick said:

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
 
Last edited by a moderator:

1. What does the sequence "sqrt(2+sqrt(sn))" converge to?

The sequence sqrt(2+sqrt(sn)) converges to s_n+1, the next term in the sequence.

2. How can we prove that "sqrt(2+sqrt(sn))" converges?

To prove that sqrt(2+sqrt(sn)) converges, we can use the Monotone Convergence Theorem which states that if a sequence is bounded and monotonically increasing, it must converge to a limit. We can also use the Squeeze Theorem to show that the sequence is bounded between two other sequences that converge to the same limit.

3. Is the convergence of "sqrt(2+sqrt(sn))" guaranteed for all values of n?

No, the convergence of sqrt(2+sqrt(sn)) is not guaranteed for all values of n. It depends on the starting value of s_0 and whether the sequence satisfies the conditions for convergence (i.e. being bounded and monotonically increasing).

4. Can we generalize the convergence of "sqrt(2+sqrt(sn))" to other similar sequences?

Yes, the convergence of sqrt(2+sqrt(sn)) can be generalized to other similar sequences. This type of sequence, called a nesting radical sequence, follows the same principles of convergence as the one given in the question. However, the specific limit and conditions for convergence may differ depending on the sequence.

5. How is the convergence of "sqrt(2+sqrt(sn))" useful in mathematics or real-world applications?

The convergence of sqrt(2+sqrt(sn)) has applications in various areas of mathematics, including calculus and number theory. It can also be used in real-world scenarios, such as in the calculation of certain physical quantities or in the analysis of algorithms. Additionally, studying the convergence of this sequence can help us understand the behavior of more complex nesting radical sequences.

Similar threads

  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
913
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
865
  • Calculus and Beyond Homework Help
Replies
20
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
998
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
Back
Top