Iterative square root? sqrt(2+sqrt(2+sqrt(

In summary, the conversation discusses nested radicals and their properties. It is mentioned that they are also known as Nested Radicals and references are provided for further reading. The conversation also touches on the topic of the Gamma function and the values of its roots. It is suggested to use a trick involving rewriting the equation and taking the square root to find the roots. However, it is noted that the limit must exist for this trick to work and it is important to consider extended real numbers.
  • #1
Damidami
94
0
The other day I was playing with my calculator and noticed that

[tex]\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} \approx 2[/tex]

But, what is that kind of expression called? How does one justify that limit?
And, to what number exactly does converge, for example:

[tex]\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}} \approx 1.6161[/tex]

[tex]\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+...}}}} \approx 2.3027[/tex]

Any references where I could read about these subjects?

Another question. Considering real [tex]x>1[/tex], we have:
[tex]\Gamma(x) - x^1 = 0[/tex] then [tex] x \approx 2[/tex]

But how does one justify that? And what are the exact values of these functions:

[tex]\Gamma(x) - x^2 = 0[/tex] then [tex] x \approx 3.562382285390898[/tex]
[tex]\Gamma(x) - x^3 = 0[/tex] then [tex] x \approx 5.036722570588711[/tex]
[tex]\Gamma(x) - x^4 = 0[/tex] then [tex] x \approx 6.464468490129385[/tex]

Thanks,
Damián.
 
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  • #2
They are called Nested Radicals. There are references in the link. I am unfortunately not familiar with their theory.
 
  • #3
slider142 said:
They are called Nested Radicals. There are references in the link. I am unfortunately not familiar with their theory.

Thank you slider142!
That answered my first question.
Does anyone know about my second question? Or any further references?
Thanks,
Damián.
 
  • #4
[tex]\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = x[/tex]

For something like this, you can rewrite the equation as

[tex]\sqrt{2+x} = x[/tex]

And then the infinite equation is captured in a finite form. From there, you simply square both sides.

[tex]2+x = x^2[/tex]

And solve for x.

But I don't know much more than that! Don't forget that square-roots are non-negative.
 
  • #5
For nested radicals of the form

[tex]\sqrt{a + \sqrt{a + \sqrt{a + \dots}}}[/tex]

using the trick

[tex]\sqrt{a + x} = x[/tex]

works very well. Two roots will emerge, but only one is positive (the other is extraneous).

In the case when a = 2, x = 2.

When a = 1, then x = [itex]\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618034[/itex] a.k.a. the golden ratio.

As to the second question regarding the Gamma function, I'm not sure much theory is available.

--Elucidus
 
  • #6
For a dose of rigor -- we have to be sure the limit really exists before we can compute it with such tricks!

In this case, it's easy: the value is the limit of an increasing sequence, and limits of increasing, extended real number-valued sequences always exist.

It's important to notice that extended real numbers come into play here! The equation
2 + x = x²​
has three relevant solutions: -1, 2, and [itex]+\infty[/itex]. We know the limit exists, so it has to have one of those three values. It's easy to rule out -1, but more work is needed to decide between 2 and [itex]+\infty[/itex].
 
  • #7
If we examine the sequence [itex]\{a_n\}_{n=0}^{\infty}[/itex] when [itex]a_0 = \sqrt{2}[/itex] and

[tex]a_{n+1}=\sqrt{2+a_n}[/tex]

Then it is possible to show by induction that [itex]a_n \leq 2 [/itex] for all n so the [itex]+\infty[/itex] case is impossible.


But you are correct, this possibility does need to be ruled out, Hurkyl.

--Elucidus
 
Last edited:

1. What is iterative square root?

Iterative square root is a mathematical process in which the square root of a number is repeatedly calculated until a desired level of accuracy is achieved. This is done by using an initial guess and refining it through a series of iterations until the difference between the calculated value and the actual value is within an acceptable margin of error.

2. How is iterative square root different from regular square root?

Regular square root involves using a mathematical formula to directly calculate the square root of a number. Iterative square root, on the other hand, uses an iterative process to refine an initial guess until the desired level of accuracy is achieved.

3. What is the purpose of using iterative square root?

The purpose of using iterative square root is to achieve a more accurate approximation of the square root of a number. It is often used in situations where the exact value of the square root is difficult to calculate, such as with irrational numbers.

4. How many iterations are needed to calculate the square root using this method?

The number of iterations needed to calculate the square root using iterative square root varies depending on the initial guess and the desired level of accuracy. In general, the more iterations that are performed, the more accurate the approximation will be.

5. Can iterative square root be used for all types of numbers?

Yes, iterative square root can be used for all types of numbers, including positive, negative, rational, and irrational numbers. However, the number of iterations needed may vary depending on the type of number being calculated.

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