Closed form of an infinitely nested radical.

In summary: No, I don't have any reason to believe there is an analytically exact solution, but I am willing to try and guess the solution, then prove that it is equal to my nested radical.
  • #1
Pzi
26
0
Hello.

Does anybody happen to know a closed form of this infinitely nested radical?
http://imageshack.us/a/img268/6544/radicals.jpg [Broken]
By any chance, maybe you even saw it somewhere?

I haven't had too much success so far. At the moment I am so desperate that I'm even willing to try and guess the solution, then prove that it is equal to my nested radical. For any real positive x the limit indeed exists (various criteria can be found for that very reason). Numerical limits can be seen in the plot:
http://imageshack.us/a/img842/876/plote.jpg [Broken]

Also here is a convergence plot:
http://img100.imageshack.us/img100/64/convergence.jpg [Broken]
It is made in a sense that using double precision variables computer sees no difference between a_{k}(x) and a_{k+1}(x) which in turn means that ~16 decimal digits have already been found. In fact it's so nasty that a{6}(50000) - a{5}(50000) < 10^(-24).
Notably the bigger my argument, the faster it converges (although I'm not sure what useful conclusions I can draw from that).

Pretty much the only known elegant cases: a(1) is equal to golden ratio, a(4)=2.

What would you suggest?



Pranas.
 
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  • #2
Hi Pzi! :smile:

You can write:
$$\sqrt{x^1 + \sqrt{x^2 + ...}} = \sqrt x \cdot \sqrt{x^0 + \sqrt{x^1 + ...}} = \sqrt x \cdot a(x)$$

If you substitute that in the expression for a(x), you get an equation that you can solve...
 
  • #3
I like Serena said:
Hi Pzi! :smile:

You can write:
$$\sqrt{x^1 + \sqrt{x^2 + ...}} = \sqrt x \cdot \sqrt{x^0 + \sqrt{x^1 + ...}} = \sqrt x \cdot a(x)$$

If you substitute that in the expression for a(x), you get an equation that you can solve...

Not really.
Your idea requires powers like 1, 2, 4, 8, 16, 32... whereas we actually have 1, 2, 3, 4, 5, 6...
 
  • #4
Still looking for any insightful ideas!
 
  • #5
I get a good fit to 1.19 x1/4
 
  • #6
Pzi said:
Still looking for any insightful ideas!

You say you are desperate for a solution. Do you have any reason to believe there is an analytically exact solution?
 

1. What is a closed form of an infinitely nested radical?

The closed form of an infinitely nested radical is an expression that represents the value of a nested radical with infinitely many terms. It is a solution that can be written using only basic arithmetic operations and finite numbers.

2. How is the closed form of an infinitely nested radical different from a regular nested radical?

The closed form of an infinitely nested radical is different from a regular nested radical because it represents the exact value of the expression, while a regular nested radical only approximates the value. The closed form is also a finite expression, while a regular nested radical has an infinite number of terms.

3. Can all infinitely nested radicals be expressed in closed form?

No, not all infinitely nested radicals can be expressed in closed form. In order for a nested radical to have a closed form, it must meet certain criteria, such as having a finite number of terms and a specific pattern in its terms. Some infinitely nested radicals do not meet these criteria and therefore cannot be expressed in closed form.

4. How is the closed form of an infinitely nested radical calculated?

The closed form of an infinitely nested radical is calculated by using mathematical techniques such as the method of undetermined coefficients or the method of Lagrange resolvents. These methods involve manipulating the expression to reveal a pattern or relationship between the terms, which can then be used to find a closed form solution.

5. What are some common examples of infinitely nested radicals with closed form solutions?

Some common examples of infinitely nested radicals with closed form solutions include the square root of 2, the golden ratio, and the cube root of 2. These expressions have closed form solutions because they have a finite number of terms and follow a specific pattern in their terms, allowing for a closed form to be calculated.

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