Finding a closed form expression given decimal approximation

[fedaykin]

Good evening. Is there a way to take a decimal approximation and see if there is a relatively simple expression?
I'm guessing there might be software for this, but I'm not sure I'm even asking the appropriate question.
If it matters, the number I'm after is $$\sqrt{1+\sqrt{2+\sqrt{4+\sqrt{8+\sqrt{16+ ... }}}}}$$ . This is the powers of 2 under a nested radical.

 

[fzero]

Good evening. Is there a way to take a decimal approximation and see if there is a relatively simple expression?
I'm guessing there might be software for this, but I'm not sure I'm even asking the appropriate question.
If it matters, the number I'm after is $$\sqrt{1+\sqrt{2+\sqrt{4+\sqrt{8+\sqrt{16+ \cdots}}}}}$$ . This is the powers of 2 under a nested radical.

You can apply Ramanujan's method to this. Set
$$ t =\sqrt{1+\sqrt{2+\sqrt{2^2+\sqrt{2^3+\sqrt{2^4+ \cdots }}}}}.$$
Then
$$ t^2 = 1 + \sqrt{2} t,$$
and we must take the positive root.

 

[fresh_42]

If I calculate $$\sqrt{2} t$$ then the powers of 2 under the roots increase exponentially:
$$ \sqrt{2+\sqrt{2^3+\sqrt{2^6+\sqrt{2^{11}+\sqrt{2^{20}+ \cdots }}}}}.$$
Am I mistaken somewhere or is $$ t^2 = 1 + \sqrt{2} t$$ meant as an approximation?

 

[fzero]

If I calculate $$\sqrt{2} t$$ then the powers of 2 under the roots increase exponentially:
$$ \sqrt{2+\sqrt{2^3+\sqrt{2^6+\sqrt{2^{11}+\sqrt{2^{20}+ \cdots }}}}}.$$
Am I mistaken somewhere or is $$ t^2 = 1 + \sqrt{2} t$$ meant as an approximation?

The first few terms in the sequence for $t$ are {1., 1.55377, 1.73205, 1.78812}, so this certainly looks convergent. You might try to prove a bound on convergence for the expression where we replace 2 by $n$.

 

[fedaykin]

Thank you guys! It's a lead!

 

[Svein]

You can apply Ramanujan's method to this. Set
$$ t =\sqrt{1+\sqrt{2+\sqrt{2^2+\sqrt{2^3+\sqrt{2^4+ \cdots }}}}}.$$
Then
$$ t^2 = 1 + \sqrt{2} t,$$
and we must take the positive root.

And then it is simple to solve: $t^{2}=1 + \sqrt{2}t$, rearrange: $t^{2}-\sqrt{2}t-1=0$ and solve: $t=\frac{\sqrt{2}\pm\sqrt{2+4}}{2}=\frac{\sqrt{2}}{2}(1\pm \sqrt{3})$. Obviously, we must use the positive sign and get $t=\frac{\sqrt{2}}{2}(1+ \sqrt{3})$ (the numerical value is 1.931852...).

 

[fedaykin]

I've made a nice little excel file to help study these. Interestingly, if you use 4 as a base, it seems to converge to exactly 2. It's the only number so far that I've seen do this. I've tried powers of 2 up to 2^16, and those appear non-rational so far.

If anyone wishes to use it, you'll have to create a module (or some other means) of implementing the following in VBA:
Option Explicit

Public Function goatVal(theInput As String) As Double

goatVal = Evaluate(theInput)

End Function

I can't attach it as macro enabled (wisely), so unfortunately, you'll have to add it manually. The previous code allows you to use the Evaluate function as a standard cell function. That is, it Excel will try to evaluate the contents of a cell as if they were a formula if you put " =goatVal(cellReference) " into them. Be careful, this can take up a decent amount of processor time. It gives me a !value error if the string is getting too large, but seems reasonable otherwise.

 

[HallsofIvy]

The basic idea is this: set $t= \sqrt{1+ \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}}$. Squaring both sides, $t^2= 1+ \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}$. So $t^2- 1= \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}$ and, factoring a $\sqrt{2}$ out of the right, $t^2- 1= \sqrt{2}\sqrt{1+ \sqrt{2+ \sqrt{2^2+ \cdot\cdot\cdot}}}= \sqrt{2}t$.

 

[fresh_42]

The basic idea is this: set $t= \sqrt{1+ \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}}$. Squaring both sides, $t^2= 1+ \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}$. So $t^2- 1= \sqrt{2+ \sqrt{2^2+ \sqrt{2^3+ \cdot\cdot\cdot}}}$ and, factoring a $\sqrt{2}$ out of the right, $t^2- 1= \sqrt{2}\sqrt{1+ \sqrt{2+ \sqrt{2^2+ \cdot\cdot\cdot}}}= \sqrt{2}t$.

I can see this delivers an upper bound and thus proves convergence. What I cannot see, as mentioned in #3, is equality by factoring out $\sqrt 2$. Since all of you insist on the equality I assume I was mistaken. Can somebody enlighten me why $\sqrt 2 t > t^2 - 1$ is wrong?

 

[HallsofIvy]

First, you want to calculate a specific numbers. So, as I and others said, we set t equal that number. How in the world did you convert that equation to an inequality?

 

[fresh_42]

$$\sqrt{2} t = \sqrt{2} \sqrt{1+\sqrt{2+\sqrt{2^2+\sqrt{2^{3}+\sqrt{2^{4}+ \cdots }}}}} = \sqrt{2 \cdot 1 + 2 \cdot \sqrt{2+\sqrt{2^{2}+\sqrt{2^{3}+ \cdots }}}} = \sqrt{2 + \sqrt {4 \cdot 2+ 4 \cdot \sqrt{2^{2}+\sqrt{2^{3}+ \cdots }}}} = \sqrt{2 + \sqrt{2^3 +\sqrt{16 \cdot 2^2 +16 \cdot \sqrt{2^{3}+ \cdots }}}} = \cdots = \sqrt{2+\sqrt{2^3+\sqrt{2^6+\sqrt{2^{11}+\sqrt{2^{20}+ \cdots }}}}}.$$

 

[fedaykin]

One thing I don't quite understand yet:

Evaluating the approximation to these (with several bases) leads to a value that appears to always be less than using Ramanujan's method.
For example: let n = 4 (where n is the base of powers under the radical) and you get:

$$t^2-2t-1=0$$
$$t=1+\sqrt{2} \approx {2.414}$$
$$t= 1 - \sqrt{2} \approx {-0.414}$$

But, if you were to guess at the value from calculation:

$$\sqrt{1+\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{256}}}}} \approx{1.999}$$

By the way, the calculated approximation does not appear to be related to the two solutions from Ramanujan's method. For the number 4, it's just a coincidence that it is the sum. Perhaps these have a very slow rate of convergence and have a 'false positive' convergence to a different value?

To everyone that's been so helpful in this thread, you've brought me a lot of joy in understanding these. I first saw the representation of the golden ratio as the powers of one under the radical, and I've been working on these on and off since college. Thank you so much!

 

[Samy_A]

I also think something is wrong here. For your first expression, the calculated value is $t=\frac{\sqrt{2}}{2}(1+ \sqrt{3})=1.93...$, but your own Excel calculation gives 1.783...
I also did a calculation in Excel, and get the same results: 1.783... for the first expression, 2 for the second expression (the one with 4).

I tried Ramajunan's method on the first expression, factoring out the $\sqrt{2}$, but didn't get the result shown above.

EDIT: here they also find 1.783... as solution, but no closed-form solution.