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## Nested Radicals Problem

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### Nested Radicals Problem

Mon Oct 14, 2013 8:51 am

Posts: 852
Location: Jaipur, India

Show that

$\displaystyle \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\cdots}}}}=2\sin \left(\frac{\pi}{18} \right)$

### Re: Nested Radicals Problem

Fri Dec 13, 2013 8:06 am

Posts: 38
Location: India, West Bengal
I present a partial solution to the problem for now :

The nested radical repeats after twice positive signs producing a negative sign, i.e., [-, +, +, -]. One can then write is as :

$\displaystyle x = \sqrt{2 - \sqrt{2 + \sqrt{2 + x}}}$

Which gives the reducible octic equation

$\displaystyle x^8 - 8x^6 + 20x^4 - 16x^2 - x + 2 = (x - 2)(x + 1)(x^3 - 3x + 1)(x^3 + x^2 - 2x - 1)$

Some of the roots get outright ruled out, for example, $\displaystyle 2$ or $\displaystyle -1$. The hard part is to prove that the real roots of the fourth factor cannot be a solution which I think can be done by invoking some acceleration methods to the nested radical and then numerically checking each root.

Hence, the real roots of $\displaystyle x^3 - 3x + 1$ are the only possible choice left. I think it wouldn't be hard to show that one root is $\displaystyle 2\sin(\pi/18)$ and is the only value of the nested radical. The desired result should follow.

### Re: Nested Radicals Problem

Sat Jul 16, 2016 9:35 am

Posts: 27
mathbalarka wrote:
I present a partial solution to the problem for now :

The nested radical repeats after twice positive signs producing a negative sign, i.e., [-, +, +, -]. One can then write is as :

$\displaystyle x = \sqrt{2 - \sqrt{2 + \sqrt{2 + x}}}$

Which gives the reducible octic equation

$\displaystyle x^8 - 8x^6 + 20x^4 - 16x^2 - x + 2 = (x - 2)(x + 1)(x^3 - 3x + 1)(x^3 + x^2 - 2x - 1)$

Some of the roots get outright ruled out, for example, $\displaystyle 2$ or $\displaystyle -1$. The hard part is to prove that the real roots of the fourth factor cannot be a solution which I think can be done by invoking some acceleration methods to the nested radical and then numerically checking each root.

Hence, the real roots of $\displaystyle x^3 - 3x + 1$ are the only possible choice left. I think it wouldn't be hard to show that one root is $\displaystyle 2\sin(\pi/18)$ and is the only value of the nested radical. The desired result should follow.

There is no need. Two of the roots are negative and the other one hovers around $1.25$.

Substituting this value of the nested radical into the equation, we obtain:

$$1.55 \approx 2-\sqrt{2+\sqrt{2+\sqrt{2-\cdots}}}$$

$$0.45 \approx \sqrt{2+\sqrt{2+\sqrt{2-\cdots}}}$$

$$0.2 \approx 2+\sqrt{2+\sqrt{2-\cdots}}$$

And we have a contradiction. Therefore, the value we are after does not belong to the solutions of $x^3 + x^2 = 2x +1$