There appears to be a lot of expertise amongst the members of this site regarding solving nested radicals, so I thought I'd share a challenging one for your radical enjoyment:

\(f(x) = \displaystyle\sqrt{x + \sqrt{\frac{x}{2} + \sqrt{\frac{x}{4} + \sqrt{\frac{x}{8} + \sqrt{\frac{x}{16} + \sqrt{\frac{x}{32} + ......}}}}}}\).

The hope is that there is an exact solution, if only for \(f(1)\) if not for \(f(x)\) in general. I suppose one interesting feature of this function is that \((f(x))^{2} - x = f(\frac{x}{2})\). I'm sure that there are many more interesting features waiting to be discovered.

## Comments

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TopNewestYour problem does not have closed form but

\[\sqrt{ x + \sqrt{\frac{x}{2} + \sqrt{\frac{x}{4} + \sqrt{\frac{x}{16} + \sqrt{\frac{x}{256} + \ldots}}}}}\]

Can have a closed form – Krishna Sharma · 2 years, 4 months ago

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– Brian Charlesworth · 2 years, 4 months ago

It does? Cool. I'll have to figure out what that is, then.Log in to reply

– Krishna Sharma · 2 years, 4 months ago

I was solving your radical and did a mistake and I solved the above radical :p, now I will post a problem on this ;)Log in to reply

– Brian Charlesworth · 2 years, 4 months ago

Haha. Well, a lot of "mistakes" have led to interesting discoveries. Ill keep an eye out for your problem.Log in to reply

My bet is that there isn't any closed form expression for this, not even in the special case of \(x=1\). – Michael Mendrin · 2 years, 4 months ago

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– Brian Charlesworth · 2 years, 4 months ago

You're probably right, but I'm getting used to seeing rabbits being pulled out of hats so I thought I'd post the problem just in case.Log in to reply

f(x) = 2sqrt(x) – Coby Tran · 2 years, 3 months ago

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For higher degree of radicals f(x)= nth root of 2x – Amit Tripathi · 2 years, 4 months ago

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Could it be written in this fashion?

\(f(x) = \sqrt{\sum_{i=0} \frac{f(x)}{2^{i}}} \) – Giovanni GuessWhat · 2 years, 4 months ago

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– Saptakatha Adak · 2 years, 4 months ago

no.. I think the 'x' terms under summation are missing and 'i' should start from 1 instead of 0.. if I'm not wrong.Log in to reply

– Giovanni GuessWhat · 2 years, 4 months ago

It's a recursive function. I thought "i" should start from 0 since the first term is "x", not "x/2"Log in to reply