We can see that if(adsbygoogle = window.adsbygoogle || []).push({});

[itex]u=\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}[/itex]

then [itex]u^2=x+u[/itex]

so [itex]u^2-u-x=0[/itex]

This has solution

[itex]\left( u-\frac{1}{2} \right)^2 -\frac{1}{4}-x=0 \Rightarrow u=\frac{1}{2} \pm \sqrt{x + \frac{1}{4}}[/itex]

This means that [itex]u \in \mathbb{R} \forall x \geq \frac{1}{4}[/itex]

In other words [itex]\sqrt{ -\frac{1}{8} + \sqrt{ - \frac{1}{8} + \sqrt{-\frac{1}{8} + \dots}}}[/itex] is real.

This is clearly true according to the above formula. However, I cannot get my head around it - to me it seems like it must be imaginary! Can anyone give an explanation of why this is turning out to be real?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Nested Radicals

**Physics Forums | Science Articles, Homework Help, Discussion**