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[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?