Crazy Iterated Square Roots

Question

I was messing around with infinite square root nesting problems like $$w_1=\sqrt{1+\sqrt{1+\sqrt{...}}}$$ which is an easy example. I decided to try one where the terms inside of the square roots form a geometric sequence: $$w_2=\sqrt{2^{-1}+\sqrt{2^{-2}+\sqrt{2^{-3}+\sqrt{...}}}}$$ but I can't figure it out. I figured that I probably have to set it equal to $x$ and define $x$ as a function of itself, but I'm not sure how to do that. A similar problem that I did figure out how to do was $$w_3=\sqrt{2^{-1}+\sqrt{2^{-2}+\sqrt{2^{-4}+\sqrt{2^{-8}+\sqrt{...}}}}}$$ because if you multiply it by $\sqrt{2}$, it collapses to a problem of the first form that I mentioned. Any ideas about that second problem?

Answer 1

If $$G_k(x) = (2x)^{k+1} + \sqrt{G_{k+1}(x)},$$ then $G_0(x)$ is the generating function of sequence A274850 and $$\sqrt{G_0(1/4)} = \sqrt{2^{-1}+\sqrt{2^{-2}+\sqrt{2^{-3}+\sqrt{...}}}}$$ but it is unlikely that this has a closed form value. More generally, let $$ f(x,q) = x + \sqrt{f(xq,q)}.$$ We can expand it in a power series $$ f(x,q) = 1 + x\frac{2}{2-q} - x^2\frac{q^2}{(2-q)^2(2-q^2)} + x^3\frac{2q^3}{(2-q)^3(2-q^2)(2-q^3)} + \dots$$ and now $G_0(x) = f(2x,2x).$