<s>[INST] Nested radical $\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$ I am studying the $f(x) = \sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$ for $x \in (0,\infty)$ and I am trying to get closed form formula for this, or at least some useful series/expansion. Any ideas how to get there?
So far I've got only trivial values, which are
$$f(1)=\sqrt{1+\sqrt{1+\cdots\sqrt{1}}}=\frac{\sqrt{5}+1}{2}$$
$$f(4) = 3$$
Second one follows from 
$$2^n+1 = \sqrt{4^n+(2^{n+1}+1)} = \sqrt{4^n+\sqrt{4^{n+1}+(2^{n+2}+1)}} = \sqrt{4^n+\sqrt{4^{n+1}+\cdots}}$$
I have managed to compute several derivatives in $x_0=1$ by using chain rule recursively on $f_n(x) = \sqrt{x^n +  f_{n+1}(x)}$, namely:
\begin{align*}
f^{(1)}(1) &= \frac{\sqrt{5}+1}{5}\\
f^{(2)}(1) &= -\frac{2\sqrt{5}}{25}\\
f^{(3)}(1) &= \frac{6\sqrt{5}-150}{625}\\
f^{(4)}(1) &= \frac{1464\sqrt{5}+5376}{3125}\\
\end{align*}
These gave me Taylor expansion around $x_0=1$ 
\begin{align*}
T_4(x) &= \frac{\sqrt{5}+1}{2} + \frac{\sqrt{5}+1}{5} (x-1) - \frac{\sqrt{5}}{25} (x-1)^2 + \frac{6\sqrt{5}-150}{3750} (x-1)^3 \\
&\ \ \ \ \ + \frac{61\sqrt{5}+224}{3125} (x-1)^4
\end{align*}
However this approach seems to be useful only very closely to the $x=1$. I am looking for something more general in terms of any $x$, but with my limited arsenal I could not get much further than this. Any ideas?
This was inspiring but kind of stopped where I did 
http://integralsandseries.prophpbb.com/topic168.html
Edit:
Thanks for the answers, i will need to go through them, looks like the main idea is to divide by $\sqrt{2x}$, so then I am getting
$$\frac{\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}}{\sqrt{2x}} = \sqrt{\frac{1}{2}+\sqrt{\frac{1}{4}+\sqrt{\frac{1}{16x}+\sqrt{\frac{1}{256x^4}+.‌​..}}}}$$
Then to make expansion from this. This is  where I am not yet following how to get from this to final expansion.
 [/INST] You can obtain an expansion in negative powers of $x$, valid for large $x$, by trying to find $$\frac{f(x)}{\sqrt{2x}}$$.  If you pull the $\frac{1}{\sqrt{2x}}$ into the square roots (note that it gets raised to progressively higher powers as it is pulled into more inner square roots) you get an expansion of the form $$f(x) = \sqrt{2x} + \frac{\sqrt{2}}{8} + \frac{\kappa}{\sqrt{x}} + \cdots$$
where calculating $\kappa$ is straightforward but a bit messy (the value I get is $-\frac{3\sqrt{2}}{32}$ but numerically it looks to be about $-\frac{1}{18}$).  Even the first order expression (without the $\frac{\kappa}{\sqrt{x}}$ term) is already accurate to an error of smaller than $0.0052$ for all $x>4$. 
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