<s>[INST] Find the value of $\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\cdots}}}}}$ . We have:
$\sqrt{(1+0)+\sqrt{(4+1)+\sqrt{(9+2)+\sqrt{(16+3)+\sqrt{(25+4)+\cdots}}}}}.$
Basically I'm not getting any clue at the moment for reducing the infinite nested radicals. Any hint would be helpful. Thanks in advance.
 [/INST] Note that we have the identity
$$n=\sqrt{(n^2-n-1)+(n+1)}$$
Which we can apply indefinitely to give
\begin{align}
2
&=\sqrt{1+3}\\
&=\sqrt{1+\sqrt{5+4}}\\
&=\sqrt{1+\sqrt{5+\sqrt{11+5}}}\\
&=\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+6}}}}\\
&=\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+7}}}}}\\
\end{align}
Note that the $n$th line above differs from the provided expression by an $O(n)$ term in the innermost square root. Due to $n$ square roots this error is reduced to zero as $n\to\infty$.
Edit: As shown above, ignoring some of the first terms gives radical expressions for every natural number. For example
$$3=\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\cdots}}}}$$
$$4=\sqrt{11+\sqrt{19+\sqrt{29+\sqrt{41+\cdots}}}}$$
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