Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges.
A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt p(x)$ for all $x\gt m$.
Take all functions $f(x)$ from $S$ and put $O(f(x))$ in $S2$.
Which function $g(x)$ in $S2$ dominates all others?
Are there asymptotic lower and upper bounds on $g(x)$?