# Find the exact value of expression $S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$

Let $$S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$$ Is it possible to write $S$ in terms of standard mathematical functions and operators? If yes, what is the exact value of $S$? If not, can it be proved?

• This looks really close to this question math.stackexchange.com/questions/837189/… – Darth Geek Jul 22 '14 at 21:40
• Is there any solution? – user164524 Jul 22 '14 at 21:43
• Not that I'm aware of. – Darth Geek Jul 22 '14 at 21:45
• One would be better off folding these into one question, I suppose---if not for the fact that I really have no expectation of their being a solution... – Semiclassical Jul 22 '14 at 22:33
• – Martin Sleziak Feb 3 '17 at 13:40