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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?

Martin Sleziak
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    This looks really close to this question http://math.stackexchange.com/questions/837189/evaluating-the-sum-lim-n-to-infty-sqrt22-sqrt32-sqrt42-cdots-s – Darth Geek Jul 22 '14 at 21:40
  • Is there any solution? –  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
  • I think that there is no solution. But, I want to see the proof. –  Jul 22 '14 at 22:55
  • Would you be satisfied by some evidence/reasoning of why this problem appears analytically intractable? – Semiclassical Jul 22 '14 at 23:22
  • The question of whether a "closed form answer" exists depends on what you admit as closed forms. For example, if one accepts zeros of arbitrary hypergeometic functions, it might be very difficult to prove that a given expression has no closed form value. I would modify this problem to that of even proving that the expression's limit is not rational; a talented mathematician could very possibly show that. Showing that it is transcendental would be a lot harder. – Mark Fischler Jul 23 '14 at 05:27
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    http://math.stackexchange.com/questions/662563/nice-way-to-express-the-radical-sqrt4-sqrt34-sqrt44-dots http://math.stackexchange.com/questions/1782032/find-sqrt4-sqrt34-sqrt44-sqrt54 http://math.stackexchange.com/questions/1843974/value-of-y-sqrt4-sqrt34-sqrt44-sqrt54-ldots – Martin Sleziak Feb 03 '17 at 13:40

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