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$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\sqrt{ \dots }}}}}}$$ I don't understand how to solve that. I mean I don't know where to begin. Tell me if this infinite radical has a solution or converge to a number. Thanks.

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  • $\begingroup$ See e.g. Nested Radical Constant and references there. $\endgroup$ – Robert Israel Dec 5 '16 at 16:06
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    $\begingroup$ Not sure, where exactly this was asked here, but I am pretty sure that it is a duplicate. $\endgroup$ – Peter Dec 5 '16 at 16:06
  • $\begingroup$ You could also see here and here. $\endgroup$ – user371838 Dec 5 '16 at 16:10
  • $\begingroup$ Is it a constant? Wow $\endgroup$ – もっと酒 Dec 5 '16 at 16:10
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i hope it can help you

Theorem (Herschfeld, 1935).

The sequence $u_{n}$=$\sqrt{a_1+\sqrt{a_2+.....+\sqrt{a_{n}}}}$

converges if and only if

$\lim_{n\to∞} sup a^{2^{-n}}_{n} \lt ∞ $

The American Mathematical Monthly, Vol. 42, No. 7 (Aug-Sep 1935), 419-429.

same question answer

more information answer

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