# How to solve this infinite radical [duplicate]

$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5+\sqrt{ \dots }}}}}}$$ 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.

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

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.