Iterated Roots Convergence

Calculus Level 2

$\sqrt{1},\quad \sqrt{ 1 + \sqrt{2} },\quad \sqrt{ 1 + \sqrt{2 + \sqrt{3 } } },\quad \cdots$

Consider the above sequence given by $a_n = \sqrt{ 1 + \sqrt{2 + \sqrt{3 + \cdots + \sqrt{n} } } }$ .

Does $a_n$ converge?