Question

For \(p>1\) let\[a_n=\sqrt[p_1]{1+\sqrt[p_2]{1+\sqrt[p_3]{1+\ldots+\sqrt[p_n]{1}}}}\]and prove that \(a_n\) converges iff\[b_n=\frac{\ln n}{\prod_{i=1}^np_i}\]is bounded.

Answer 1

Edited question again, lol.

Answer 2

I think we can write a general case that \(a_n\in\mathbb R\) and \(\sqrt{1+\sqrt{1+...+\sqrt{1}}}\geq a_n\forall p>2\) converges. But how do I show this?