TITLE: Find the value of $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+ ....... }}}}$
QUESTION [7 upvotes]: I have tried solving this question. It would be great if someone gives some idea about how to go about solving this question.

REPLY [3 votes]: Two long comment, this question is related to other question, but it seems that there is no closed form from thier realively partiel answers:


Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$


Problem 6 - IMO 1985


How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$


Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$