TITLE: On the convergence of $\sqrt{1+\sqrt[3]{2+...+\sqrt[n+1]{n}}}$.
QUESTION [1 upvotes]: I would like to study the convergence of the following sequence, 
$$\sqrt{1+\sqrt[3]{2+...+\sqrt[n+1]{n}}}$$
I don't know how to deal with it. I appreciate any kind of help.

REPLY [2 votes]: The sequence 
$$a_n:=\sqrt{1+\sqrt[3]{2+\sqrt[4]{3+\sqrt[5]{\dots+\sqrt[n+1]{n}}}}}$$
is increasing and bounded above by
$$b_n:=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$$
which is convergent by How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
Hence $a_n$ converges to a finite real number $L\in (1,2)$ (see achille hui's answer).
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