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.
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Do you need the explicit value of the limit or just the convergence? – Robert Z Oct 17 '17 at 05:47
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@RobertZ Both can be, if you get the exact value better. – JamesJ Oct 17 '17 at 05:50
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is the answer anyhow converging to $\lim_{n\to\infty} \sqrt{n}$ ? – The Dead Legend Oct 17 '17 at 05:55
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@TheDeadLegend This obviously does not converge – JamesJ Oct 17 '17 at 05:59
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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).
Robert Z
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