<s>[INST] Determine whether $\sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}}}$ diverges or not 
Determine whether $\{a_n\}$ is convergent or not, where
  $$a_n:=\sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}}}.$$

At least, we can obtain$$\sqrt[n]{1+\sqrt[n]{2+\sqrt[n]{3+\cdots+\sqrt[n]{n}}}}\le\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt[n]{n}}}}<2,$$which imlies $\{a_n\}$ is bounded. But is it monotonic?
 [/INST] We have
$$
\sqrt[n]{n} \le n,
$$
$$
\sqrt[n]{{n - 1 + \sqrt[n]{n}}} \le \sqrt[n]{{n + \sqrt[n]{n}}} \le \sqrt[n]{{2n}} \le 2n,
$$
$$
\sqrt[n]{{n - 2 + \sqrt[n]{{n - 1 + \sqrt[n]{n}}}}} \le \sqrt[n]{{n + \sqrt[n]{{n - 1 + \sqrt[n]{n}}}}} \le \sqrt[n]{{n + 2n}} \le 3n,
$$
etc. Hence $$
\sqrt[n]{{1 + \sqrt[n]{{2 +  \cdots  + \sqrt[n]{n}}}}} \le \sqrt[n]{{1 + (n - 1)n}} \to 1.
$$
On the other hand $a_n >1$. Hence, the limit is $1$.
</s>