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$ \lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ? $
Either closed answer or an upper bound would help.

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    $\begingroup$ In this forum, it is customary to give your own thoughts and attempts, in addition to a problem statement. $\endgroup$ – GEdgar Oct 19 '13 at 23:36
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    $\begingroup$ This limit converges: see math.stackexchange.com/a/61055/73324 $\endgroup$ – vadim123 Oct 19 '13 at 23:37
  • $\begingroup$ @GEdgar, I calculated 1st 7 terms. Seems it's converging to ~1.758. $\endgroup$ – Chin Yeh Oct 20 '13 at 0:06
  • $\begingroup$ Let $\rho_n$ denote the $n$th nested radical given above, and let $k_n$ be given by the recurrence $k_1 = 3,$ and $k_n = {k_{n-1}}^2 - n$ for $(n \ge 2)$. Replacing $n$ with $n + k_n$ gives a sequence whose $n$th term is larger than $\rho_n$, and which is identically $2$ when the radicals are all simplified. So, your sequence does converge. $\endgroup$ – Chris Oct 20 '13 at 0:06
  • $\begingroup$ @vadim123, your link points to another link math.stackexchange.com/questions/170858/…, which seems to be very helpful. $\endgroup$ – Chin Yeh Oct 20 '13 at 0:07
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Unfortunately, our problem is apparently quite a difficult one.

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  • $\begingroup$ You put a period to the question mark. I'll take it. The question was answered thoroughly by Hirschfeld in 1935. pballew.net/1935Herschfeld.pdf. $\endgroup$ – Chin Yeh Oct 20 '13 at 1:07

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