# Infinite square-rooting

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

• In this forum, it is customary to give your own thoughts and attempts, in addition to a problem statement. – GEdgar Oct 19 '13 at 23:36
• This limit converges: see math.stackexchange.com/a/61055/73324 – vadim123 Oct 19 '13 at 23:37
• @GEdgar, I calculated 1st 7 terms. Seems it's converging to ~1.758. – Chin Yeh Oct 20 '13 at 0:06
• 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. – Chris Oct 20 '13 at 0:06
• @vadim123, your link points to another link math.stackexchange.com/questions/170858/…, which seems to be very helpful. – Chin Yeh Oct 20 '13 at 0:07