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This question already has an answer here:

I recently came across this problem

Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$

After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists.

So my question is

Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges.

Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help .

If the solution from that book is required I can post it but it is not complete as I mentioned.

Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist.

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marked as duplicate by YuiTo Cheng, mrtaurho, Adrian Keister, José Carlos Santos sequences-and-series Jul 2 at 21:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Did you look at all the links in all the answers to that earlier question? $\endgroup$ – Gerry Myerson Jul 2 '12 at 12:27
  • $\begingroup$ Thanks I found the answer in this article Herschfeld: On infinite radicals. $\endgroup$ – sabertooth Jul 2 '12 at 12:59
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    $\begingroup$ @GerryMyerson I think it's hard to search for the earlier posts. $\endgroup$ – Yai0Phah Jul 2 '12 at 13:18
  • $\begingroup$ @Frank While it is true that the SE search function leaves much to be desired (instead try googling with site:MSE), it does work well for reasonable unique terms like "nested radical". Indeed, it yields the cited duplicate question as first match. But, of course, one does need to know the English buzzwords for these objects. $\endgroup$ – Bill Dubuque Jul 2 '12 at 14:46
  • $\begingroup$ @Frank, yes, it's hard to search for earlier posts, and no criticism of OP is implied when someone points out that a question has been asked and answered before. But perhaps I miss the point of your comment? Anyway, now that we all know it's a duplicate, why has no one else voted to close? $\endgroup$ – Gerry Myerson Jul 2 '12 at 23:52
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Let $f_n(0)=\sqrt{1+n}$ and $f_n(k)=\sqrt{1+(n-k)f_n(k-1)}$. Then $0<f_n(0)<n+1$ when $n>0$. Assume that $f_n(k)<n+1-k$ and we can show by induction that $$ f_n(k+1) < \sqrt{1+(n-k-1)(n-k+1)} = \sqrt{1+(n-k)^2-1} = n+1-(k+1) $$ for all k. Your expression is $f_n(n-2)$ which is increasing in $n$ and bounded above by $3$, so converges.

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Note that $$ x+1=\sqrt{1+x(x+2)}\tag{1} $$ Iterating $(1)$, we get $$ x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{(x+5)}}}}}\tag{2} $$ Note that $$ s_3=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\color{#C00000}{\sqrt{1}}}}}}\tag{3} $$ Instead of $\color{#C00000}{\sqrt{1}}$ as in the last term of $(3)$, $(2)$ has $\color{#C00000}{(x+n+2)}$. Thus, the increasing sequence in $(3)$ is bounded above by $x+1$. Thus, the sequence in $(3)$ has a limit.

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    $\begingroup$ I try to compliment this every time - but I love the custom-dulled colors. $\endgroup$ – davidlowryduda Dec 15 '12 at 19:40
  • $\begingroup$ @mixedmath Yes, they feel quite soft and seems fitting for math work. $\endgroup$ – Simply Beautiful Art Dec 6 '16 at 22:15
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$x + 1 = \sqrt {1 + x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{}.....}}}$. Put $x=2$ gives you the solution. For proof see http://zariski.files.wordpress.com/2010/05/sr_nroots.pdf

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  • $\begingroup$ It is the same proof that I know but discussion of convergence is not complete still +1 for dicussion of numerical convergence. $\endgroup$ – sabertooth Jul 2 '12 at 12:52
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Note that $\frac{x+y}{x+z} < \frac{y}{z}$ for positive $x, y, z$; thus $\frac{\sqrt{x+y}}{\sqrt{x+z}} < \frac{\sqrt{y}}{\sqrt{z}}$ for $y > z$.

Thus we get $\frac{a_{n+1}}{a_n} = \frac{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1+(n+1)\sqrt{1}}}}}}}{\sqrt{\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}} < \frac{\sqrt{\sqrt{\cdots\sqrt{n+2}}}}{\sqrt{\sqrt{\cdots\sqrt{1}}}} = O(n^{(2^{-n})})$.

Next, $\ln{a_{n+1}}-\ln{a_n} = O(\frac{\ln_n}{2^n})$. Summing the equations we get $\ln{a_n} - \ln{a_1} = O(\frac{\ln_1}{2^1} + \cdots + \frac{\ln_n}{2^n})$; letting $n$ to approach $\infty$, we get $\ln{a} - \ln{a_1} = O(1)$, thus there is a finite limit of $a_i$.

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This is a known question of Putnam competition: I give a complete proof with details:

$f_n(x)=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+n-1)\sqrt{1+(x+n)}}}}$

it is clear that for $m>n$, we have $f_m(x)>f_n(x)$. Moreover

$x+1=\sqrt{1+x(x+2)}=\sqrt{1+x\sqrt{1+(x+1)(x+2)}}=\cdots=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+\cdots+(x+n-1)(x+n+1)}}}\geq f_n(x)$.

So this tells us that $\lim_{n\to \infty}f_n(x)$ exists. Now let $\lim_{n\to \infty}f_n(x)=f(x)$ hence $f(x)\leq x+1$ and $f(x)>\sqrt{1+x\sqrt{1+x\sqrt{1+x\sqrt{1+...}}}}=\frac{x+\sqrt{x^2+4}}{2}>x$ hence $x<f(x)<x+1$. Now, let $g(x)=x+1-f(x)$ then $0\leq g(x)<1$, it is easy to see that $(f(x))^2=1+xf(x+1)$ hence $(x+1+f(x))g(x)=xg(x+1)$, so

$$\frac{g(x)}{x}\leq \frac{g(x+1)}{x+1}\leq \cdots\leq \frac{g(x+n)}{x+n}<\frac{1}{x+n}\to 0$$ hence $f(x)=x+1$, so $f(2)=3$

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