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I'm trying to prove that this sequence $(x_n)$, where $x_1 =\sqrt a$ and $x_{n+1}=\sqrt{a +x_n}$ has a limit, then I would like to find the limit of

$L=\sqrt {a+\sqrt{a+\sqrt{a+\sqrt{a...}}}}$

It's easy to find this limit and prove that this sequence is monotone (induction over $\mathbb N$). What I found difficult is prove that this sequence is bounded.

I need help in this part.

Thanks a lot.

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marked as duplicate by MJD, njguliyev, Henry T. Horton, Thomas Andrews, Daniel Rust Sep 5 '13 at 16:05

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2 Answers 2

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When $a\ge1$, we have $x_1=\sqrt{a}\le a<2a$ and by mathematical induction, $$x_{n+1}^2 = a + x_n < a+2a=3a < 4a^2\ \Rightarrow\ x_{n+1}<2a.$$ When $a<1$, the sequence is dominated by the analogous sequence with $a=1$ and hence it is bounded.

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I didn't understand, when a=2, we have $\sqrt {2+\sqrt{2+\sqrt{2+\sqrt{2…}}}}$, it isn't a constant sequence. –  user42912 Feb 19 '13 at 10:36
    
@user42912 Oops, I misread the question. Thanks for pointing out the error. Will delete this answer in a couple of minutes. –  user1551 Feb 19 '13 at 10:37
    
no problem, thank you for trying to help me :) –  user42912 Feb 19 '13 at 10:38
    
@user42912 I've tried to fix the proof. Please see if it's OK now. –  user1551 Feb 19 '13 at 10:47
    
it seems ok! thanks again. –  user42912 Feb 19 '13 at 10:54
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If the limit exists, it is obvious to see that $$ \lim_{n\rightarrow \infty} a_{n+1} = \lim_{n \rightarrow \infty} a_n$$ So lets call the limit $x$ than $$x=\sqrt{a+x}$$ For the bound, i would use banach fix point theorem. The fix point theorem together with the fact, that the root is a strict contraction. So we know $$x_1\leq \sqrt{a} + \sqrt{\sqrt{a}}$$ For $a$ sufficiently large $a$ will be a upper bound and $0$ a lower bound. For $a$ not sufficiently large we don't need a proof, since we know it is lower bounded by $0$ and monotone increasing, so we can take the upper bound of the sufficiently large $a$ (monotonicity of the root).

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How would you use this theorem? I couldn't figure out how to apply in this question. –  user42912 Feb 19 '13 at 10:18
    
the root is a strict contraction –  Dominic Michaelis Feb 19 '13 at 10:25
    
I didn't understand, please add a little bit more information in your answer. Thank you very much –  user42912 Feb 19 '13 at 10:32
    
better now ? sry if it was to short –  Dominic Michaelis Feb 19 '13 at 10:50
    
sorry but I didn't understand yet. Maybe I don't have the maturity to understand your answer yet. Anyway, thank your for trying to help me. –  user42912 Feb 19 '13 at 11:00
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