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Math Help - Find the limit of the sequence

  1. #1
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    Find the limit of the sequence

    A sequence is defined by a1=2, an+1=(2an+1)​1/2

    I have proven by induction that an is bounded above by 1+(2)1/2, and that an is monotone increasing. By the monotone convergence theorem I can deduce that the sequence converges but I cannot figure out how to prove what it converges to. I know that it converges to 1+(2)1/2 but I don't know how to show that. Could someone help me finish this question?
    Last edited by Shakarri; March 11th 2013 at 08:15 AM.
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  2. #2
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    Re: Find the limit of the sequence

    Quote Originally Posted by Shakarri View Post
    A sequence is defined by a1=2, an+1=(2an+1)​1/2
    I have proven by induction that an is bounded above by 1+(2)1/2, and that an is monotone increasing. By the monotone convergence theorem I can deduce that the sequence converges but I cannot figure out how to prove what it converges to. I know that it converges to 1+(2)2 but I don't know how to show that. Could someone help me finish this question?
    Can you solve L=\sqrt{2L+1}~?
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    Re: Find the limit of the sequence

    Actually that's what I did to find the limit but I believe this doesn't work in the general case and so cannot be considered a proof. Are you sure it is a valid proof?
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    Re: Find the limit of the sequence

    Quote Originally Posted by Shakarri View Post
    Actually that's what I did to find the limit but I believe this doesn't work in the general case and so cannot be considered a proof. Are you sure it is a valid proof?
    Validity depends upon what you are trying to do.

    I understood you to be asking how to find the limit, not to prove anything.

    Monotone, increasing, bounded sequences converge to its least upper bound.

    I think it tedious to prove that 1+\sqrt2 is the LUB of that sequence. But it can be done (I will not do it).
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    Re: Find the limit of the sequence

    One way to prove that the limit is the fixpoint is to use the Banach fixed-point theorem. To apply it, you need to do two things: (1) find a closed segment X such that f(x)=\sqrt{2x+1} maps X to X, and (2) prove that f(x) is a contraction on X. Concerning (1), you can choose X=[2,1+\sqrt{2}]. From the facts that f(x) increases on X and f(x)<1+\sqrt{2} for x\in X it follows that f(x):X\to X. Concerning (2), it is sufficient to show that for some constant q, |f'(x)|\le q<1 for all x\in X. Then the fact that f(x) is a contraction follows by the the mean value theorem.

    The Banach fixed-point theorem says that the fixpoint of f(x) is unique and is the limit of a_n. So, to find the limit of f(x) it is sufficient to solve x = f(x).
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    Re: Find the limit of the sequence

    Весело -)
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