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Math Help - Convergence Proof

  1. #1
    Member thaopanda's Avatar
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    Convergence Proof

    Suppose that A > 0 is a given real number. Define the sequence { a_n} n \in N recursively by:

    a_1 is a positive real number; a_{n+1} = \frac{1}{2} (a_n + \frac{A}{a_n}).

    Prove that { a_n} n \in N converges to \sqrt{A}

    No clue where to start...
    Last edited by Plato; October 12th 2009 at 09:38 AM. Reason: Fix LaTeX subscript
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  2. #2
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    Quote Originally Posted by thaopanda View Post
    Suppose that A > 0 is a given real number. Define the sequence { a_n} n \in N recursively by:
    a_1 is a positive real number; a_{n+1} = \frac{1}{2} (a_n + \frac{A}{a_n}).
    Prove that { a_n} n \in N converges to \sqrt{A}
    Use induction to prove the sequence becomes decreasing bounded below.
    From that you know that the sequence converges to say L.
    Then solve L=\frac{1}{2}\left(L+\frac{A}{L}\right) for L
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  3. #3
    Member thaopanda's Avatar
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    thanks! that was a lot easier than I thought
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