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Math Help - Show convergence

  1. #1
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    Show convergence

    if a _{n+2} = (a _{n+1} + a _n)/2 for all n greater or equal to 1. show that
    a _n converges to (a _n + 2a _2)/3.

    Please help me. I know that we want to show that the absolute value of a _n minus(a _n + 2a _2)/3 is less than epsilon but I am not sure how to get there. Please help.

    Ok so am i supposed to know what the first few terms are? can somebody please help me?
    Last edited by janae77; April 21st 2010 at 07:00 AM. Reason: Re-titled.
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  2. #2
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    The question itself makes no sense. If a_n is sequence indexed by "n" then its limit cannot involve "n". Are you sure you have copied this correctly?
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  3. #3
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    Quote Originally Posted by janae77 View Post
    if a_{n+2}=\frac{ a_{n+1} + a_n}{2} for all n greater or equal to 1. show that
    a_n converges to \frac{a_1+2a_2}{3}.
    (I corrected the question)

    You can notice that the sequence u_n=a_{n+1}-a_{n} satisfies u_{n+1}=-\frac{1}{2}u_n and u_1=a_2-a_1 hence by induction u_{n+1} = \frac{(-1)^n}{2^n}(a_2-a_1) (this is a geometric sequence), and you can then compute a_{n+1}-a_1=u_n+u_{n-1}+\cdots+u_1=\cdots (sum of terms from a geometric sequence) and conclude by taking the limit as n\to\infty.
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