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Series question

  1. Dec 23, 2005 #1
    Hi, I'm trying to figure out the average age in a solera process. The equations are:

    An = (2/3)(An-1 + 1) + (1/3)(0)
    Bn = (2/3)(Bn-1 + 1) + (1/3)(An-1 + 1)
    Cn = (2/3)(Cn-1 + 1) + (1/3)(Bn-1 + 1)

    With initial state:

    A0 = 0
    B0 = 0
    C0 = 0

    The question is, what is C as n goes to infinity?
  2. jcsd
  3. Dec 27, 2005 #2


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    Homework Helper

    Find the limiting value of An as n -> infinity.
    Substitute this value into the difference eqn for Bn.

    Find the limiting value of Bn as n -> infinity.
    Substitute this value into the difference eqn for Cn.

    And finally, find the limiting value of Cn as n -> infinity.

    I also wrote a small program to work out the values of the series, and got the limiting value for Cn as Cn = 8.00000....
    Last edited: Dec 27, 2005
  4. Dec 28, 2005 #3

    matt grime

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    Homework Helper

    If you assume that they all converge to a, b and c respectively then they satisfy

    a=(2/3)(a+1) + 1/3

    b=(2/3)(b+1) +(1/3)(a+1)

    c = (2/3)(c + 1) + (1/3)(b + 1)

    which you can solve.

    So, if there is a solution, that is what it is. You might need to prove that a limit exists, though.
  5. Dec 28, 2005 #4
    Hey, thanks for the help guys.

    Ya, I figured out later to just substitute a_n = a_n+1 and solve. I suppose I should prove that the series converges, but this is good enough for now.


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