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Math Help - iterating weighted serie to convergence

  1. #1
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    iterating weighted serie to convergence

    I have basic weighted function

    a = a*(1- alpha) + b*alpha
    a = a + alpha*( b - a )

    where
    0 < alpha < 1

    I should figure out how many ( n ) times I have to iterate before
    abs( a_n - b) is almost zero.

    I have calculated formulas for four iterations if it helps to solve the problem:

    a1 = a + alpha*(b-a)
    a2 = a + 2*alpha*(b-a) - 1*alpha^2*(b-a)
    a3 = a + 3*alpha*(b-a) - 3*alpha^2*(b-a) + alpha^3*(b-a)
    a4 = a + 4*alpha*(b-a) - 6*alpha^2*(b-a) + 4*alpha^3*(b-a) - alpha^4*(b-a)

    thank you,
    Markus
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  2. #2
    MHF Contributor
    Joined
    Nov 2008
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    France
    Posts
    1,458
    Hi

    You can show that a_n-b = (a_0-b)(1-\alpha)^n

    Therefore to get |a_n-b| < \epsilon for some given \epsilon > 0 you must have n \geq \frac{\ln \frac{\epsilon}{|a_0-b|}}{\ln(1-\alpha)}
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  3. #3
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    Great, thanks!
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