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The invariance principle

  1. Mar 20, 2012 #1
    Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
    xn+1 =xn +yn , yn+1 = 2xnyn .Find the limits of {xn} and {yn}.

    why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?
     
  2. jcsd
  3. Mar 21, 2012 #2

    mathman

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    The problem as stated is wrong (e.g. x0 = 2, y0 =1. they just keep getting bigger).
     
  4. Mar 21, 2012 #3
    Oh! Sorry, I transcribed the question erroneously. It should be:

    Given x0 and y0 such that x0 > y0 > 0. Define, for n = 0,1,2,,
    xn+1 =(xn +yn)/2 , yn+1 = 2xnyn/(xn + yn) .Find the limits of {xn} and {yn}.

    why is the answer lim{xn} = lim{yn} = sqrt(x0y0)?
     
  5. Mar 21, 2012 #4

    disregardthat

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    1) Show that x_n is decreasing, and y_n is increasing.
    2) Show that x_n has a lower bound, and y_n has an upper bound.

    Hint: Show that x_n < y_n by induction, and then that x_n+1 < x_n, and y_n+1 > y_n.

    Do you know why they must converge in this case? It is a well known fact.

    3) Find the limits by using the equations.
     
  6. Mar 25, 2012 #5
    It is a well-known fact? Please expand on it and enlighten me! Thank you for the hints though!
     
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