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Math question on limits; Invariance Principle

  1. Apr 26, 2015 #1
    1. The problem statement, all variables and given/known data

    Hi Guys,

    This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help.

    "E1. Starting with a point S (a, b) of the plane with 0 < b < a, we generate a sequence of points (xn, yn) according to the rule x0 = a, y0 = b, xn+1 =(xn + yn) / 2 and yn+1 = (2xnyn) / (xn + yn).

    Here it is easy to find an invariant. From (xn+1yn+1) = xnyn, for all n we deduce xnyn = ab for all n. This is the invariant we are looking for. Initially, we have y0 < x0. This relation also remains invariant. Indeed, suppose yn < xn for some n. Then xn+1 is the midpoint of the segment with endpoints yn, xn. Moreover, yn+1 < xn+1 since the harmonic mean is strictly less than the arithmetic mean.

    Thus,
    0 < xn+1 − yn+1 = [(xn − yn) / (xn + yn)] * [(xn − yn) / 2] < (xn − yn) / 2
    for all n. So we have limxn = lim yn = x with x2 = ab or x = √ab.

    Here the invariant helped us very much, but its recognition was not yet the
    solution, although the completion of the solution was trivial."

    3. The attempt at a solution

    I cannot figure out the bit in bold at all. It says lim xn = lim yn, but where does this come from? From a cursory look at the definition of a limt, is it simply since |xn+1 - yn+1| < (xn-yn)/2, we find that for all N>n, that xn+1 and yn+1 are limits of each other?
     
    Last edited: Apr 26, 2015
  2. jcsd
  3. Apr 26, 2015 #2
    Shouldn't this be in the math section ? I don't see any harmonic motion here.................
    [EDIT:- I now realize that perhaps you intend those n's and (n+1)'s to be in the subscript, you can easily do that from the post template......]
     
  4. Apr 26, 2015 #3
    I made the changes.

    I was not aware this is not harmonic motion question (I assumed the sequence was following a harmonic motion) but I didn't realise there was a precise definition. I will move this to math section.
     
  5. Apr 26, 2015 #4
    Quite alright. "To err is to human" eh!
    Cheers :)
     
  6. Apr 26, 2015 #5
    Thanks
     
  7. Apr 26, 2015 #6
    Now, do you know anything about limits?
     
  8. Apr 26, 2015 #7
    Think geometrically. What is feature of the two points ##x_{n+1}## and ##y_{n+1}## is different (or has changed) from the points ##x_n## and ##y_n##
    [EDIT:- and what happens to this "feature" if you keep creating these new points ad infinitum ? that is to say what happens "in the limit".]
     
  9. Apr 26, 2015 #8
    Xn gets larger while Yn gets smaller to preserve the fact that XnYn =ab right?
     
  10. Apr 26, 2015 #9
    Precisely........now if you keep doing this what will happen eventually i.e "what will happen in the limit" ?
    [EDIT:- Geometrically speaking you can imagine ##X_n## and ##Y_n## as two points slowly approaching each other from opposite directions on the number line, one gets bigger, the other gets smaller....... they keep getting closer to one another after each iteration]
     
    Last edited: Apr 26, 2015
  11. Apr 26, 2015 #10
    Not much I am afraid. I know that a sequence goes to a limit if it gets closer and closer as the index goes up
     
  12. Apr 26, 2015 #11
    It will get closer and closer to something for the limit to exist, otherwise it would just go to infinity....... am I right?
    [EDIT:- now think of what that "something" is in the original question. Also see edit to post #9]
     
  13. Apr 26, 2015 #12
    Ah ok! I get it now, they will approach each other due to the invariant nature of their relation ship XnYn=ab. So their limit can be denoted arbitrarily by X. And the limit of their products is then just the product of their limits which implies X2 = ab

    Thank you for your help you have helped in my journey; much appreciated!
     
  14. Apr 26, 2015 #13
    Well done......you solved it.
    Also this was a good intro to limits ;)
     
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