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Convergence of Series

  1. Nov 26, 2008 #1
    What is the name of the test that allows you to find convergence for such series as (0,1,0,1,0,1...)? It makes that series converge to 1/2
     
  2. jcsd
  3. Nov 26, 2008 #2
    That series diverges. You got something wrong. Or maybe I don't understand what you are asking.
     
  4. Nov 26, 2008 #3

    lurflurf

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    It depends what you mean by diverges.
     
  5. Nov 26, 2008 #4
    The arithmetic average of all limit points.
     
  6. Nov 28, 2008 #5

    daniel_i_l

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    Maybe you mean the series (1,-1,1,-1,...). In that case Euler used the formula for the power series (that (1,x,x^2...) converges to 1/(1-x)) where x=-1. Then he got 1/(1-(-1)) = 1/2. But with the modern definition of convergence, the formula is only true when -1<x<1.
     
  7. Nov 28, 2008 #6
    I believe the technique you're looking for is called Cesaro Summation...check it out on Wikipedia.
     
  8. Nov 29, 2008 #7
    Yes, Cesaro summation. That's what I was looking for.
     
  9. Dec 1, 2008 #8
    That's a sequence, not a series.
     
  10. Dec 1, 2008 #9
    It's the sequence of partial sums for the series whose terms are (-1)^n (or something close to it), which is not summable by the normal definition, but has Cesaro sum 1/2.
     
  11. Dec 1, 2008 #10
    I suppose I was only considering the first post or so, so never mind. :)

    Series are usually written as summations though.
     
  12. Dec 1, 2008 #11
    You're absolutely right about that summation notation would be vastly more appropriate...you definitely have to read a bit into the first post before it makes sense.
     
  13. Dec 1, 2008 #12
    Same thing
     
  14. Dec 2, 2008 #13

    HallsofIvy

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    No, it's not the same thing. Please try to learn enough mathematics that you can at least ask an intelligible question without people trying to guess what you are really asking.
     
  15. Dec 2, 2008 #14
    The example I gave was a harmonic series
     
  16. Dec 2, 2008 #15
    No, it was a sequence.

    You have a bunch of numbers, a series is a bunch of numbers with an operator.
     
  17. Dec 2, 2008 #16

    HallsofIvy

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    And even if you had written it as a series is still would not be a harmonic series! The harmonic series is 1+ 1/2+ 1/3+ ...+ 1/n+ ....
     
  18. Dec 3, 2008 #17
    Dude, this is trivial nomenclature. It's obvious he knows what he means, and it's obvious everyone else knew what he meant, and were able to answer his question. In math it is not the name that is important but rather the meaning and structure.
     
  19. Dec 3, 2008 #18

    lurflurf

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    That is true we do use mophisms, but in the communication of math name is very important. If you do not communicate clearly the meaning and structure of the objects you describe will be changed.
    What is the sum of [sin(x)]^2 by [cos(x)]^2?
    [tanh(x)]^2
     
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