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I Reversing the order of summation

  1. Apr 15, 2016 #1
    Is ∑f from a to b the same as ∑f from b to a?
    In other words, does the order of summation matter?
     
  2. jcsd
  3. Apr 15, 2016 #2

    BvU

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    No, since a+b = b+a
     
  4. Apr 15, 2016 #3
    Thank you.
     
  5. Apr 15, 2016 #4
    I'll add that, if there is only a finite number of terms, or if all but finitely many nonzero terms are of the same sign, then any order of summation gives the same result.

    But (and I hope this is not too much information):
    -----------------------------------------------------------

    For any convergent infinite summation

    cj = K​

    that does not converge absolutely:

    |cj| = ∞,

    then there is an surprising theorem that suggests how important it is to be cautious:

    Theorem: For such a summation as cj, and any real number L, there is some rearrangement ∑' of the order of summation such that

    ∑' cj = L.
     
  6. Apr 16, 2016 #5
    Thank you.
     
  7. Apr 16, 2016 #6

    look this video
     
  8. Apr 16, 2016 #7

    BvU

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    In post #1 there is.
     
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