Exchanging the order of summation

Discussion in 'Math Research' started by John Washburn, Nov 7, 2011.

  1. Are there conditions other than uniform convergence or absolut
    convergence, which permit the order of summation to interchanged?

    I have a double summation over n = 1 to \infty and q= 1 to \infty of
    the summand f(n,q). The limit processess are q first, then n, but i
    would like to evaluate n first then q. If it matters f(n,q) is finite
    and real for positive integers, n and q.

    I have sum with a definite when there is a single limit process
    involved. Namely, I have two non-decreasing functions g(Q) and h(Q)
    and a well define limit as Q increases without bound:

    limit_{Q \to \infty} sum_{n=1}^{g(Q)} sum_{q=1}^{g(Q)} = K.

    I seems to me I am very close to the Fubini-Tonelli theorem and that
    if the double summation with a single limit process has a finite limit
    the iterated sum has the same finite limit regardless of the order of
    summation.

    Or is the proper conclusion that if a finite, limit exists, then all
    three limits are the same. No guarantee that a finite limit exist,
    jsut that if it does all three limit processes lead to the same value.

    So my question in another form is this:
    Is the existence of a finite value of the double sum using a single
    limit process (functions of Q), sufficient to permit the interchanging
    the order of the limit processes; q tends to infinity and n tending to
    infinity?

    Thanks for any time you might give to this question.
    John Washburn
     
    John Washburn, Nov 7, 2011
    #1
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