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Topic: Re: Exchanging the order of summation
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G. A. Edgar

Posts: 2,497
Registered: 12/8/04
Re: Exchanging the order of summation
Posted: Nov 9, 2011 11:08 AM
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In article
<d76d43c30a634c5cb29e6ab44da4d4e9@CITESHT4.ad.uillinois.edu>, John
Washburn <Math@WashburnResearch.org> wrote:

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


A simple example to consider is this one: f(n,n)=1, f(n,n+1)=-1,
for all n, and everything else 0.  Interchanged sums are not equal.

>
> 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




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