Date: Nov 10, 2011 6:52 AM
Author: JohnWashburn
Subject: Re: Exchanging the order of summation

On Nov 9, 11:08 am, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>

wrote:

> In article

> <d76d43c30a634c5cb29e6ab44da4d...@CITESHT4.ad.uillinois.edu>, John

>

> Washburn <M...@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

I should have wrote:

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

as:

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

Sorry for the typing error.

It looks like uniform convergence though is required, but the Moore-

smith theorem may be applicable as the sequence as n tends to infinity

coverges uniformly and point-wise as q tends to infinity.