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Re: Exchanging the order of summation
Posted:
Nov 10, 2011 6:49 AM
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Resent-From: <bergv@illinois.edu>
From: Dan Luecking <LookInSig@uark.edu>
Date: November 9, 2011 2:10:02 PM MST
To: "sci-math-research@moderators.isc.org"
<sci-math-research@moderators.isc.org>
Subject: Re: Exchanging the order of summation
On Mon, 7 Nov 2011 01:59:46 +0000, John Washburn
<Math@WashburnResearch.org> wrote:
> 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.
There are two general result with slightly different hypotheses:
1. If the terms f(n,q) are all positive, then all limit processes
produce the same sum (whether finite or infinite). (Tonelli's
Theorem).
2. Suppose the sum of all |f(n,q)| is finite (by Tonellis' Theorem,
any processes can be used for this test) then the sum of f(n,q) is
the same by any limit process. (Fubini's Theorem).
If neither hypotheses is satisfied, then there are two
extreme possibilities:
a. All sums produce +infinity or -infinity. This happens if
the terms of one sign have finite sum and the terms of the
other sign have an infinite sum
b. Any real (or infinite) value can be obtained as a result of
some limit process. This happens if the positive terms sum
to +infinity and the negative terms sum to -infinity AND the
terms f(n,q) tend to 0 as n+q tends to infinity.
And there are other possibilities illustrated by the following
example
f(n,q) = (-1)^{n+q}
For some limiting processes, the sum of these produce limits
(say summing an even number of terms in n and then an even
number in q to get 0 for all such sums) others produce
infinity or nothing at all (e.g., partial sums alternating
between +1 and -1).
Dan
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