# Exchanging the order of summation

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

1. ### John WashburnGuest

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