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Re: Exchanging the order of summation
Posted:
Nov 9, 2011 11:06 AM


Am 07.11.2011 02:59, schrieb John Washburn: > 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 nondecreasing 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. >
Something seems to be off here: is g(Q) supposed to be the upper index limit twice with no summand term f(n,q)?
If you are not aware of it, have a look at "Integration and modern analysis" by Benedetto and Czaja, page 451 (MooreSmith theorem). It should be available online via Google books.
The interchange condition there includes uniformity, though, so it may be old news to you.
Best Kruno



