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Topic: Re: Exchanging the order of summation
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Kruno Sever

Posts: 1
Registered: 11/9/11
Re: Exchanging the order of summation
Posted: Nov 9, 2011 11:06 AM
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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 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.
>


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 (Moore-Smith 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



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