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Suppose I have $\sum\limits_{n = 1}^{\infty}\sum\limits_{m = 1}^{\infty} a_{m, n}$ where $a_{m, n} \geq 0$ for all $m$ and $n$. Can I interchange the two summations? If so why?

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    These two questions seem to be related: [Under what condition we can interchange order of a limit and a summation?](http://math.stackexchange.com/questions/23057/under-what-condition-we-can-interchange-order-of-a-limit-and-a-summation) and [When can you switch the order of limits?](http://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits/)2011-12-09

2 Answers 2

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It's worth knowing that rearrangements can change the value of sums or integrals only if the positive and negative parts both diverge to infinity.

Fubini's theorem says rearrangements are fine if both parts are finite.

Tonelli's theorem says rearrangements are fine if what's being summed or integrated is everywhere non-negative. (It follows that it also works if it's everywhere non-positive, since the minus sign pulls out.)

Putting the two together gives you what I said in the first paragraph.

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    what about the formal series?2013-08-07
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Yes. Because either way it is equal to $\displaystyle{\sup\limits_{M,N}\;\sum_{n=1}^N\sum_{m=1}^M a_{m,n}}$.

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    @Michael: Yes, that is the point I meant to express in my comment, along with a way to see that it is so. (And one could allow $\sup$ to take on the value $+\infty$ to apply the suggestion in my answer.)2011-12-10