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I am trying to highlight to my friend that the change of order of summation/integrals should be done with care. In that regard, the conversation moved towards the following question.

An example of a double summation with $f(m,n) > 0$ of the form $$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} f(m,n)$$ which diverges but $$\sum_{m=1}^{\infty} f(m,n)$$ converges for all $n$, and $$\sum_{n=1}^{\infty} f(m,n)$$ converges for all $m$.

I am not able to construct an example immediately of my head.

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Qiaochu's example is a good one for what you asked, but based on the intro text it would seem you want to ask for $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} f(m,n)$ to converge and $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} f(m,n)$ to diverge. Not that I have any suggestions for this. – Ross Millikan Jun 1 '11 at 20:48
@Ross: My intro is probably misguiding. I was explaining her the change of order of integration/summation and then along the discussion for some reason landed up on this question. – user17762 Jun 1 '11 at 21:28
up vote 8 down vote accepted

Take $f(m, n) = \delta_{mn}$ plus something very small.

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I'm missing something. What does "plus something very small" mean? Why doesn't $f(m,n) = \delta_{mn}$ work by itself? – Jason DeVito Jun 1 '11 at 19:40
@Jason: the OP asked for $f(m, n)$ to be positive everywhere. (I'm not sure why.) – Qiaochu Yuan Jun 1 '11 at 20:45
Ah, that'll teach me to read closer. Thank you! – Jason DeVito Jun 1 '11 at 23:14

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