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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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    @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.2011-06-01

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

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    Ah, that'll teach me to read closer. Thank you!2011-06-01