An example of a divergent double sum 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.

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