Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a divergent series. Having said that, I'm currently interested in the following:



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> Have you encountered in your own research or do you recall from someone's research paper "interesting" double sums that are convergent where reversing order of summation does not work? Please provide references.
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No Tonelli or Fubini
Since there's a "number theory" tag, I suggest the quasimodular form 
$E\_2(\tau)$, defined for $\tau$ in the upper half-plane as a multiple of
$\sum\_{m\in\bf Z} {\sum\_{n\in\bf Z}}' (m\tau+n)^{-2}$
where the $\prime$ indicates omission of the term $(m,n)=(0,0)$.
For even $k>2$, the corresponding sum
$\sum\_{m\in\bf Z} {\sum\_{n\in\bf Z}}' (m\tau+n)^{-k}$
converges absolutely and yields modularity of $E\_k$.
But for $k=2$, switching the sums yields $\tau^{-2} E\_2(1/\tau)$,
which is not the same thing as $E\_2(\tau)$! (But you can still recover the
formula for the difference by carefully keeping track of how
switching $\sum\_m$ and $\sum\_n$ changes the sum).


