So the first guess of the simplest elliptic function (of order 2) associated to the lattice $L:=\mathbb Z+i\mathbb Z$ would be $$\sum_{m,n\in\mathbb Z}\frac{1}{(z-m-in)^2}$$ This sum is doubly periodic. But it is not considered to be an elliptic function since is it not absolutely convergent at the point of $L$. But I don't understand why do we require this while we we want the series to be unbounded in $L$ as the elements of $L$ are the poles?
For example, $$\sum_{m,n\in\mathbb Z}\frac{1}{(z-m-in)^2}=\frac1{z^2}+\sum_{m,n\in\mathbb Z\setminus\{0\}}\frac{1}{(z-m-in)^2}$$ so in fact when $z=0$ the second summand diverges. But why we don't say that $0$ is a pole of the series?