Order of evaluation for summing over integers

Question

In Fourier analysis, how should I interpret sums of the form $$ \sum_{n \in \mathbb{Z}} a_n = \sum_{n=-\infty}^\infty a_n?$$ Is it $$\lim_{N \to \infty} \sum_{|n| < N} a_n$$ or $$\sum_{n=1}^\infty a_{-n} + \sum_{n=0}^\infty a_n?$$ When do I not have to care about the order of summation?

Answer 1

The first is sometimes called the principal value since it is similar to the Cauchy Principal Value for sums. The second is the analog of the standard convergence of integrals. In any case, describing the kind of convergence is best.

The sum $$ \pi\cot(\pi z)=\sum_{k\in\mathbb{Z}}\frac1{k+z} $$ requires the principal value sum to converge, but it is often written as above.