# Order of evaluation for summing over integers

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?

• I would use the latter interpretation. – copper.hat Dec 27 '16 at 19:13
• I have most often seen the former used in discussion above convergence of Fourier series (for example as seen here en.wikipedia.org/wiki/Convergence_of_Fourier_series). – Winther Dec 27 '16 at 19:38
• Note that summation order is irrelevant if and only if either $\sum_{n \in \mathbb{Z}} \max[a_n,0] < \infty$ or $\sum_{n \in \mathbb{Z}} \min[a_n,0] > -\infty$. (This allows the sum to be $\infty$ or $-\infty$). If you want to ensure a finite sum that is independent of order, then you need $\sum_{n\in \mathbb{Z}} |a_n| < \infty$. – Michael Dec 27 '16 at 20:36

## 1 Answer

The first, sometimes called the principal value since it is similar to the Cauchy Principal Value for sums, implies the second, but not vice-versa. 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.