Is there a standard definition of $\sum_{-\infty}^{+\infty} a_n$?

Question

In my class of Time Series, the professor mentioned about the sequence of real numbers $(a_n)_{n \in \mathbb Z}$ and the sum $$\sum_{-\infty}^{+\infty} a_n$$

I've searched on the Internet but could not find a definition for the sum from $-\infty$ to $+\infty$.
Is there a standard definition for this kind of sum?

Answer 1

In general, for a given index set $I$, we speak of the sum of the elements $\{x_i\}_{i\in I}$ and write it ass $$\sum_{i\in I} x_i.$$

However, it would be a good practice for you to show that $\sum_{i\in I} x_i$ converges in the a normed space if and only if $I$ is at most countable (i.e. finite or countable infinite).

The definition of the sum is as follows: Let $T$ be a countable set, then we write $$\sum_{n\in T} x_n=s$$ if for $\varepsilon > 0$ there is a finite subset $T'$ of $Τ$ such that for all finite sets $T''$ for which $T'\subset T'' \subset Τ$ we have $$\left|s-\sum_{n\in T''}x_n \right|< \varepsilon $$ (For definition, see e.g. Ultrametric calculus, W. H. Schikhof. CUP)

Answer 2

I would say that this is just equal to $\sum_{n=0}^\infty a_n + \sum_{n=1}^\infty a_{-n}$, as long as both series are convergent. I don't know if this is a standard definition; it just seems the obvious way to go.

Answer 3

$\mathbb Z$ is measure space with the counting measure, and the sum $$\sum_{n\in \mathbb Z} a_n$$ may be defined as the integral of the function $a$ w.r.t. this measure (when it 'converges absolutely', i.e. when $a$ is integrable.)