Cube Sums of Families on the Integer Lattice

Let $n\in\mathbb{N}$, let $\mathbb{Z}^{n}$ be the integer lattice, and for $N\in\mathbb{N}$ let $\Gamma_{N}$ be the cube of lattice points whose components lie between $-N$ and $N$; it is a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite. Let $a:\mathbb{Z}^{n}\to\mathbb{R}$ be a family of real numbers.

1. (Cube sums)¶ For $N\in\mathbb{N}$ the $N$th cube sum of $a$ is the sum over the finite index set $\Gamma_{N}$,

$$S_{N}(a)=\sum_{k\in\Gamma_{N}}a(k).$$

2. (Lattice sum)¶ The family $a$ is cube-summable if the sequence $(S_{N}(a))_{N\in\mathbb{N}}$ converges. Its limit, unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, is the lattice sum of $a$, written

$$\sum_{k\in\mathbb{Z}^{n}}a(k)=\lim_{N\to\infty}S_{N}(a).$$