Is this rigorous notation?

Question

$$\sum_{n=-\infty}^{+\infty} f(x,n)$$

Is it rigorous to write this ?

It feels weird to write "$n = -\infty$" ...

Answer 1

It is okay. However, you should know that

$$\sum_{n=-\infty}^{+\infty} f(x,n)=\lim_{n\to +\infty} \sum_{k=-n}^{n}f(x,k)$$

Answer 2

On second thought, this is not weirder than writing $+\infty$ on the top. Recall that the notation $\sum_{n=n_0}^\infty$ is not a sum but a series, i.e. already there we have a notational distinction from $\sum_{n=n_0}^{n_1}$ where asummand with $n=n_1$ does occur in the sum, whereas no summand with $n=\infty$ "occurs" in the series $\sum_{n=n_0}^\infty$. A more "neutral" notation such as $\sum_{n\in\mathbb Z}$ instead of $\sum_{n=-\infty}^\infty$ might look preferable sometimes, but that would suppres the limit character.

At any rate, there are different interpretations possible regarding the precise definition of the symbol in terms of limits (for an algebraists, the world is simpler: all these things are only defined if at most finitely many summands are nonzero ;) ), that is, does it stand for $\lim_{N\to\infty}\sum_{n=-N}^N$ or for $\lim_{(N,M)\to(\infty,\infty)}\sum_{n=-M}^N$ or ...?

You can also view things this way: There are a couple of distinct (complex) symbols with different (but subtly related) meaning, for which we might have introduced totally different notations, such as : $$\begin{align}\sum_{\color{blue}{var}=\color{green}{expr_1}}^{\color{green}{expr_2}}\color{green}{expr_3}(\color{blue}{var})\qquad &= \qquad \operatorname{FiniteSum}(\color{blue}{var},\color{green}{expr_1},\color{green}{expr_2},\color{green}{expr_3}(\color{blue}{var})) \\ \sum_{\color{blue}{var}=\color{green}{expr_1}}^{\infty}\color{green}{expr_2}(\color{blue}{var}) \qquad &= \qquad \operatorname{Series}(\color{blue}{var},\color{green}{expr_1},\color{green}{expr_2}(\color{blue}{var}))\\\sum_{\color{blue}{var}=\infty}^{\infty}\color{green}{expr}(\color{blue}{var}) \qquad &= \qquad \operatorname{BiSeries}(\color{blue}{var},\color{green}{expr}(\color{blue}{var})) \end{align}$$ Yeah, with such notation, no confusion could arise and nobody would have to wonder what that symbol $\infty$ stands for.