# Notation for summation

I have a function $f(x)$ that I want to sum in two separate ways:

• across integer values of $x\geqslant0$
• across all real values of $x\geqslant0$

I am interested in the notation for both situations. Is it legitimate to say something like

$$\sum_{x \in \mathbb{Z}\geqslant0} f(x)$$

and $$\sum_{x \in \mathbb{R}\geqslant0} f(x)$$

I realise that this second example is also equivalent to a partial integral, but since the expression isn't algebraically integrable, I want to explore alternative notations.

• Both are not really sums, since you can only sum finitely many summands. For the first I can guess a definition as the limit of $\sum_{x=0}^n f(x)$ as $n\to\infty$. For the second you have to give a definition. – Christoph Sep 19 '18 at 15:42
• OK, understood. Thanks. – Richard Burke-Ward Sep 19 '18 at 15:53

The way summation is defined on a finite set $S$ like below$$\sum\limits_{x\in S} f(x)$$ is to first order the set with some bijective map from the set of numbers from $1$ to $n$, then evaluate the summation as $$f(x_1)+f(x_2)+...+f(x_n)$$Now, for countably infinite sets, we can generalize this procedure. We begin by creating a bijective map from the natural numbers to the set $S$, and then, we compute the sum $$f(x_1)+f(x_2)+..+f(x_n)$$ for all $n$. After doing so, we can take the limit as $n\to\infty$.
As such, your first summation doesn't make much sense, unless we create a mapping between your set $\mathbb{N}\cup\{0\}$ and $\mathbb{N}$, which is relatively trivial to do. Your second summation is not well defined, as there exists no bijective mapping between the set and the natural numbers.