Real Summation

Why

We want to succinctly denote the sum of a list of real numbers.

Definition

Suppose $x = (x_1, \dots , x_n)$ is a list of real numbers. The sequence sum of $x$ is the result of first summing the first two numbers, then summing the result with the third number, and so on, until we have summed all the numbers.

Notation

We denote the sequence sum of $x$ by

\[ \sum_{i = 1}^{n} x_i \]

Summing over finite sets

Suppose $A$ is a finite set and $f: A \to \R $ is a function. We write

\[ \sum_{a \in A} f(a) \]

to denote the sum $\sum_{i = 1}^{n} f(\sigma (i))$ where $\sigma : \set{1, \dots , \num{A}} \to A$ is any numbering of $A$. As a result of the commutativity of addition, the particular numbering we choose is inconsequential.