Natural Summation

Why

We want a notation for expressing the sum of several natural numbers.

Definition

Let $s = (m_1, \dots , m_n)$ be a sequence of natural numbers. The sequence sum of $s$ 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 sum of a sequence using by using the $\sum$ symbol. $\sum$ is the capital greek letter “sigma” and is a mnemonic for “sum.”

Let $(m_1, \dots , m_n)$ be a sequence of natural numbers. Let us denote by $m_i$ an element of the sequence, where $i = 1, \dots , n$.

We denote that the sum ranges over an (ordered) index set $\set{1, \dots , n}$ by writing $\sum_{i = 1}^{n}$. We denote the sequence sum

\[ \sum_{i = 1}^{n} m_i. \]

Summing over finite sets

Suppose $A$ is a finite set and $f: A \to \N $ is a function. Then we the notation

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

is notation for $\sum_{i = 1}^{n} f(\sigma (i))$ where $\sigma : \set{1, \dots , \num{A}} \to A$ is any numbering of $A$. The numbering is inconsequential because addition is commutative.