# Definition:Series

## Definition

### General Definition

Let $\left({S, \circ}\right)$ be a semigroup.

Let $\left \langle {a_n} \right \rangle$ be a sequence in $S$.

Informally, a series is what results when an infinite product is taken of $\left \langle {a_n} \right \rangle$:

$\displaystyle s := \sum_{n \mathop = 1}^\infty a_n = a_1 \circ a_2 \circ a_3 \circ \cdots$

Formally, a series is a sequence in $S$.

### Series in a Standard Number Field

The usual context for the definition of a series occurs when $S$ is one of the standard number fields $\Q, \R, \C$.

The series is what results when $\sequence {a_n}$ is summed to infinity:

$\displaystyle \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

## Sequence of Partial Sums

The sequence $\sequence {s_N}$ defined as the indexed summation:

$\displaystyle s_N: = \sum_{n \mathop = 1}^N a_n = a_1 + a_2 + a_3 + \cdots + a_N$

is the sequence of partial sums of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

## Tail of a Series

Let $N \in \N$.

The expression $\displaystyle \sum_{n \mathop = N}^\infty a_n$ is known as a tail of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

## Notation

When there is no danger of confusion, the limits of the summation are implicit and the notations:

$\displaystyle \sum a_n$

and

$\displaystyle \sum_n a_n$

are often seen for $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

## Also known as

Some sources use the term infinite series, but the adjective is technically redundant in that series are defined as being infinite.