# Definition:Series

## Contents

## 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 $\Q, \R, \C$.

A **series** is what results when $\left \langle {a_n} \right \rangle$ 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 $\left \langle {s_N} \right \rangle$ 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.