# Definition:Convergent Series

## Definition

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

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

This series is said to be **convergent** if and only if its sequence of partial sums $\left \langle {s_N} \right \rangle$ converges in the topological space $\left({S, \tau}\right)$.

If $s_N \to s$ as $N \to \infty$, the series **converges to the sum $s$**, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

### Convergent Series in a Normed Vector Space (Definition 1)

Let $V$ be a normed vector space.

Let $d$ be the induced metric on $V$.

Let $\displaystyle S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

$S$ is **convergent** if and only if its sequence $\left \langle {s_N} \right \rangle$ of partial sums converges in the metric space $\left({V, d}\right)$.

### Convergent Series in a Normed Vector Space (Definition 2)

Let $\struct {V, \norm {\, \cdot \,}}$ be a normed vector space.

Let $\displaystyle S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

$S$ is **convergent** if and only if its sequence $\sequence {s_N}$ of partial sums converges in the normed vector space $\struct {V, \norm {\, \cdot \,}}$.

### Convergent Series in a Number Field

Let $S$ be one of the standard number fields $\Q, \R, \C$.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

Let $\sequence {s_N}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$.

If $s_N \to s$ as $N \to \infty$, the series **converges to the sum $s$**, and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

A series is said to be **convergent** if and only if it converges to some $s$.

### Divergent Series

A series which is not convergent is **divergent**.

## Also known as

A **convergent series** is also known as a **summable series**.