Definition:Convergent Series
Definition
Let $\struct {S, \circ, \tau}$ be a topological semigroup.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
This series is said to be convergent if and only if its sequence of partial sums $\sequence {s_N}$ converges in the topological space $\struct {S, \tau}$.
If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
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 $\ds S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.
$S$ is convergent if and only if its sequence $\sequence {s_N}$ of partial sums converges in the metric space $\struct {V, d}$.
Convergent Series in a Normed Vector Space (Definition 2)
Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\ds S := \sum_{n \mathop = 1}^\infty a_n$ 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 $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
Let $\sequence {s_N}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
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 $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
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.