Definition:Convergent Series
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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.
