Definition:Convergent Series

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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.


Also see

  • Results about convergent series can be found here.


Sources