Definition:Convergent Series

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

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

This series is said to be convergent iff its sequence $\left \langle {s_N} \right \rangle$ of partial products 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 $\displaystyle \sum_{n=1}^\infty a_n = s$.

Convergent Series in a Normed Vector Space
Let $V$ be a normed vector space.

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

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

This series is said to be convergent iff its sequence $\left \langle {s_N} \right \rangle$ of partial sums converges in the metric space $\left({V, d}\right)$.

Convergent Series in a Number Field
Let $S$ be one of the standard number fields $\Q, \R, \C$.

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

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

It follows that $\left \langle {s_N} \right \rangle$ 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 $\displaystyle \sum_{n=1}^\infty a_n = s$.

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

Divergent Series
A series which is not convergent is divergent.