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 $\ds \sum_{n \mathop = 1}^\infty a_n = s$.

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

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

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 $s$.

A series which is not convergent is divergent.