Definition:Series

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

Let $\left \langle {a_n} \right \rangle$ be a sequence in $S$.

Let $\left \langle {s_N} \right \rangle$ be the sequence defined as:
 * $\displaystyle s_N = \sum_{n \mathop = 1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$

Then $\left \langle {s_N} \right \rangle$ is called the sequence of partial products of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Series in a Number Field
The usual context for the definition of a series occurs when $S$ is one of the standard number fields $\Q, \R, \C$.

Then $\left \langle {s_N} \right \rangle$ is the sequence defined as:
 * $\displaystyle s_N = \sum_{n \mathop = 1}^N a_n = a_1 + a_2 + \cdots + a_N$

Then we refer to $\left \langle {s_N} \right \rangle$ as the sequence of partial sums of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Notation
When there is no danger of confusion, the limits of the summation are implicit and the notations:


 * $\displaystyle \sum a_n$

and


 * $\displaystyle \sum_n a_n$

are often seen for $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.