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:
 * $$s_N = \sum_{n=1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$$.

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

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:
 * $$s_N = \sum_{n=1}^N a_n = a_1 + a_2 + \cdots + a_N$$.

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

Notation
When there is no danger of confusion, the limits of the summation are implicit and the notation $$\sum a_n$$ is often seen for $$\sum_{n=1}^\infty a_n$$.