Definition:Series

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

Let $$\left \langle {s_N} \right \rangle$$ be the sequence defined as $$s_N = \sum_{n=1}^N a_n = a_1 + a_2 + \cdots + a_N$$.

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

If $$s_N \to s$$ as $$n \to \infty$$, the series converges to the sum $$s$$.

Thus in this case we can write $$s = \sum_{n=1}^\infty a_n$$.

Note that this can only make any sense when $$\left \langle {s_N} \right \rangle$$ converges.

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