Definition of series

Given a sequence \(\{u_n\}\), we can calculate the sum \(S_n\) of its first \(n\) terms, which we called a series.
When we are referring specifically to a given sequence \(\{u_n\}\), the symbol \(S_n\) unambiguously stands for
\[ S_n = u_1 + u_2 + \ldots + u_n. \]

The difference between a sequence and a series is that a sequence has terms separated by commas, while a series has terms of the sequence separated by addition, \(+\), i.e., the terms are added together.

Convention: Define \(S_0\ = 0\) since no terms are summed when \(n = 0\).

As you increase the size of \(n\), \(S_n\) involves more terms (known as summands):

\[ \begin{align*} S_1 & = u_1 \\ S_2 & = u_1 + u_2 \\ S_3 & = u_1 + u_2 + u_3 \\ S_4 & = u_1 + u_2 + u_3 + u_4 \\ \vdots & \\ S_{n-1} & = u_1 + u_2 + u_3 + u_4 + \ldots + u_{n-1} \\ S_{n} & = u_1 + u_2 + u_3 + u_4 + \ldots + u_{n} \\ \end{align*} \]

This observation leads us to a very important relationship between \(u_n\) and \(S_n\):

\[ S_n = \big( u_1 + u_2 + \ldots + u_{n-1} \big) + u_n = S_{n-1} + u_n. \]