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&nbsp;\(n = 0\).
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 \(S_n\): 
\[
S_n = \big( u_1 + u_2 + \ldots + u_{n-1} \big) + u_n = S_{n-1} + u_n.
\]

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 \(S_n\): 

Definition of series

Definition of series