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\section{Series\label{Series}}

A (\emph{finite})\emph{\ series} is the sum of some of the terms of a
sequence. In adding the terms of a sequence, it is helpful to have a more
compact way of writing long sums.

Given a sequence 
\[
a_{1},a_{2},a_{3},a_{4}... 
\]
we can write the sum of the first $n$ terms using sigma notation. This
notation derives its name from the Greek letter $\sum $ (capital sigma,
corresponding to our $S$ for sum). Sigma notation is used as follows: 
\[
\sum_{k=1}^{n}a_{k}=a_{1}+a_{2}+a_{3}+a_{4}+\cdot \cdot \cdot +a_{n} 
\]

The left side of this expression is read ``The sum of $a_{k}$ from $k=1$ to $%
k=n$.'' The letter $k$ is called the \textbf{index of summation}, or the 
\textbf{summation variable}, and the idea is to replace $k$ in the
expression after the sigma by the integers $1,2,3,...n,$ and add the
resulting expressions, arriving at the right side of the equation.

\subsection{Properties of Sums\label{Properties of Sums}}

Let $a_{1},a_{2},a_{3},a_{4}...$ and $b_{1},b_{2},b_{3},b_{4}...$ be
sequences. Then for every positive integer $n$ and any real number $c$,

\begin{enumerate}
\item  $\sum\limits_{k=1}^{n}\left( a_{k}+b_{k}\right)
=\sum\limits_{k=1}^{n}a_{k}+\sum\limits_{k=1}^{n}b_{k}$

\item  $\sum\limits_{k=1}^{n}\left( a_{k}-b_{k}\right)
=\sum\limits_{k=1}^{n}a_{k}-\sum\limits_{k=1}^{n}b_{k}$

\item  $\sum\limits_{k=1}^{n}ca_{k}=c\left(
\sum\limits_{k=1}^{n}a_{k}\right) $
\end{enumerate}

\subsection{Definition of a Series\label{Definition of a Series}}

Let $a_{1},a_{2},a_{3},a_{4}...$ be a sequence. A sum of the form 
\[
a_{1}+a_{2}+a_{3}+a_{4}+\cdot \cdot \cdot +a_{N} 
\]
is called a \textbf{(finite series)}. The number $a_{1}$ is called the 
\textbf{first term} of the series, $a_{2}$ the \textbf{second term}, and so
on. The number that the series adds to is called the \textbf{sum} of the
series.

\subsection{The Partial Sums of a Series\label{The Partial Sums of a Series}}

For the series 
\[
a_{1}+a_{2}+a_{3}+a_{4}+\cdot \cdot \cdot +a_{n} 
\]

the \textbf{partial sums} are 
\begin{eqnarray*}
S_{1} &=&a_{1} \\
S_{2} &=&a_{1}+a_{2} \\
S_{3} &=&a_{1}+a_{2}+a_{3} \\
S_{4} &=&a_{1}+a_{2}+a_{3}+a_{4} \\
&&\cdot \\
&&\cdot \\
&&\cdot \\
S_{4} &=&a_{1}+a_{2}+a_{3}+\cdot \cdot \cdot +a_{n} \\
&&\cdot \\
&&\cdot \\
&&\cdot
\end{eqnarray*}

$S_{1}$ is called the \textbf{first partial sum}, $S_{2}$ is the \textbf{%
second partial sum}, and so on. $S_{n}$ is called the $n$\textbf{th partial
sum}. The sequence $S_{1},S_{2},S_{3},...,S_{n}...$ is called the \textbf{%
sequence of partial sums}.

\subsection{\ Related Reference Topics\hfill}

{\small \hyperref{Summation rules and formulas}{}{}{Cal05_01.tex#summation
notation}}

{\small \hyperref{Series}{}{}{Cal09_02.tex#series}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Series}{}{}{../../Help/DM7-7.tex#Series}}

{\small \hyperref{Summation notation}{}{}{../../Help/Summation.tex}}\hfill 

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