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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Sequences">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\begin{document}


\section{Sequences\label{sequences}}

A \textbf{sequence} is a list of numbers written in a definite order: 
\[
a_{1},a_{2},...,a_{n},... 
\]
The number $a_{1}$ is called the first term, $a_{2}$ is called the second
term, and $a_{n}$ is called the $nth$ term. A sequence can be defined as a
function whose domain is the set of positive integers. The sequence $\left\{
a_{1},a_{2},...\right\} $ is also denoted by $\left\{ a_{n}\right\} $ or $%
\left\{ a_{n}\right\} _{n=1}^{\infty }$.

\begin{example}
The\textbf{\ Fibonacci sequence\label{fibonacci sequence}} $\left\{
f_{n}\right\} $ is defined recursively by the conditions 
\[
f_{1}=1\qquad f_{2}=1\qquad f_{n}=f_{n-1}+f_{n-2}\qquad n\geq 3
\]
Each term in the sequence is the sum of the preceding two terms. The first
few terms of the sequence are 
\[
\left\{ 1,1,2,3,5,8,13,21,...\right\} 
\]
\end{example}

\begin{definition}
A sequence $\left\{ a_{n}\right\} $ has the limit $L$ . Write\label%
{convergent sequence}\label{divergent sequence} 
\[
\lim\limits_{n\rightarrow \infty }a_{n}=L\qquad \text{or}\qquad
a_{n}\rightarrow L\text{ as }n\rightarrow \infty 
\]
if for every $\epsilon >0$ there is a corresponding integer $N$ such that 
\[
\left| a_{n}-L\right| <\epsilon \qquad \text{whenever}\qquad n>N
\]
If $\lim\nolimits_{n\rightarrow \infty }a_{n}$ exists, the sequence \textbf{%
converges}. If there is no limit $L$, the sequence \textbf{diverges}.
\end{definition}

\begin{theorem}
If $\lim\nolimits_{n\rightarrow \infty }f\left( x\right) =L$ and $f\left(
n\right) =a_{n}$ when $n$ is an integer, then $\lim\nolimits_{n\rightarrow
\infty }a_{n}=L$.
\end{theorem}

\begin{definition}
$\lim\nolimits_{n\rightarrow \infty }a_{n}=\infty $ means that for every
positive number $M$ there is an integer $N$ such that 
\[
a_{n}>M\qquad \text{whenever}\qquad n>N
\]
\end{definition}

\subsubsection{Limit Laws for Sequences\label{limit laws for sequences}}

If $\left\{ a_{n}\right\} $ and $\left\{ b_{n}\right\} $ are convergent
sequences and $c$ is a constant, then

\begin{enumerate}
\item  $\lim\limits_{n\rightarrow \infty }\left( a_{n}+b_{n}\right)
=\lim\limits_{n\rightarrow \infty }a_{n}+\lim\limits_{n\rightarrow \infty
}b_{n}$

\item  $\lim\limits_{n\rightarrow \infty }\left( a_{n}-b_{n}\right)
=\lim\limits_{n\rightarrow \infty }a_{n}-\lim\limits_{n\rightarrow \infty
}b_{n}$

\item  $\lim\limits_{n\rightarrow \infty }ca_{n}=c\lim\limits_{n\rightarrow
\infty }a_{n}$

\item  $\lim\limits_{n\rightarrow \infty }\left( a_{n}b_{n}\right)
=\lim\limits_{n\rightarrow \infty }a_{n}\cdot \lim\limits_{n\rightarrow
\infty }b_{n}$

\item  $\lim\limits_{n\rightarrow \infty }\frac{a_{n}}{b_{n}}=\frac{%
\lim\limits_{n\rightarrow \infty }a_{n}}{\lim\limits_{n\rightarrow \infty
}b_{n}}$ if $\lim\limits_{n\rightarrow \infty }b_{n}\neq 0$

\item  $\lim\limits_{n\rightarrow \infty }c=c$
\end{enumerate}

\begin{theorem}[Squeeze Theorem for Sequences]
\label{squeeze theorem for sequences}If $a_{n}\leq b_{n}\leq c_{n}$ for $%
n\geq n_{0}$ and $\lim\nolimits_{n\rightarrow \infty
}a_{n}=\lim\nolimits_{n\rightarrow \infty }c_{n}=L$, then $%
\lim\nolimits_{n\rightarrow \infty }b_{n}=L$.
\end{theorem}

\begin{theorem}
If $\lim\nolimits_{n\rightarrow \infty }\left| a_{n}\right| =0$, then $%
\lim\nolimits_{n\rightarrow \infty }a_{n}=0$.
\end{theorem}

The sequence $\left\{ r^{n}\right\} $ is convergent if $-1<r\leq 1$ and
divergent for all other values of $r$. 
\[
\lim\limits_{n\rightarrow \infty }r_{n}=\left\{ 
\begin{array}{lll}
0 & \text{if} & -1<r<1 \\ 
1 & \text{if} & r=1
\end{array}
\right. 
\]

\begin{definition}
\label{increasing sequence}\label{decreasing sequence}\label%
{monotonic sequence}A sequence $\left\{ a_{n}\right\} $ is called \textbf{%
increasing} if $a_{n}\leq a_{n+1}$ for all $n\geq 1$; that is, it is
increasing if $a_{1}\leq a_{2}\leq a_{3}\leq ...$. It is called \textbf{%
decreasing} if $a_{n}\geq a_{n+1}$ for all $n\geq 1$. It is called \textbf{%
monotonic} if it is either increasing or decreasing.
\end{definition}

\begin{definition}
\label{bounded sequence}A sequence $\left\{ a_{n}\right\} $ is \textbf{%
bounded above} if there is a number $M$ such that 
\[
a_{n}\geq M\qquad \text{for all }n\geq 1
\]
It is \textbf{bounded below} if there is a number $m$ such that 
\[
m\leq a_{n}\qquad \text{for all }n\geq 1
\]
If it is bounded above and below, then $\left\{ a_{n}\right\} $ is a \textbf{%
bounded sequence}.
\end{definition}

\begin{axiom}[Completeness Axiom]
\label{completeness axiom}If $S$ is a nonempty set of real numbers that has
an upper bound $M$ ($x\leq M$ for all $x$ in $S$), then $S$ has a \textbf{%
least upper bound} $b$.
\end{axiom}

\begin{theorem}
Every bounded, monotonic sequence is convergent.
\end{theorem}

\subsection{Related Reference Topics}

{\small \hyperref{Sequences}{}{}{al09_01.tex}}

{\small \hyperref{Arithmetic and Geometric Sequences}{}{}{al09_02.tex}}

\subsection{Related Scientific Notebook Topics}

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

{\small \hyperref{Sequences in Scientific Notebook}{}{}{%
../../Help/DM7-7.tex#Sequences}}

{\small \hyperref{Subscripts on functions}{}{}{%
../../Help/DM7-7.tex#Subscripted function}}

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