\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{multicol}


\title{Math 1b.  Lecture 21 \\
Power Series}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply the idea of a power series.

\item
To be able to determine the interval of convergence of a power series.


\end{itemize} 

\section{Definition of a Power Series}

A {\em power series\/} is a series of the form
$$
\sum_{n = 0}^\infty c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots.
$$
A {\em power series centered at \/} $x =a$ is a series of the form
$$
\sum_{n = 0}^\infty c_n (x - a)^n = c_0 + c_1 (x - a) + c_2 (x - a)^2 + c_3 (x - a)^3 + \cdots.
$$
The most natural example of a power series is the geometric series
$$
\frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots.
$$

\section{Convergence of Power Series}

We can usually find where a power series converges using the ratio test.

\section{Theorem}

For a given power series
$$
\sum_{n = 0}^\infty c_n (x - a)^n,
$$
there are three possibilities.
\begin{enumerate}

\item
The series only converges for $x = a$.

\item
The series converges for all $x$.

\item
There is a positive number $R$ called {\em the radius of convergence\/} such that the series converges if $|x - a| < R$ and diverges if $|x - a| > R$.


\end{enumerate}
In case (3), the endpoints of the interval must be tested for convergence separately.

\section{Examples}

\begin{center}
\begin{tabular}{|c|c|c|} \hline
Series & Radius of Convergence $R$ & Interval of Convergence \\ \hline
$\displaystyle \sum_{n=0}^\infty a x^n$ & $R=1$ & $(-1, 1)$ \\
$\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!}$ & $R= \infty$ & $(-\infty, \infty)$ \\
$\displaystyle \sum_{n=0}^\infty n! x^n$ & $R=0$ & $\{ 0 \}$ \\
$\displaystyle \sum_{n=1}^\infty \frac{(x - 3)^n}{n}$ & $R=1$ & $[2, 4)$ \\
$\displaystyle \sum_{n=1}^\infty \frac{x^n}{n^2 5^n}$ & $R=5$ & $[-5, 5]$ \\ \hline
\end{tabular}
\end{center}



\section{Worksheet Problems}

\begin{enumerate}

\item
Find the interval and radius of convergence of
$\displaystyle \sum_{n=1}^\infty (x - 2)^n$

\item
Suppose that the power series $\sum_{n = 0}^\infty a_n x^n$ converges if $x = -3$ and diverges if $x = 7$.  Indicate which of the following statements must be true, cannot be true, or may be true.
\begin{enumerate}

\item
The power series converges if $x = -10$.

\item
The power series diverges if $x = 3$.

\item
The power series converges if $x = 6$.

\item
The power series diverges if $x = 2$.

\item
The power series diverges if $x = -7$.

\item
The power series converges if $x = -4$.

\end{enumerate}

\item
Find the interval and radius of convergence of
$\displaystyle \sum_{n=2}^\infty \frac{(x - 3)^{2n}}{n^4}$

\item
Find the interval and radius of convergence of
$\displaystyle \sum_{n=2}^\infty \frac{(x + 5)^{n}}{n \ln n}$

\item
Find the interval and radius of convergence of
$\displaystyle \sum_{n=1}^\infty \frac{(x + 1)^{n}}{n}$

\item
Give an example of a power series that converges on the interval $[-11, -3)$.




\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.5 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{April 5, 2006}



\end{document}

