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


\title{Math 1b.  Lecture 18 \\
Other Convergence Tests 
(Alternating Series, Absolute and Conditional Convergence)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply the definition of an alternating series.

\item
To understand and be able to apply the concepts of conditional and absolute convergence.


\end{itemize} 

\section{Alternating Series}

An alternating series is a series of the form
$$
\sum_{n=1}^\infty (-1)^{n+1} a_n = a_1 - a_2 + a_3 - a_4 + \cdots.
$$

\section{The Alternating Series Test}

Let
$$
\sum_{n=1}^\infty (-1)^{n+1} a_n = a_1 - a_2 + a_3 - a_4 + \cdots
$$
be an alternating series satisfying the following conditions.
\begin{enumerate}

\item 
$a_1 \geq a_2 \geq a_3 \geq \cdots$

\item
$\displaystyle \lim_{n \to \infty} a_n = 0$

\end{enumerate}
Then the series converges.

For example, the alternating harmonic series
$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
$$
converges.

\section{Proof of the Alternating Series Test}

Let
$$
\sum_{n=1}^\infty (-1)^{n+1} a_n = a_1 - a_2 + a_3 - a_4 + \cdots.
$$
The sequence of  even partial sums,
\begin{gather*}
S_2 = a_1 - a_2 \geq 0 \\
S_4 = a_1 - a_2 + a_3 - a_4 = S_2 +  a_3 - a_4 \geq S_2 \\
S_6 = S_4 +  a_5 - a_6 \geq S_4 \\
\vdots \\
S_{2n} = S_{2n - 2} +  a_{2n-1} - a_{2n} \geq S_{2n - 2}, 
\end{gather*}
is increasing.  This sequence is also bounded since
$$
S_{2n} = a_1 - (a_2 - a_3 ) - (a_2 - a_3 ) - \cdots - (a_{2n-1} - a_{2n-2} ) - a_{2n} \leq a_1.
$$
Thus, the sequence $\{  S_{2n} \}$ is increasing and bounded above and
$$
\lim_{n \to \infty} S_{2n} = S.
$$

We claim that the limit of odd partial sums also converges to $S$,
\begin{eqnarray*}
\lim_{n \to \infty} S_{2n + 1}
& = &
\lim_{n \to \infty} S_{2n} + a_{2n+1} \\
& = &
\lim_{n \to \infty} S_{2n} + \lim_{n \to \infty} a_{2n+1} \\
& = &
S + 0 = S.
\end{eqnarray*}

\section{Estimating Sums}


If
$$
\sum_{n=1}^\infty (-1)^{n+1} a_n = a_1 - a_2 + a_3 - a_4 + \cdots
$$
is an alternating series satisfying
\begin{enumerate}

\item 
$a_1 \geq a_2 \geq a_3 \geq \cdots$,

\item
$\displaystyle \lim_{n \to \infty} a_n = 0$,

\end{enumerate}
then
$$
|R_n| = |S - S_n| \leq a_{n+1}.
$$


This is true since
$$
|S - S_n| \leq |S_{n+1} - S_n| = a_{n+1}.
$$

\section{Absolute and Conditional Convergence}

A series $\sum a_n$ is {\em absolutely convergent\/} if  $\sum |a_n|$.  If  $\sum a_n$ converges, but $\sum |a_n|$ diverges, then $\sum a_n$ is {\em conditionally convergent}.  For example,
\begin{itemize}

\item
$\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}$ is absolutely convergent.

\item
$\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ is conditionally convergent.


\end{itemize}

An absolutely convergent series is convergent.  Suppose that $\sum |a_n|$ converges.  Then $\sum 2|a_n|$ is convergent.  Since
$$
0 \leq a_n + |a_n| \leq 2 |a_n|,
$$
the series $\sum (a_n + |a_n|)$ converges by the comparison test.  Thus,
$$
\sum a_n = \sum (a_n + |a_n| )- \sum |a_n|
$$
converges.

\section{Worksheet Problems}

\begin{enumerate}

\item 
Determine the convergence of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}$.



\item 
Determine the convergence of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2 + 2^n}$.

\item 
Suppose that $a_n \geq 0$ for all $n \geq 1$.  Is is possible for $\sum_{n=1}^\infty a_n$ to converge conditionally?

\item 
For what values of $p$ does the series $\displaystyle \sum_{n=2}^\infty (-1)^{n+1}\frac{\ln k}{n^p}$ converge absolutely?

\item 
For what values of $p$ does the series $\displaystyle \sum_{n=2}^\infty (-1)^{n+1}\frac{\ln k}{n^p}$ converge conditionally?


\item 
Determine the convergence of $\displaystyle 1 - \frac{1}{2^3} + \frac{1}{3^2} - \frac{1}{4^3} + \frac{1}{5^2} - \frac{1}{6^3} +  \frac{1}{7^2} - \frac{1}{8^3} +\cdots $.


\end{enumerate}



\section*{References}

\begin{itemize}

\item
\S 8.4 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*{March 22, 2006}



\end{document}

