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\title{Math 1b.  Lecture 15 \\
Series---Definitions and Properties}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand the definition of a series.

\item
To understand and be able to apply the basic properties of series.

\item
To understand and be able to apply the Divergence Test.


\end{itemize} 

\section{An Example}

Suppose that the country of Pottsylvania spends \${2} billion and that each recipient of a fraction of this wealth spends 90\% of the dollars that he or she receives.  In turn, the secondary recipients spend 90\% of the dollars that they receive, and so on.  What is the total spending that results from the original injection of \${2}~ billion dollars into the economy?
$$
2 + 0.9 \cdot 2 + (0.9)^2 \cdot 2 + (0.9)^3 \cdot 2 + \cdots
$$
This is an example of an infinite series.

\section{Definitions}

\begin{itemize}

\item
An infinite series is an infinite sum of the form
$$
\sum_{k = 1}^\infty a_k = a_1 + a_2 + a_3 + \cdots + a_k + \cdots.
$$
Consider the example
$$
\sum_{k = 1}^\infty \frac{1}{2^k} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^k} + \cdots.
$$
If we draw a picture of a unit square and consider each term in the series to be an area, then it is easy to conclude that $\sum_{k = 1}^\infty \frac{1}{2^k} = 1$.

\item
We need a formal definition of convergence.  Let
$$
S_n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^n} 
$$
be a {\em partial sum}.  We say the series {\em converges\/} if 
$$
\lim_{n \to \infty} S_n = L
$$ 
exists.

\end{itemize}

\section{Examples}

\begin{itemize}

\item
$\displaystyle \sum_{n = 1}^\infty \frac{1}{n(n + 1)} = \sum_{n = 1}^\infty \frac{1}{n}- \frac{1}{n+1}$ converges to 1.  This is an example of a telescopic series.

\item
The {\em Harmonic series\/}  $\displaystyle \sum_{n = 1}^\infty \frac{1}{n}$ diverges.  Compare $\displaystyle 1 + \int_1^\infty \frac{dx}{x}$ to a right-hand sum for the integral.

\end{itemize}

\section{Properties}

\begin{itemize}

\item
$\displaystyle \sum_{k = 1}^\infty ca_k = c\sum_{k = 1}^\infty a_k$

\item
$\displaystyle \sum_{k = 1}^\infty a_k \pm b_k = \sum_{k = 1}^\infty a_k \pm \sum_{k = 1}^\infty b_k$


\end{itemize}

\section{Test for Divergence}

If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n = 1}^\infty a_n$ diverges.

\section{Worksheet Problems}

\begin{enumerate}

\item
Determine the convergence or divergence of each of the following series.  If the series converges, try to find the sum.
\begin{enumerate}

\item 
$\displaystyle \sum_{n = 1}^\infty \tan n$


\item 
$\displaystyle \sum_{n = 1}^\infty \frac{n+1}{2n - 3}$


\item 
$\displaystyle \sum_{n = 1}^\infty  \frac{2}{n^2 + 4n + 3}$


\item 
$\displaystyle \sum_{n = 1}^\infty \ln\left(  \frac{n}{n+1} \right)$





\end{enumerate}

\item
Let $\{ a_n\}$ be an increasing sequence such that $a_1 >0$ and $a_n \leq 100$ for all $n \geq 1$.
\begin{enumerate}

\item
Does $\displaystyle \lim_{n \to \infty} a_n$ converge?

\item
Does $\displaystyle \sum_{n=1}^\infty a_n$ converge?


\end{enumerate}

\item
Suppose that the partial sums of $\sum_{n=1}^\infty a_n$ satisfy the inequality
$$
\frac{6 \ln n}{\ln(n^2 + 1)} < S_n < 3 + ne^{-n}
$$
for all $n > 100$.
\begin{enumerate}

\item
Does $\displaystyle \sum_{n=1}^{\infty} a_n$ converge?  If so, what is the limit? 

\item
Can anything be said about   $\displaystyle \lim_{n \to \infty} a_n$?


\end{enumerate}



\end{enumerate}


\section*{References}

\begin{itemize}

\item
\S 8.2 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 13, 2006}



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