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\title{Math 1b.  Lecture 14 \\
Series---Geometric Series}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply geometric series.

\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?
$$
2 + 0.9 \cdot 2 + (0.9)^2 \cdot 2 + (0.9)^3 \cdot 2 + \cdots
$$

\section{Geometric Series}

Let
$$
\sum_{k = 0}^\infty a r^k = a + ar + ar^2 + ar^3 + \cdots.
$$
The $n$th partial sum is
$$
S_n = a + ar + ar^2 + ar^3 + \cdots + a r^n = a \frac{1 - r^{n+1}}{1 - r} .
$$
\begin{itemize}

\item
If $|r| < 1$, then $\displaystyle \sum_{k = 0}^\infty a r^k = \frac{a}{1-r}$.

\item
If $|r| \geq 1$, the series diverges.


\end{itemize}

\section{Worksheet Problems}


\begin{enumerate}

\item 
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?

\item
Determine the convergence of each of the following geometric series.  If the series converges, find its sum.
\begin{enumerate}

\item 
$1 + 0.4 + 0.16 + 0.064 + \cdots$

\item
$\displaystyle \sum_{n = 0}^\infty \frac{(-6)^n}{5^n}$

\item
$\displaystyle \sum_{n = 2}^\infty \frac{3^n}{\pi^n}$


\end{enumerate}


\item
Express $6.254254\ldots$ as a ratio of two integers.

\item
A ball is dropped from a height of 6 ft.  Each time the ball bounces, it comes back up to one-half of its previous height.  What is the total distance that the ball travels?




\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 15, 2006}



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