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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsubsection{Series}

There are many oddities involved in this infinite summation process, which we
call an \textit{infinite series}. An infinite series is an infinite sum. There
are a few examples where we really can see the infinite sum, like the one mentioned
in the section on Achilles and the tortoise.  

\begin{example}
\( 1+1/2+1/4+1/8+\dots =2 \) 
\end{example}
You can see that this sum really does add up to 2 by marking a line segment
of two {}``units{}'' length

\vspace{0.3cm}
{\par\centering \includegraphics{two-units.ps} \par}
\vspace{0.3cm}

In order to measure from 0 to 2, first go from 0 to 1. Then, from 1, add 1/2
more to get 3/2.

\vspace{0.3cm}
{\par\centering \includegraphics{three-haves.ps} \par}
\vspace{0.3cm}

Then, add 1/4 to get 7/4:

\vspace{0.3cm}
{\par\centering \includegraphics{seven-fourths.ps} \par}
\vspace{0.3cm}

Add 1/8 to get 15/8:

\vspace{0.3cm}
{\par\centering \includegraphics{fifteen-eighths.ps} \par}
\vspace{0.3cm}

And so on. Each time you add \( 1/2^{n} \) to the previous sum, you get half
of the total distance remaining to 2, something like football penalties near
the goal line. In the limit, as the number of steps (or penalties) goes to infinity,
you reach 2.

\begin{example}
\[
\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots =1.\]
 
\end{example}
This series, called the \textit{telescoping series}, is the easiest of all to
see, once you know the trick: \[
\frac{1}{n\cdot (n+1)}=\frac{1}{n}-\frac{1}{n+1}.\]
 So, you add them together: \begin{eqnarray*}
\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots  & = & (\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+\dots \\
 & = & \frac{1}{1}+(-\frac{1}{2}+\frac{1}{2})+(-\frac{1}{3}+\frac{1}{3})+(-\frac{1}{4}+\frac{1}{4})+(-\frac{1}{5}+\dots )+\dots \\
 & = & 1.
\end{eqnarray*}



\subsubsection{Sequences}

We need to get some terminology straight here. A \textit{series}, or \textit{infinite
series}, is the infinite sum. The things you add up are the \textit{terms} of
the series. They form an \textit{infinite sequence}. A sequence is just the
listing, one number after another. A series is a summing of those numbers. 

Mathematicians think of sequences as being easier to understand. Just one number
after another. You can usually tell if the sequence is approaching something,
like \( 1/n \) approaches 0, as \( n \) goes to \( \infty  \). Technically,
a sequence is a function of the positive integers, but we write one as \( \{a_{n}\} \).
We say that a sequence \( \{a_{n}\} \) has a \textit{limit} \( L \) as \( n\rightarrow \infty  \),
if the numbers (\textit{terms}) approach \( L \) as \( n \) gets large. 

Let's look at some examples. We usually really only care about \textit{limits}
of sequences, and whether they exist. For example, find the limits of the following
sequences, if they exist:  

\begin{example}
\( a_{n}=1/2^{n} \) 
\end{example}
\begin{description}
\item [Solution]~
\end{description}
\begin{example}
\( b_{n}=2^{n} \) 
\end{example}
\begin{description}
\item [Solution]~
\end{description}
\begin{example}
\( \{c_{n}\}=\{3,\, 3.1,\, 3.14,\, 3.141,\, 3.1415,\, 3.14159,\ldots \} \)
(This \textit{should} remind you of something). 
\end{example}
\begin{description}
\item [Solution]~\end{description}

\end{document}
