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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{Achilles and the Tortoise -- Sequences and Series}

I always begin a discussion of series and sequences (these two topics are really
one idea, viewed from two different sides) with Greek mythology. This keeps
alive the belief that we are fostering liberal education. 

Today's myth is that of Achilles and the tortoise. The story is, for reasons
that I'll never figure out, they are going to have a race. (This is also told
with a hare in the role of Achilles, but then it doesn't seem so Greek.) The
paradox involved in this story is that, assuming Achilles gives the tortoise
a head start, which is only fair, he can never win, since in order to cover
the distance from Achilles to the tortoise (the head start), Achilles must first
go half that distance, then half the remaining distance, and then half of that,
\textit{et cetera}, so that he has to do \textit{infinitely} many things in
order to catch up to the tortoise. 

\vspace{0.3cm}
{\par\centering \includegraphics{achilles-tortoise.ps} \par}
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As with all the Greek paradoxes, this has an explanation. Let's assume that
the distance from Achilles to the tortoise is 1 mile. Then, in order to get
to the tortoise, Achilles must first go a distance 1/2 mile, which would take
him, say, 2 minutes. He then has to go the next 1/4 mile, which would take 1
minute, then the next 1/8 mile, taking 1/2 minute, then the next 1/16 mile,
taking 1/4 minute, \textit{et cetera}. 

\vspace{0.3cm}
{\par\centering \includegraphics{achilles-tortoise-line.ps} \par}
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So, adding this all up, he has to run:\[
1/2+1/4+1/8+1/16+\dots \: miles,\]
which takes:\[
2+1+1/2+1/4+\dots \; minutes.\]
These totals may seem infinite, since they do go on forever, like some demented
math professor, but they actually do approach a finite value as a limit. 

When you think about it, Achilles only has to run the one mile (the tortoise
hasn't even taken a step yet), which is \( 1/2+1/4+1/8+\dots =1 \), and it
takes \( 2+1+(1/2+1/4+\dots )=4 \) minutes (he \textit{is} fast). There is
really no paradox, since the infinitely many things he has to do can be done
infinitely fast at the end. 
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