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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle
\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]This one isn't quite fair, since I don't tell you a specific formula
to compute \( c_{n} \) for each \( n \). In exams like the SAT, this sort
of description of a sequence is common, but not so in calculus. Without a formula,
there is no reason to suspect that the next term is not 37. However, we are
pretty good at pattern recognition, and you probably see that \( c_{n} \) is
supposed to be \( \pi  \) to \( n \) significant figures. Actually, the whole
concept of infinite decimal expansions is really the same as a sequence (and
can also be expressed as a series), where the final number is the limit of the
process of taking more and more terms of the decimal expansion. So, \[
\lim _{n\rightarrow \infty }c_{n}=\pi .\]
Of course, there is a way to write a formula for \( c_{n} \), it just isn't
paticularly helpful. One way to do that would be to say that \( c_{n}=[10^{n-1}\pi ]/10^{n-1} \),
where \( [x] \) is called the {}``greatest integer{}'' of \( x \), for example,
\( [2.1]=2 \).
\item [Back]~\end{description}

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