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\begin{document}


\title{Math 21, Fall, 2000}


\address{Lehigh University}


\email{david.johnson@lehigh.edu}

\maketitle

\subsection{Sigma notation.}

\begin{example}
Find \[
\sum _{i=2}^{100}(\frac{1}{i}-\frac{1}{i-1}).\]
 
\end{example}

\subsubsection*{Solution}

This is more sophisticated than the others. It is what is called a telescoping
sum, named after the kind of telescope you would see on an old pirate movie,
which the captain would expand to look through, then collapse it back down to
a fraction of its original length. Write out the first few terms of that sum:\[
\sum _{i=2}^{100}(\frac{1}{i}-\frac{1}{i-1})=\left( \frac{1}{2}-\frac{1}{1}\right) +\left( \frac{1}{3}-\frac{1}{2}\right) +\left( \frac{1}{4}-\frac{1}{3}\right) +\cdots ,\]
and you see that the second part of the second term cancels the first part of
the first, etc., so\begin{eqnarray*}
\sum _{i=2}^{100}(\frac{1}{i}-\frac{1}{i-1}) & = & \left( \frac{1}{2}-\frac{1}{1}\right) +\left( \frac{1}{3}-\frac{1}{2}\right) +\left( \frac{1}{4}-\frac{1}{3}\right) +\cdots +\left( \frac{1}{100}-\frac{1}{99}\right) \\
 & = & \left( 0-\frac{1}{1}\right) +\left( \frac{1}{3}-0\right) +\left( \frac{1}{4}-\frac{1}{3}\right) +\cdots +\left( \frac{1}{100}-\frac{1}{99}\right) \\
 & = & \left( -\frac{1}{1}\right) +\left( 0\right) +\left( \frac{1}{4}-0\right) +\cdots +\left( \frac{1}{100}-\frac{1}{99}\right) \\
 & = & \cdots \\
 & = & -\frac{1}{1}+\frac{1}{100}\\
 & = & -\frac{99}{100}.
\end{eqnarray*}

\end{document}
