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\begin{document}


\title{Math 21, Fall, 2000}


\address{Lehigh University}


\email{david.johnson@lehigh.edu}

\maketitle

\subsection{Sigma notation}


\subsection*{Summation formulas}

\begin{example}
Evaluate \[
\sum _{i=1}^{m}\left( \sum _{j=1}^{n}\left( i+j\right) \right) .\]

\end{example}

\subsubsection*{Solution}

There is a bit of a trick to this one. First sum over one index (\( j \)),
for each value of the other (\( i \)).\begin{eqnarray*}
\sum _{i=1}^{m}\left( \sum _{j=1}^{n}\left( i+j\right) \right)  & = & \sum _{i=1}^{m}\left( i\sum _{j=1}^{n}\left( 1\right) +\sum _{j=1}^{n}\left( j\right) \right) \\
 & = & \sum _{i=1}^{m}\left( i\cdot n+\frac{n(n+1)}{2}\right) \\
 & = & n\sum _{i=1}^{m}\left( i\right) +\frac{n(n+1)}{2}\sum _{i=1}^{m}\left( 1\right) \\
 & = & \frac{n\cdot m(m+1)}{2}+\frac{m\cdot n(n+1)}{2}.
\end{eqnarray*}


\begin{xca}
Find \[
\lim _{n\rightarrow \infty }\left( \sum _{i=1}^{n}\frac{1}{n}\left( \frac{i}{n}\right) ^{2}\right) \]
\end{xca}

\end{document}
