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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}
Find \[
\sum _{i=1}^{n}\frac{3}{n}\left( \left( \frac{i}{n}\right) ^{2}+1\right) .\]

\end{example}

\subsubsection*{Solution}

\begin{eqnarray*}
\sum _{i=1}^{n}\frac{3}{n}\left( \left( \frac{i}{n}\right) ^{2}+1\right)  & = & \sum _{i=1}^{n}\frac{3}{n}\left( \frac{i}{n}\right) ^{2}+\sum _{i=1}^{n}\frac{3}{n}\cdot 1\\
 & = & \frac{3}{n^{3}}\sum _{i=1}^{n}i^{2}+\frac{3}{n}\sum _{i=1}^{n}1\\
 & = & \frac{3}{n^{3}}\frac{n(n+1)(2n+1)}{6}+\frac{3}{n}n\\
 & = & \frac{(n+1)(2n+1)}{2n^{2}}+3.
\end{eqnarray*}
Note how the formulas for the sums of powers of the first \( n \) integers
are used. This is a common theme here. Also don't forget about the sum of 1's
as being \( 1+1+1+\cdots  \), \( n \) times. 
\end{document}
