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\noindent
{\large\bf Notes on Sequence and Series}

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We think of a {\it sequence} as a list (finite or infinite)
of objects: for example, $\langle 1, 2, 3, 4, \ldots\rangle$ or 
$\langle \mbox{\rm apple, banana, cherry, date,}\ldots\rangle$.
The sequences we will consider will be sequences of numbers.
A {\it series} is the sum of terms in a sequence.
To remember the difference between a sequence and a series
think: if mathematicians ruled the world we would have
to change the name of the baseball championship
to ``The World Sequence''!

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\noindent
{\bf How might sequences be presented?:}
\begin{itemize}
\item as a list: $\langle 3, 5, 17, 257, 65537 \rangle$
\item as a function with domain equal to an initial segment of {\bf
N}: $a_n = n!$
\item recursively: $a_1 = a_2 = 1$, $a_{n+1} = a_n + a_{n-1}$
\end{itemize}

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\noindent
{\bf Special sequences:}
\begin{itemize}
\item arithmetic sequence: $\langle a, a+d, a+2d, a+3d,\ldots\rangle$
\item geometric sequence: $\langle a, ar, ar^2, ar^3,\ldots\rangle$
\end{itemize}

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\noindent
{\bf Series:}
\begin{itemize}
\item sigma notation
\item sum of a finite arithmetic series: $\displaystyle 
\sum_{i=1}^n (a+(i-1)d) = 
\frac{n}{2}\left(\mbox{\rm first}+\mbox{\rm last}\right)$
\item sum of a finite geometric series: $\displaystyle 
\sum_{i=1}^n ar^{i-1} = \frac{a(1-r^n)}{1-r}$
\item sum of an infinite geometric series:
if $|r|<1$ then  $\displaystyle 
\sum_{i=1}^{\infty} ar^{i-1} = \frac{a}{1-r}$
\end{itemize}

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\noindent
{\bf Applications:}
\begin{itemize}
\item repeating decimals are rational
\item annuities and installment buying
\end{itemize}

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