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\noindent
{\Large MA 114, Section 21 \hfill Name:\underline{\hspace{2.8in}}\\
Project 2 (25 points) \\
Due August 3, 2000 \\ 
}

\vspace{.2in}

\noindent
You may have heard about the Fibonacci sequence of numbers.  It starts
with the numbers
$$1, 1, 2, 3, 5, 8, 13, \dots$$
this sequence was introduced around A.D. 1200 by Leonardo Fibonacci.  We
can define this sequence $\{f_n\}$ by the recursive formula
$$f_1=f_2=1, \hspace{1in} f_{n+2} = f_{n+1}+f_n.$$
However, this definition is not very useful if we need to know what $f_{223}$
is.  In this project, we will find an explicit formula for $f_n$.  
Complete the steps below.

\be
\item
Find $f_1$ through $f_{12}$.
\item
Consider the power series $F(x) = f_1x+f_2x^2+f_3x^3+\cdots$.  Show that
the radius of convergence of this power series is at least $\frac{1}{2}$.
\item
Find a second degree polynomial $p(x)$ such that $p(x)F(x) = x$.
\item
Write a partial fraction decomposition for $\frac{x}{p(x)}$.
\item
Write the power series expansion for $\frac{1}{ax+b}$ and determine its
radius of convergence.
\item
Write the power series expansion for your partial fraction decomposition
for $\frac{x}{p(x)}$ and use it to determine the coefficients for $F(x)$.
\item
What is the radius of convergence for $F(x)$?
\item
Use your result in problem 6 to determine a formula for $f_k$ for
each value of $k$.  Use this formula and a calculator to find the first 8 terms
of the sequence $\{f_n\}$ and verify your answers by comparing them
with problem 1.
\item
Find $f_{223}$.
\ee

You will be expected to justify each statement you make.  For example, when
finding $p(x)$ above, you must convince me that $p(x)F(x)=x$.  Don't just
tell me what $p(x)$ is.  The same rules apply as last time:  

You should work in groups of 1 to 3 students.  Each
group will hand in one report for the group and all group members will
receive the SAME GRADE.  By signing your name to the report you submit,
you are attesting
that you participated in the work and that you accept the grade of the group.
When you use results from the
text (or from other sources) be sure to give  precise
reference by page and equation number.


You should feel free to ask for help in my office hours or by e-mail.  Little to no
time will be spent on this in class.  You are free to consult with others
outside your group; however, you should not copy directly from another paper,
obviously.











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