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\CourseNumber \hfill \coursename \\
Project   \hfill \today
\medskip 

The goal of this project is to establish an amusing  formula relating
$\pi$ to an infinite   product, which is known as Wallis's product.
(Of course this infinite product has 
to be defined as a limit.) 
The  formula is:
\begin{equation}\label{wallis}
\frac \pi 2 = \frac  2 1\cdot \frac 2 3\cdot \frac 4 3\cdot \frac 4
5\cdot \frac 6 5 \cdot \frac 6 7 \dots.
\end{equation}
The right-hand side of this expression can also be written as
$$
\lim _{n  \rightarrow \infty } \prod _{k=1} ^n \frac {(2k)^2} {
(2k-1)(2k+1)}.
$$
Though we have not used the symbol $\prod$ before, you should be able
to guess what it means, once I tell you that $\prod$ is related to
products in the same way that $\sum$ is related to sums.

Your assignment is to carry out the steps below in order to see why
(\ref{wallis}) is  true. The tools you will use are integration by
parts (which is probably the most useful technique for studying
integrals) and the comparison properties for integrals.

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. Each of you should participate in preparing the report and on
the front of your paper please write ``All group members participated
in the preparation of this project''.  Each group member should sign
this statement indicating they agree with this statement.


In your writeup, you should not just give the computations without
explanation. The best papers will probably not give every detail of
every step of
the computation but
will give a sentence or two describing what is being done and then the
result of several steps.   When you use results from the text (or from
other sources) be sure to give a precise reference by page and
equation number.  

You should feel free to ask for help--tell us where you are stuck, and
we will try to point you in the right direction. You are also free to
consult with your fellow students. However, you should not copy
directly from another paper, obviously.




\begin{enumerate}

\item Derive the reduction formula for $\cos$, \#34  inside the back cover of
Varberg, Purcell and Rigdon.  Hint: What I really wanted to do was ask you to derive the
reduction formula for $\sin$, \#73. Unfortunately, this is done for
you in section 8.4. You should imitate the argument used in example 7
p. 389  to establish the reduction formula for $\cos$. 


\item Use the reduction formula  for
$\sin$ to show that for $n=2,3,\dots$, 
$$
\int_0 ^{\pi /2} \sin^{n } x \, dx = \frac {n-1} n\int _0 ^{\pi/2} \sin^{n-2 } x
\,d x .
$$

\item Show that for $n= 1,2,\dots$
$$
\int_0 ^ {\pi /2} \sin^{2n+1} x \, dx  = \frac 2 3 \cdot \frac 4 5
\cdot \dots \cdot \frac {2n } { 2n+1}  =\prod_{k=1}^n \frac{2k}{2k+1}
$$
and 
$$
\int_0 ^ {\pi /2} \sin ^{2n} x \,dx =  \frac \pi 2 \cdot \frac 1 2
\cdot \frac 3 4 \cdot \dots \cdot \frac {2n-1} {2n}=\frac \pi 2
\prod_{k=1}^n
\frac  {2k-1}{2k} .
$$
Hint: Before you try the general formula, you might want to try a few
concrete examples, say $n=0,1$ and $2$. After you see the pattern, use
mathematical induction to 
establish the formulae for all $n$. 

\item  Divide the two formulae in part 3  and show
$$
\frac \pi 2 \frac {\int_0 ^ {\pi /2} \sin^{2n+1} x \, dx }{\int_0 ^
{\pi /2} \sin ^{2n} x \,dx }   = \frac 2 1 \cdot \frac 2 3 \cdot \frac
4 3\cdot \frac 4 
5\cdot \frac 6 5 \cdot \frac 6 7 \cdot \dots\cdot  \frac { 2n } {
2n-1} \cdot \frac {2n} 
{2n+1}  .
$$

\item To finish, we need to show that 
\begin{equation}\label{lim}
\lim _{n\rightarrow\infty }   \frac {\int_0 ^ {\pi /2} \sin^{2n+1} x
\, dx }{\int_0 ^ 
{\pi /2} \sin ^{2n} x \,dx } = 1. 
\end{equation}


To do this, we begin with the inequalities
$$
 \sin^{2n+2 } x \leq \sin^{2n+1}x \leq \sin^{2n} x, \qquad n=0,1,2\dots.
$$
For which $x$ are  these true? Why?  Using these inequalities and part
2, show that 
$$
f(n)\leq  \frac {\int_0 ^ {\pi /2} \sin^{2n+1} x
\, dx }{\int_0 ^ 
{\pi /2} \sin ^{2n} x \,dx } \leq h(n)
$$
where 
$$
\lim _{n\rightarrow \infty} f(n) = \lim_{n\rightarrow \infty} h(n) =1.
$$
Now the squeeze theorem tells us that (\ref{lim}) and hence (\ref{wallis})
holds.  In your report, you should find explicit formulae for the
functions $f$ and $h$. 

\item 
  Who was Wallis and how did this product arise?
\end{enumerate}


The above is adapted from the book {\em Calculus}, by Michael Spivak.
In your copious spare time, you look through this book to see a
different view of what a calculus book should be.

\end{flushleft}


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