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\textbf{Math 475 Big Problems, Batch 3}
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\noindent \textbf{Big Problem 9: Tulie Number Flipping}.
Tulie has just discovered the Reciprocal button on her calculator (it's labeled ``$x^{-1}$''￼on the TI-83+). First she noticed that if you enter a number and take the reciprocal, and then take the reciprocal of the result, you get back the original number.
This was a little boring, so she tried the following. First she picked some positive number. Then:
\begin{itemize}
\item (a) She subtracted the whole number part from it. If the result was zero, she stopped.
\item (b) If it wasn't zero, she took the reciprocal. The result always had a whole number part so she went back to step (a).
\end{itemize}
For instance: She started with the number 1.4 (she typed in 1.4 and hit Enter).
\begin{itemize}
\item a. She subtracted 1 to get .4
\item b. Then she took the Reciprocal to get 2.5.
\item a \& b again. Now she subtracted 2 to get .5, and hit the Reciprocal button to get 2.
\item a again. Now she took away 2 and got 0, so she stopped.
\end{itemize}
She discovered that a lot of numbers she tried were \underline{dull} because she would eventually get zero in step `a again' and have to stop. But some numbers like $\sqrt{3}$ were \underline{endless}, because she never seemed to get zero.
\begin{enumerate}
\item Prove $\sqrt{3}$￼ is really endless.
\item Which choices of natural numbers $n$ make $\sqrt{n}$ an endless number?
\item Tulie discovered that for some numbers $n$ the expansion of $n$ started repeating after only two steps. Which numbers are these? Do you see any patterns in their expansions and in the expansions of nearby numbers? Test some of your patterns.
\item What happens to $e$? Can you find a pattern to what happens to $e^{1/n}$ ? \\[1in]
\end{enumerate}
\noindent \textbf{Big Problem 10: Irrationality of $e$}.
Let us use a famous series expression for $e$ to prove $e$ is irrational.
\begin{enumerate}
\item Write down the Taylor Series for the function $e^x$. You can look it up... I'll assume you derived it in Calculus 2 and can prove it converges everywhere.
\item Write down an infinite series expression for $e$.
\item Assuming $e$ is genuinely equal to the infinite series, prove $e$ is irrational. (Hint: if $e = p/q$, multiply through by $(q!)$. This should lead to an integer part plus something you can prove is less than 1.)
\end{enumerate}
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