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\homework{15: Transcendental Numbers}{Due Monday, Week 9}


Solve \textbf{one} of the following \textbf{four} problems.    As always, prove your claims/have fun!  Also, this set is \textbf{extra-credit}, because of the holiday!  Take care, and I'll see you all in a week!
--Paddy


\begin{enumerate}
\item Prove that the constant
\begin{align*}
L = \sum_{n=1}^\infty 10^{-(n!)}
\end{align*} 
is transcendental.
\item Let $A$ be the collection of all algebraic real numbers.  What is the cardinality of $A$?  What is the measure of $A$?  (Prove both claims.)
\item A \textbf{Liouville number} $x$ is an irrational number with the following property: for any positive $n \in \mathbb{N}$, there is some $p, q \in \mathbb{Z}$, $q > 1$, such that
\begin{align*}
0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^n}.
\end{align*}
In a sense, Liouville numbers are irrational numbers that can be very closely approximated by rational numbers!  

Show that the constant in problem 1 is a Liouville number.
\item Show that any Liouville number is transcendental.
\end{enumerate}


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