\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
C_{n}
 & = & \frac{(2n)!}{(n + 1) (n!)^{2}} \\
& = & \frac{(2n)!}{n! (n + 1)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
C_{n}
 & = & \frac{(2n)!}{n! (n + 1)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
C_{n}
 & = & \frac{1}{n + 1} \left ( {2n \atop n} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\left ( {p \atop q} \right )
 & = & \frac{p!}{q! (p - q)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{C_{n + 1}}{C_{n}}
 & = & \frac{\frac{(2 (n + 1))!}{(n + 1)! (n + 1 + 1)!}}{\frac{(2n)!}{n! (n + 1)!}} \\
& = & \frac{\frac{(2n + 2)!}{(n + 1)! (n + 2)!}}{\frac{(2n)!}{n! (n + 1)!}} \\
& = & \frac{\frac{(2n)! (2n + 1) (2n + 2)}{n! (n + 1) (n + 1)! (n + 2)}}{\frac{(2n)!}{n! (n + 1)!}} \\
& = & \frac{(2n + 1) (2n + 2)}{(n + 1) (n + 2)} \\
& = & \frac{2 (2n + 1) (n + 1)}{(n + 1) (n + 2)} \\
& = & \frac{2 (2n + 1)}{n + 2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \frac{C_{n + 1}}{C_{n}}
 & = & \lim_{n \to \infty} \frac{2 (2n + 1)}{n + 2} \\
& = & \lim_{n \to \infty} \frac{2 \left ( 2 + \frac{1}{n} \right )}{1 + \frac{2}{n}} \\
& \approx & \frac{2 (2 + 0)}{1 + 0} \\
& = & \frac{4}{1} \\
& = & 4 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{C_{n - 1}}{C_{n}}
 & = & \frac{\frac{(2 (n - 1))!}{(n - 1)! (n - 1 + 1)!}}{\frac{(2n)!}{n! (n + 1)!}} \\
& = & \frac{\frac{(2n - 2)!}{(n - 1)! n!}}{\frac{(2n)!}{n! (n + 1)!}} \\
& = & \frac{\frac{n (2n)!}{2n (2n - 1) n!}}{\frac{(2n)!}{n! (n + 1)}} \\
& = & \frac{n + 1}{2 (2n - 1)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \frac{C_{n - 1}}{C_{n}}
 & = & \lim_{n \to \infty} \frac{n + 1}{2 (2n - 1)} \\
& = & \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2 (2 - \frac{1}{n})} \\
& \approx & \frac{1 + 0}{2 (2 - 0)} \\
& = & \frac{1}{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} C_{n}
 & = & 4^{n} + \, \mbox{constante} \\
\end{eqnarray*}
\end{document}
