\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
F_{n}
 & = & \frac{(3n)!}{(2n + 1) n! (2n)!} \\
& = & \frac{(3n)!}{n! (2n + 1)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F_{n}
 & = & \frac{(3n)!}{n! (2n + 1)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F_{n}
 & = & \frac{1}{2n + 1} \left ( {3n \atop n} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\left ( {p \atop q} \right )
 & = & \frac{p!}{q! (p - q)!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{F_{n + 1}}{F_{n}}
 & = & \frac{\frac{(3 (n + 1))!}{(n + 1)! (2 (n + 1) + 1)!}}{\frac{(3n)!}{n! (2n + 1)!}} \\
& = & \frac{\frac{(3n + 3)!}{(n + 1)! (2n + 3)!}}{\frac{(3n)!}{n! (2n + 1)!}} \\
& = & \frac{\frac{(3n)! (3n + 1) (3n + 2) (3n + 3)}{n! (n + 1) (2n + 1)! (2n + 2) (2n + 3)}}{\frac{(3n)!}{n! (2n + 1)!}} \\
& = & \frac{(3n + 1) (3n + 2) (3n + 3)}{(n + 1) (2n + 2) (2n + 3)} \\
& = & \frac{3 (3n + 1) (3n + 2)}{2 (n + 1) (2n + 3)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}
 & = & \lim_{n \to \infty} \frac{3 (3n + 1) (3n + 2)}{2 (n + 1) (2n + 3)} \\
& = & \lim_{n \to \infty} \frac{3 \left ( 3 + \frac{1}{n} \right ) \left ( 3 + \frac{2}{n} \right )}
{2 \left ( 1 + \frac{1}{n} \right ) \left ( 2 + \frac{3}{n} \right )} \\
& \approx & \frac{3 (3 + 0) (3 + 0)}{2 (1 + 0) (2 + 0)} \\
& = & \frac{27}{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{F_{n - 1}}{F_{n}}
 & = & \frac{\frac{(3 (n - 1))!}{(n - 1)! (2 (n - 1) + 1)!}}{\frac{(3n)!}{n! (2n + 1)!}} \\
& = & \frac{\frac{(3n - 3)!}{(n - 1)! (2n - 1)!}}{\frac{(3n)!}{n! (2n + 1)!}} \\
& = & \frac{\frac{(3n - 3)!}{(n - 1)! (2n - 1)!}}{\frac{(3n - 3)! (3n - 2) (3n - 1) (3n)}{(n - 1)! n (2n - 1)! (2n) (2n + 1)}} \\
& = & \frac{n (2n) (2n + 1)}{(3n - 2) (3n - 1) (3n)} \\
& = & \frac{2n (2n + 1)}{3 (3n - 2) (3n - 1)} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \frac{F_{n - 1}}{F_{n}}
 & = & \lim_{n \to \infty} \frac{2n (2n + 1)}{3 (3n - 2) (3n - 1)} \\
& = & \lim_{n \to \infty} \frac{2 \left ( 2 + \frac{1}{n} \right )}{3 \left ( 3 - \frac{2}{n} \right ) \left ( 3 - \frac{1}{n} \right )} \\
& \approx & \frac{2 (2 + 0)}{3 (3 - 0) (3 - 0)} \\
& = & \frac{4}{27} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} F_{n}
 & = & \left ( \frac{27}{4} \right )^{n} + \, \mbox{constante} \\
\end{eqnarray*}
\end{document}
