\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
y
 & = & \frac{1}{\sqrt{1 + (ax)^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & \frac{u}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
u
 & = & ax \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \frac{1}{\sqrt{1 + (ax)^{2}}} \\
& = & \frac{1}{\sqrt{1 + \left ( a \frac{u}{a} \right )^{2}}} \\
& = & \frac{1}{\sqrt{1 + u^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \frac{1}{\sqrt{1 + x^{2}}} \\
& = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (- x^{2})^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \frac{1}{\sqrt{1 + u^{2}}} \\
& = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (- u^{2})^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (- u^{2})^{n} \\
& = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (- (ax)^{2})^{n} \\
& = & 1 - \frac{1}{2} (ax)^{2} + \frac{3}{8} (ax)^{4} - \frac{5}{16} (ax)^{6} + \frac{35}{128} (ax)^{8} - \frac{63}{256} (ax)^{10} + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\mbox{Term } (n + 1)}{\mbox{Term } (n)}
 & = & \frac{\frac{(2 (n + 1))!}{2^{2 (n + 1)} ((n + 1)!)^{2}} (- (ax)^{2})^{n + 1}}{\frac{(2n)!}{2^{2n} (n!)^{2}} (- (ax)^{2})^{n}} \\
& = & \frac{\frac{(2n + 2)!}{2^{2n + 2} (n!)^{2} (n + 1)^{2}} (- (ax)^{2})}{\frac{(2n)!}{2^{2n} (n!)^{2}}} \\
& = & - \frac{\frac{(2n)! (2n + 1) (2n + 2)}{4 (n + 1)^{2}} (ax)^{2}}{(2n)!} \\
& = & - \frac{2 (2n + 1) (n + 1)}{4 (n + 1)^{2}} (ax)^{2} \\
& = & - \frac{2n + 1}{2 (n + 1)} (ax)^{2} \\
& = & - \frac{2n + 1}{2n + 2} (ax)^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \left | \, \frac{\mbox{Term } (n + 1)}{\mbox{Term } (n)} \, \right |
 & < & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\lim_{n \to \infty} \left | \, \frac{\mbox{Term } (n + 1)}{\mbox{Term } (n)} \, \right |
 & = & \lim_{n \to \infty} \left | \, - \frac{2n + 1}{2n + 2} (ax)^{2} \, \right | \\
& = & \lim_{n \to \infty} \left | \, \frac{2 + \frac{1}{n}}{2 + \frac{2}{n}} (ax)^{2} \, \right | \\
& \approx & \left | \, \frac{2 + 0}{2 + 0} (ax)^{2} \, \right | \\
& = & | \, (ax)^{2} \, | \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
y
 & = & \sum_{n = 0}^{n = \infty} \frac{(2n)!}{2^{2n} (n!)^{2}} (- (ax)^{2})^{n} \\
& & | \, x \, | \, \, < \, \, \frac{1}{a} \\
\end{eqnarray*}
\end{document}
