\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \sqrt{1 - p x} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sqrt{1 - p x}
 & = & \sqrt{1 - x} \times \mbox{een of andere reeks die een functie is van } p \mbox{ en } x \\
f (x)
 & = & g (x) \, h (x) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sqrt{1 - x}
 & = & - \sum_{n = 0}^{n = \infty} \frac{(2n)!}{(2 n - 1) 2^{2n} (n!)^{2}} x^{n} \\
& = & 1 - \frac{1}{2} x - \frac{1}{8} x^{2} - \frac{1}{16} x^{3} - \frac{5}{128} x^{4} - \frac{7}{256} x^{5} - \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \sqrt{1 - p x} \\
& = & \sum_{i = 0}^{i = \infty} a_{i} x^{i} \\
& = & - \sum_{i = 0}^{i = \infty} \frac{(2i)! p^{i}}{(2 i - 1) 2^{2i} (i!)^{2}} x^{i} \\
& = & 1 - \frac{1}{2} p x - \frac{1}{8} p^{2} x^{2} - \frac{1}{16} p^{3} x^{3} - \frac{5}{128} p^{4} x^{4} - \frac{7}{256} p^{5} x^{5} - \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
g (x)
 & = & \sqrt{1 - x} \\
& = & \sum_{j = 0}^{j = \infty} b_{j} x^{j} \\
& = & - \sum_{j = 0}^{j = \infty} \frac{(2j)!}{(2 j - 1) 2^{2j} (j!)^{2}} x^{j} \\
& = & 1 - \frac{1}{2} x - \frac{1}{8} x^{2} - \frac{1}{16} x^{3} - \frac{5}{128} x^{4} - \frac{7}{256} x^{5} - \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
h (x)
 & = & \frac{f (x)}{g (x)} \\
& = & \frac{\sum_{i = 0}^{i = \infty} a_{i} x^{i}}{\sum_{j = 0}^{j = \infty} b_{j} x^{j}} \\
& = & \sum_{n = 0}^{n = \infty} c_{n} x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{n}
 & = & \frac{1}{b_{0}} \left ( a_{n} - \sum_{k = 1}^{k = n} b_{k} c_{n - k} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{0}
 & = & \frac{1}{b_{0}} \left ( a_{0} - \sum_{k = 1}^{k = 0} b_{k} c_{0 - k} \right ) \\
& = & \frac{1}{b_{0}} (a_{0}) \\
& = & \frac{1}{1} (1) \\
& = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{1}
 & = & \frac{1}{b_{0}} \left ( a_{1} - \sum_{k = 1}^{k = 1} b_{k} c_{1 - k} \right ) \\
& = & \frac{1}{1} (a_{1} - b_{1} c_{0}) \\
& = & - \frac{1}{2} p + \frac{1}{2} \cdot 1 \\
& = & - \frac{1}{2} p + \frac{1}{2} \\
& = & \frac{1}{2} (1 - p) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{2}
 & = & \frac{1}{b_{0}} \left ( a_{2} - \sum_{k = 1}^{k = 2} b_{k} c_{2 - k} \right ) \\
& = & \frac{1}{1} (a_{2} - b_{1} c_{1} - b_{2} c_{0}) \\
& = & - \frac{1}{8} p^{2} + \frac{1}{2} \frac{1}{2} (1 - p) + \frac{1}{8} \cdot 1 \\
& = & - \frac{1}{8} p^{2} + \frac{1}{4} (1 - p) + \frac{1}{8} \\
& = & - \frac{1}{8} p^{2} + \frac{1}{4} - \frac{1}{4} p + \frac{1}{8} \\
& = & \frac{3}{8} - \frac{1}{4} p - \frac{1}{8} p^{2} \\
& = & \frac{1}{8} (3 - 2 p - p^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{3}
 & = & \frac{1}{b_{0}} \left ( a_{3} - \sum_{k = 1}^{k = 3} b_{k} c_{3 - k} \right ) \\
& = & \frac{1}{1} (a_{3} - b_{1} c_{2} - b_{2} c_{1} - b_{3} c_{0}) \\
& = & - \frac{1}{16} p^{3} + \frac{1}{2} \frac{1}{8} (3 - 2 p - p^{2}) + \frac{1}{8} \frac{1}{2} (1 - p) + \frac{1}{16} \cdot 1 \\
& = & - \frac{1}{16} p^{3} + \frac{1}{16} (3 - 2 p - p^{2}) + \frac{1}{16} (1 - p) + \frac{1}{16} \\
& = & - \frac{1}{16} p^{3} + \frac{3}{16} - \frac{1}{8} p - \frac{1}{16} p^{2} + \frac{1}{16} - \frac{1}{16} p + \frac{1}{16} \\
& = & \frac{5}{16} - \frac{3}{16} p - \frac{1}{16} p^{2} - \frac{1}{16} p^{3} \\
& = & \frac{1}{16} (5 - 3 p - p^{2} - p^{3}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{4}
 & = & \frac{1}{b_{0}} \left ( a_{4} - \sum_{k = 1}^{k = 4} b_{k} c_{4 - k} \right ) \\
& = & \frac{1}{1} (a_{4} - b_{1} c_{3} - b_{2} c_{2} - b_{3} c_{1} - b_{4} c_{0}) \\
& = & - \frac{5}{128} p^{4} + \frac{1}{2} \frac{1}{16} (5 - 3 p - p^{2} - p^{3}) + \frac{1}{8} \frac{1}{8} (3 - 2 p - p^{2})
 + \frac{1}{16} \frac{1}{2} (1 - p) + \frac{5}{128} \cdot 1 \\
& = & - \frac{5}{128} p^{4} + \frac{1}{32} (5 - 3 p - p^{2} - p^{3}) + \frac{1}{64} (3 - 2 p - p^{2})
 + \frac{1}{32} (1 - p) + \frac{5}{128} \\
& = & - \frac{5}{128} p^{4} + \frac{5}{32} - \frac{3}{32} p - \frac{1}{32} p^{2} - \frac{1}{32} p^{3}
 + \frac{3}{64} - \frac{1}{32} p - \frac{1}{64} p^{2} + \frac{1}{32} - \frac{1}{32} p + \frac{5}{128} \\
& = & \frac{35}{128} - \frac{5}{32} p - \frac{3}{64} p^{2} - \frac{1}{32} p^{3} - \frac{5}{128} p^{4} \\
& = & \frac{1}{128} (35 - 20 p - 6 p^{2} - 4 p^{3} - 5 p^{4}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{5}
 & = & \frac{1}{b_{0}} \left ( a_{5} - \sum_{k = 1}^{k = 5} b_{k} c_{5 - k} \right ) \\
& = & \frac{1}{1} (a_{5} - b_{1} c_{4} - b_{2} c_{3} - b_{3} c_{2} - b_{4} c_{1} - b_{5} c_{0}) \\
& = & - \frac{7}{256} p^{5} + \frac{1}{2} \frac{1}{128} (35 - 20 p - 6 p^{2} - 4 p^{3} - 5 p^{4})
 + \frac{1}{8} \frac{1}{16} (5 - 3 p - p^{2} - p^{3}) \\
 & & \qquad \qquad + \, \frac{1}{16} \frac{1}{8} (3 - 2 p - p^{2}) + \frac{5}{128} \frac{1}{2} (1 - p) + \frac{7}{256} \cdot 1 \\
& = & - \frac{7}{256} p^{5} + \frac{1}{256} (35 - 20 p - 6 p^{2} - 4 p^{3} - 5 p^{4})
 + \frac{1}{128} (5 - 3 p - p^{2} - p^{3}) \\
 & & \qquad \qquad + \, \frac{1}{128} (3 - 2 p - p^{2}) + \frac{5}{256} (1 - p) + \frac{7}{256} \\
& = & - \frac{7}{256} p^{5} + \frac{35}{256} - \frac{5}{64} p - \frac{3}{128} p^{2} - \frac{1}{64} p^{3} - \frac{5}{256} p^{4}
 + \frac{5}{128} - \frac{3}{128} p - \frac{1}{128} p^{2} - \frac{1}{128} p^{3} \\
 & & \qquad \qquad + \, \frac{3}{128} - \frac{1}{64} p - \frac{1}{128} p^{2}
 + \frac{5}{256} - \frac{5}{256} p + \frac{7}{256} \\
& = & \frac{63}{256} - \frac{35}{256} p - \frac{5}{128} p^{2} - \frac{3}{128} p^{3} - \frac{5}{256} p^{4} - \frac{7}{256} p^{5} \\
& = & \frac{1}{256} (63 - 35 p - 10 p^{2} - 6 p^{3} - 5 p^{4} - 7 p^{5}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{0}
 & = & 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{1}
 & = & - \frac{1}{2} p \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{1}
 & = & - \frac{1}{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{1}
 & = & \frac{1}{2} (1 - p) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{2}
 & = & - \frac{1}{8} p^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{2}
 & = & - \frac{1}{8} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{2}
 & = & \frac{1}{8} (3 - 2 p - p^{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{3}
 & = & - \frac{1}{16} p^{3} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{3}
 & = & - \frac{1}{16} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{3}
 & = & \frac{1}{16} (5 - 3 p - p^{2} - p^{3}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{4}
 & = & - \frac{5}{128} p^{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{4}
 & = & - \frac{5}{128} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{4}
 & = & \frac{1}{128} (35 - 20 p - 6 p^{2} - 4 p^{3} - 5 p^{4}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{5}
 & = & - \frac{7}{256} p^{5} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{5}
 & = & - \frac{7}{256} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{5}
 & = & \frac{1}{256} (63 - 35 p - 10 p^{2} - 6 p^{3} - 5 p^{4} - 7 p^{5}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
a_{m}
 & = & - \frac{(2m)! p^{m}}{(2 m - 1) 2^{2m} (m!)^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
b_{m}
 & = & - \frac{(2m)!}{(2 m - 1) 2^{2m} (m!)^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
c_{m}
 & = & \frac{1}{b_{0}} \left ( a_{m} - \sum_{k = 1}^{k = m} b_{k} c_{m - k} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sqrt{1 - p x}
 & = & \sqrt{1 - x} \biggl ( 1 + \frac{1}{2} (1 - p) x + \frac{1}{8} (1 + 2 p - 3 p^{2}) x^{2} \\
 & & \quad \quad \quad \quad \quad + \, \frac{1}{16} (1 + p + 3 p^{2} - 5 p^{3}) x^{3} \\
 & & \quad \quad \quad \quad \quad \, \, + \, \frac{1}{128} (5 + 4 p + 6 p^{2} + 20 p^{3} - 35 p^{4}) x^{4} \\
 & & \quad \quad \quad \quad \quad \, \, \, \, + \, \frac{1}{256} (7 + 5 p + 6 p^{2} + 10 p^{3} + 35 p^{4} - 63 p^{5}) x^{5} \\
 & & \quad \quad \quad \quad \quad \quad \quad + \, \ldots \biggr ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \sqrt{1 - x} \biggl ( 1 + \frac{1}{2} (1 - p) x + \frac{1}{8} (1 + 2 p - 3 p^{2}) x^{2} \\
 & & \quad \quad \quad \quad \quad + \, \frac{1}{16} (1 + p + 3 p^{2} - 5 p^{3}) x^{3} \\
 & & \quad \quad \quad \quad \quad \, \, + \, \frac{1}{128} (5 + 4 p + 6 p^{2} + 20 p^{3} - 35 p^{4}) x^{4} \\
 & & \quad \quad \quad \quad \quad \, \, \, \, + \, \frac{1}{256} (7 + 5 p + 6 p^{2} + 10 p^{3} + 35 p^{4} - 63 p^{5}) x^{5} \\
 & & \quad \quad \quad \quad \quad \quad \quad + \, \ldots \biggr ) \\
\end{eqnarray*}
\end{document}
