12

12. [Maximum mark: 20]

(a) Let $f(x) = (1 - ax)^{-\frac{1}{2}}$, where $ax < 1, a \neq 0$. The $n^{\text{th}}$ derivative of $f(x)$ is denoted by $f^{(n)}(x), n \in \mathbb{Z}^+$. Prove by induction that
$f^{(n)}(x) = \frac{a^n(2n-1)! \,(1-ax)^{-\frac{2n+1}{2}}}{2^{2n-1}(n-1)!},\; n \in \mathbb{Z}^+.$

(b) By using part (a) or otherwise, show that the Maclaurin series for $f(x) = (1 - ax)^{-\frac{1}{2}}$ up to and including the $x^2$ term is $1 + \frac{1}{2}ax + \frac{3}{8}a^2x^2$.

(c) Hence, show that $(1 - 2x)^{-\frac{1}{2}}(1 - 4x)^{-\frac{1}{2}} \approx \frac{2 + 6x + 19x^2}{2}$.

(d) Given that the series expansion for $(1 - ax)^{-\frac{1}{2}}$ is convergent for $|ax| < 1$, state the restriction which must be placed on $x$ for the approximation
$(1 - 2x)^{-\frac{1}{2}}(1 - 4x)^{-\frac{1}{2}} \approx \frac{2 + 6x + 19x^2}{2}$ to be valid.

(e) Use $x = \frac{1}{10}$ to determine an approximate value for $\sqrt{3}$. Give your answer in the form $\frac{c}{d}$, where $c, d \in \mathbb{Z}^+$.