Find the expansion of ${{\left( \frac{1+2{{x}^{2}}}{4-x} \right)}^{\frac{1}{2}}}$ in ascending pow

Question

Find the expansion of ${{\left( \frac{1+2{{x}^{2}}}{4-x} \right)}^{\frac{1}{2}}}$ in ascending powers of $x$, up to and including the term in ${{x}^{2}}$.[4] Find the set of values of $x$ for which the expansion is valid.[2] By putting $x=\frac{1}{4}$, show that $\sqrt{30}\approx \frac{a}{b}$, where $a$ and $b$ are integers to be determined.[2]

Accepted Answer

$\frac{1}{2}\left( 1+\frac{1}{8}x+\frac{131}{128}{{x}^{2}} \right)+...$ (i) $\left\{ x\in \mathbb{R}:-\frac{1}{\surd 2}< x < \frac{1}{\surd 2} \right\}$ (ii) $\frac{12288}{2243}$ or $\frac{11215}{2048}$

2013 IJC P1 Q8

2013 IJC P1 Q8