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

1/2+x/16+131x^2/256+...; (i) -1/sqrt(2)<x<1/sqrt(2); (ii) sqrt(30)≈11215/2048 (or 12288/2243).

2013 IJC P1 Q8

2013 IJC P1 Q8

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