For $|x| \lt \frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1-4 x)^2\left(1-2 x^2\right)^{1 / 2}}{(4-x)^{3 / 2}}$ is

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Aicharya · Accepted Answer

$\frac{(1-4 x)^2\left(1-2 x^2ight)^{1 / 2}}{(4-x)^{3 / 2}}$ $\begin{aligned} & =\left(1+16 x^2-8 xight)\left(1-2 x^2ight)^{1 / 2} 4^{-3 / 2}\left(1-\frac{x}{4}ight)^{-3 / 2} \ & =\frac{1}{8}\left(1-8 x+16 x^2ight)\left(1-\frac{2}{2} x^2+\ldotsight)\left(1+\frac{3 x}{2 	imes 4} \ldotsight) \end{aligned}$ So, co-efficient of $x=\frac{1}{8}\left\{\frac{3}{8}-8ight\}=\frac{-61}{64}$.

For $|x| \lt \frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1-4 x)^2\left(1-2…

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