First slide

Binomial theorem for negative integral and rational Index

Question

The coefficient of x⁴ in the expansion of ${\{\sqrt{1 + x^{2}} - x\}}^{- 1}$ in ascending powers of x, when $| x | < 1$ , is

Moderate

A

0
B

$\frac{1}{2}$
C

$- \frac{1}{2}$
D

$- \frac{1}{8}$

Solution

$\begin{matrix} {[\sqrt{1 + x^{2}} - x]}^{- 1} = \frac{1}{\sqrt{1 + x^{2}} - x} \times \frac{(\sqrt{1 + x^{2}} + x)}{(\sqrt{1 + x^{2}} + x)} \\ = \frac{\sqrt{1 + x^{2}} + x}{1 + x^{2} - x^{2}} = x + \sqrt{1 + x^{2}} = x + {(1 + x^{2})}^{1 / 2} \\ = x + 1 + \frac{1}{2} x^{2} + \frac{1}{2} (- \frac{1}{2}) \frac{x^{4}}{2!} + \dots \end{matrix}$
Therefore, the coefficient of x⁴ is -1/8.

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