The coefficient of `x^4` in the expansion of `{sqrt(1+x^2)-x}^(-1)` in ascending powers of `x ,` when `|x|<1,i s` `0` b. `1/2` c. `-1/2` d. `-1/8`

Options

A. 0
B. `1/2`
C. `-1/2`
D. `-1/8`

Answer

D

Explanation

`[sqrt(1+x^(2))-x]^(-1)=(1)/(sqrt(1+x^(2))-x) xx ((sqrt(1+x^(2))+x))/((sqrt(1+x^(2))+x))`
`= (sqrt(1+x^(2))+x)/(1+x^(2)-x^(2))=x+sqrt(1+x^(2))=x+(1+x^(2))^(1//2)`
`= x+1+1/2 x^(2) + 1/2(-1/2) (x^(4))/(2!) + "….."`
Therefore, the coefficient of `x^(4)` is `-1//8`.