Question

Define the function \(f:(-1,1)\rightarrow (-\frac{\pi}{2},\frac{\pi}{2})\) by
\(f(x)=\sin^{-1}x\)
Let a₆ denote the coefficient of x⁶ in the Taylor series of (f(x))² about x = 0. Then, the value of 9a₆ equals ____________ (rounded off to two decimal places).

Answer 1

Correct Answer: 1.5

The Taylor series of \( f(x) = \sin^{-1} x \) around \( x = 0 \) is known, and we can square this series to find the Taylor series of \( (f(x))^2 \). The coefficient \( a_6 \) corresponds to the term involving \( x^6 \) in this expanded series. Through a detailed expansion, the value of \( a_6 \) can be calculated, and multiplying this by 9 gives the result: \[ 9a_6 \approx 1.50. \] Thus, the correct answer is \( 1.50 \)