Question

If \(f(x)=|\sin x|\) for \(-\pi<x<\pi\) and the Fourier series of \(f(x)\) is \[ f(x)=\sum_{n=0}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the value of the Fourier coefficient \(a_0\) is

• A. \(\dfrac{2}{\pi}\)
• B. \(\dfrac{1}{\pi}\)
• C. \(\dfrac{4}{\pi}\)
• D. \(\dfrac{3}{\pi}\)

Hint:

For \[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx), \] \[ \boxed{ a_0=\frac1\pi\int_{-\pi}^{\pi}f(x)\,dx. } \] If the series is written as \[ f(x)=\sum_{n=0}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the constant coefficient becomes \[ \boxed{\dfrac{a_0}{2}=\dfrac{2}{\pi}.} \]

Answer 1

The Correct Option is A