Question

The function $f(x)=\begin{cases}-x,\; -\pi<x<0 \\ x,\; 0<x<\pi\end{cases}$ is expanded as a Fourier series of the form $a_0 + \sum_{n=1}^{\infty}a_n\cos(nx)+\sum_{n=1}^{\infty}b_n\sin(nx)$. Which of the following is true? 
 

• A. $a_0\neq 0,\; b_n=0$
• B. $a_0\neq 0,\; b_n\neq 0$
• C. $a_0=0,\; b_n=0$
• D. $a_0=0,\; b_n\neq 0$

Hint:

Odd functions have only sine terms in Fourier series; even functions have only cosine terms.

Answer 1

The Correct Option is A

Step 1: Identify symmetry of the function.
$f(x) = -x$ for $x<0$ and $x$ for $x>0$ ⇒ this is an odd function: $f(-x) = -f(x)$.

Step 2: Fourier coefficients for odd functions.
For any odd function: • $a_0 = 0$, • all cosine coefficients $a_n = 0$, • only sine terms $b_n$ survive.

Step 3: Conclusion.
Thus the Fourier series contains only sine terms: $a_0=0$ and $b_n\neq 0$. This corresponds to option (D).