Question

The Fourier series expansion of \( x^3 \) in the interval \( -1 \leq x \leq 1 \) with periodic continuation has

• A. only sine terms
• B. only cosine terms
• C. both sine and cosine terms
• D. only sine terms and a non-zero constant

Hint:

For odd functions, the Fourier series consists of only sine terms, since sine functions are odd.

Answer 1

The Correct Option is A

Step 1: Analyze the function \( x^3 \).
The function \( x^3 \) is an odd function, meaning \( f(-x) = -f(x) \). The Fourier series expansion of an odd function contains only sine terms, as sine functions are odd. Step 2: Conclusion.
Thus, the Fourier series expansion of \( x^3 \) will have only sine terms. Final Answer: \[ \boxed{\text{Only sine terms}} \]