Question

If $f(x) = x^3$, $0 \le x \le 4$, $f(x+4) = f(x)$ $\forall x \in \mathbb{R}$ and the Fourier series of $f(x)$ is $f(x) = \sum_{n=0}^{\infty} \left(a_n \cos \frac{n\pi x}{2} + b_n \sin \frac{n\pi x}{2}\right)$, then $a_0 =$}

• A. \(8\)
• B. \(32\)
• C. \(16\)
• D. \(24\)

Hint:

Always double check whether a Fourier question writes the constant term as $\frac{a_0}{2}$ or simply as part of a summation $\sum a_n \cos(\dots)$. If written inside the summation directly, $a_0$ represents the exact absolute mathematical average $\frac{1}{T}\int f(x)dx$ instead of $\frac{2}{T}\int f(x)dx$.

Answer 1

The Correct Option is C