List of practice Questions

Which one of the following options is the correct Fourier series of the periodic function \( f(x) \) described below: \[ f(x) = \begin{cases} 0, & \text{if } -2 < x < -1, \\ 2k, & \text{if } -1 \leq x < 1, \quad \text{period} = 4, \\ 0, & \text{if } 1 \leq x < 2. \end{cases} \]

The following function is defined over the interval \([-L, L]\): \[ f(x) = px^4 + qx^5 \] If it is expressed as a Fourier series, \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \sin \left( \frac{n \pi x}{L} \right) + b_n \cos \left( \frac{n \pi x}{L} \right) \right), \] which options amongst the following are true?