Question

The Fourier series expansion of the function \( f(x) = |\cos(x)| \) in the interval \( (-\pi, \pi) \) is \( \alpha + \beta \left[ \dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots \right] \).
The values of \( \alpha \) and \( \beta \) respectively are ______.

• A. \( 2/\pi \) and \( 4/\pi \)
• B. \( 4/\pi \) and \( 2/\pi \)
• C. \( \pi/2 \) and \( \pi/4 \)
• D. \( \pi/4 \) and \( \pi/2 \)

Hint:

Compute the constant term and the coefficient of cos(2x) in the Fourier series of |cos x| and compare with the given form.

Answer 1

The Correct Option is A

Step 1: Set up the Fourier coefficient formulas.
\( f(x) = |\cos(x)| \) is even and has period \( \pi \), so its Fourier series on \( (-\pi,\pi) \) has only cosine terms. The constant term is \( a_0 = \dfrac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\,dx \), and the other coefficients are \( a_n = \dfrac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx \).

Step 2: Work out the constant term \( a_0 \).
Because \( f \) is even, \( a_0 = \dfrac{1}{\pi}\int_{0}^{\pi} |\cos(x)|\,dx = \dfrac{1}{\pi}\left[\int_{0}^{\pi/2}\cos(x)\,dx - \int_{\pi/2}^{\pi}\cos(x)\,dx\right] \). Each of those integrals gives \( \sin(x) \) evaluated at the limits, which comes out to 1, so \( a_0 = \dfrac{1}{\pi}(1+1) = \dfrac{2}{\pi} \). This constant term is \( \alpha \), so \( \alpha = \dfrac{2}{\pi} \).

Step 3: Work out the coefficient \( a_2 \) of \( \cos(2x) \).
Carrying out \( a_2 = \dfrac{1}{\pi}\int_{-\pi}^{\pi} |\cos(x)|\cos(2x)\,dx \) using the same even-function split gives \( a_2 = \dfrac{4}{3\pi} \). Comparing this to the given series, the coefficient of \( \cos(2x) \) is written as \( \beta \times \dfrac{1}{3} \), so \( \dfrac{\beta}{3} = \dfrac{4}{3\pi} \), which gives \( \beta = \dfrac{4}{\pi} \).

Step 4: Check consistency with the \( \cos(4x) \) term.
The same integral pattern for \( a_4 \) gives \( a_4 = -\dfrac{4}{15\pi} \). The given series writes this as \( \beta \times \left(-\dfrac{1}{15}\right) \), so \( -\dfrac{\beta}{15} = -\dfrac{4}{15\pi} \), giving \( \beta = \dfrac{4}{\pi} \) again, matching Step 3.

Final Answer:
The constant term is \( \alpha = 2/\pi \) and the shared factor is \( \beta = 4/\pi \).
\[ \boxed{\alpha = \dfrac{2}{\pi}, \ \beta = \dfrac{4}{\pi}} \]