Question

For the Fourier series of the following function of period \( 2\pi \): 

The ratio (to the nearest integer) of the Fourier coefficients of the first and the third harmonic is: 
 

• A. 1
• B. 2
• C. 3
• D. 6

Hint:

The Fourier coefficients can be computed using integration, and the ratio of the first and higher harmonics reveals the relative strengths of the frequencies in the signal.

Answer 1

The Correct Option is D

Step 1: Fourier coefficients.
The Fourier series of a function \( f(x) \) is given by the sum of sines and cosines of integer multiples of the fundamental frequency. The first and third Fourier coefficients correspond to the fundamental frequency and its third harmonic, respectively. By calculating the Fourier coefficients, we find the ratio of the first and third coefficients to be 6.
Step 2: Conclusion.
Thus, the correct answer is option (D), with the ratio being 6.