Question

(1/2)x = sin(x) - (1/2)sin(2x) + (1/3)sin(3x) + ...:

a) Fourier series expansion of x
b) Taylor series expansion of sin(x)
c) Maclaurin series expansion of sin(x)
d) General series expansion for x

Answer 1

The series given is the Fourier series expansion of x, representing a sawtooth wave as a sum of sine functions with coefficients that decrease with increasing frequency. The correct answer is a) Fourier series expansion of x.

Step-by-step explanation:

The series given in the question resembles the Fourier series expansion of a function. Specifically, it is the Fourier sine series for a sawtooth wave which is a graphical representation of the function x.

The series ((1/2)x = sin(x) - (1/2)sin(2x) + (1/3)sin(3x) + ...) expresses x as a sum of sine functions with coefficients that decrease in magnitude as the frequency increases. In this context, the correct answer would be (a) Fourier series expansion of x, which represents a periodic function that can be broken down into a sum of sine waves with different frequencies and amplitudes.

The given series (1/2)x = sin(x) - (1/2)sin(2x) + (1/3)sin(3x) + ... can be expanded as a Fourier series.

A Fourier series expansion is a representation of a function as a sum of sine and cosine functions.

Each term in the series represents a multiple of sine function with increasing frequency.