Final answer:
The Fourier series expansion of a function f(x) is expressed as a sum of sine and cosine functions with specific coefficients, a_n and b_n, which are calculated using definite integrals over one period of the function.
Step-by-step explanation:
The Fourier series expansion formula of a function f(x) is given by:
f(x) = a_0/2 + ∑ (a_n × cos(nωx) + b_n × sin(nωx))
where n ranges from 1 to infinity, and ω is the angular frequency. The coefficients a_0, a_n, and b_n are given by:
a_0 = (1/π) ∫_{-π}^{π} f(x) dx
a_n = (1/π) ∫_{-π}^{π} f(x) cos(nx) dx
b_n = (1/π) ∫_{-π}^{π} f(x) sin(nx) dx
The integral bounds may change depending on the interval over which the function f(x) is defined. If the interval is not symmetric about the origin, the bounds would be adjusted accordingly to cover one full period of the function.