Fourier Series Expansion of f(x) = π * sin(πx)

To find the Fourier series expansion of the given function, f(x) = π * sin(πx), we need to determine the coefficients of the cosine and sine terms. The Fourier series expansion can be written as:

f(x) = a₀ + Σ [aₙ * cos(nπx) + bₙ * sin(nπx)]

We can calculate the coefficients aₙ and bₙ using the following formulas:

a₀ = (1/L) * ∫[f(x)]dx
aₙ = (2/L) * ∫[f(x) * cos(nπx)]dx
bₙ = (2/L) * ∫[f(x) * sin(nπx)]dx

where L is the period of the function, which in this case is 1.

Calculation of a₀:
To calculate a₀, we integrate the function f(x) = π * sin(πx) over the period (0, 1):

a₀ = (1/1) * ∫[π * sin(πx)]dx
= [-cos(πx)] from 0 to 1
= -cos(π) + cos(0)
= -1 + 1
= 0

Calculation of aₙ:
To calculate aₙ, we integrate the function f(x) = π * sin(πx) multiplied by cos(nπx) over the period (0, 1):

aₙ = (2/1) * ∫[π * sin(πx) * cos(nπx)]dx
= [2π/(nπ)] * ∫[sin(πx) * cos(nπx)]dx
= [2π/(nπ)] * [-1/(n+1) * cos((n+1)πx)] from 0 to 1
= [2π/(nπ)] * [-1/(n+1) * cos((n+1)π) + 1/(n+1) * cos(0)]
= [2π/(nπ)] * [-1/(n+1) * (-1)^n + 1/(n+1)]
= 2/(nπ) * [(-1)^n - 1/(n+1)]

Calculation of bₙ:
To calculate bₙ, we integrate the function f(x) = π * sin(πx) multiplied by sin(nπx) over the period (0, 1):

bₙ = (2/1) * ∫[π * sin(πx) * sin(nπx)]dx
= [2π/(nπ)] * ∫[sin(πx) * sin(nπx)]dx
= [2π/(nπ)] * [1/(n-1) * cos((n-1)πx)] from 0 to 1
= [2π/(nπ)] * [1/(n-1) * cos((n-1)π) - 1/(n-1) * cos(0)]
= [2π/(nπ)]