Find the fourier series

Question

The Fourier series up to the fifth harmonic of f(t), which is periodic of period 4, and which has Fourier coefficients a₀=1, a_n=0 for n≥1 and bn={0     n even; 10/ nπ   n odd

Answer 1

If f(t) has period 4, harmonics are multiples of 2π/4 = π/2

Then, as a₀ = 1, a_n = 0 for n >= 1 and bn = 0, n even and 10/nπ, n odd, we have b₁ = 10/π, b₃ = 10/(3π), and b5 = 2/π

Then, f(t) ≈ 1 + 10/π sin π/2 t + 10/(3π) sin 3π/2 t + 2/π sin 5π/2 t