Q1, 2: Options are Odd or Even extensions.
My attempt.
1. Odd extension
2. Even extension (guessing, just looking at graph)
Please help me out, cheers.
3) b(n) = (2/π) ∫(t = 0 to π) (π - t) sin(nπt/π) dt
............= (2/π) [(π - t) * -cos(nt)/n - (-1)(-sin(nt)/n^2)] {for t = 0 to π}
............= 2/n.
4) a(0) = (2/π) ∫(t = 0 to π) (π - t) dt
............= (2/π) [(πt - (1/2)t^2) {for t = 0 to π}
............= π.
5 )For n > 0:
a(n) = (2/π) ∫(t = 0 to π) (π - t) cos(nπt/π) dt
.......= (2/π) [(π - t) * sin(nt)/n - (-1)(-cos(nt)/n^2)] {for t = 0 to π}
.......= (2/π) * -((-1)^n - 1)/n^2
.......= 0 if n is even and 4/(πn^2) if n is odd.