Why is a fourier-legendre expansion of a polynomial function

Question

Assignment:

1. The first three Legendre polynomials are P₀(x) = 1, P₁(x) = x, and P₂(x) = 1/2(3x²- 1). If x = cosθ , then P₀( cosθ ) = 1 and P₁( cosθ ) = cos θ . Show that P₂( cosθ ) = 1/4( 3cos2θ + 1 ).

2. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (θ) = ^_n=0∞∑ c_n p_n cos(θ) of F( θ ) = 1 - cos2θ .

3. Why is a Fourier-Legendre expansion of a polynomial function that is defined on the interval ( -1, 1 ) necessarily a finite series.

4. Using only your conclusions from problem 10, find the finite Fourier-Legendre series of f(x) = x2