Question

Find the sum (Hint: Consider the Fourier series for the function f(t)nandf(t +k)-f(t) for all integer k.)

Answer 1

The Fourier series of F(x) in [-1/2, 1/2] is f(x) = a_0/2 + sigma^infinity_n = 1 an cos n pix/L + sigma^infinity_n = 1 bn sin n pi x/L (1) Because 2L = 1 L = 1/2 a_0 = 1/L integral^1/2_-1/2 f(x) dx = 1/1/2 integral^1/2_-1/2 x^2 dx = 2 integral^1/2_-1/2 x^2 dx = 2 middot [x^3/3]^1/2_-1/2 = 2/3 [(1/2)^3 - (-1/2)^3] = 2/3 [1/8 - (-1/8)] = 2/3 [1/8 + 1/8] = 2/3 [2/8] = 4/3.8 a_0 = 4^1/3.8_2 = 1/6 Therefore a_0 = 1/6 (2) an = 1/L integral^L_-L F(x) middot cos (n pi x/1/2) dx = 1/1/2 integral^1/2_-1/2 x^2_u cos_v(2n pi x) dx = 2 [x^2 sin 2n pi x/2n pi -2x (-cos 2 n pi x/2^2 n^2 pi^2) + 2 (-sin 2n pi x/2^3 n^3 pi^3)]^1/2_-1/2 \r\n Therefore a_n = [(1/2)^2 sin 2n pi 1/2/2n pi + 2 times 1/2 cos (2n pi times 1/2)/4n^2 pi^2 -2 sin 2n pi times 1/2/8 n^3