推 kyoiku : 謝啦 01/14 16:10
※ 引述《kyoiku (生死間有大恐怖)》之銘言：
: f(x) = / -1, -pi <= x < 0
: \ 1, 0 <= x <= pi
: (a) Find the Fourier series of f(x) on [-pi , pi]
n = 0
a_0 = 0
n > 0
π
a_n = (1/π)∫f(x) cos(nx) dx
-π
= (1/π) 0
π
b_n = (1/π) ∫f(x) sin(nx) dx
-π
π
= (2/π) (-cos(nx))/n |
0
= (2/π) [(-1)^(n+1) + 1]/n
∞
f(x) = Σ 4/[π(2k+1)] * sin((2k+1)x)
k=0
: (b) Use (a) to find the sum of the series
: 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +-......
原式 = (π/4)f(π/2)
= π/4
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