>>381

フーリエ級数展開する。
図を書くと、奇関数だから、a(n)=0
b(n) =(1/π)∫[-π→-π/2] (-x-π)sin(nx) dx
　　　+(1/π)∫[-π/2→π/2] x sin(nx) dx
　　　+(1/π)∫[π/2→π] (-x+π) sin(nx) dx
=
(2/n)cos(nπ) -(1/n)cos(nπ/2) +(2/π)(1/n^2)sin(nπ/2)
-(1/n)cos(nπ/2) +(2/π)(1/n^2)sin(nπ/2)
-(2/n)cos(nπ) +(2/n)cos(nπ/2)
=(4/π)(1/n^2) sin(nπ/2)

f(x)〜納n=1→∞] b(n) sin(nx) =(4/π){(1/1^2)sinx -(1/3^2)sin3x +(1/5^2)sin5x -(1/7^2)sin7x +････}
x=π/2を代入
f(π/2)=π/2=(4/π){(1/1^2)+(1/3^2)+(1/5^2)+(1/7^2)+･･･}

∴(1/1^2)+(1/3^2)+(1/5^2)+(1/7^2)+･･･=(π/2)(π/4)=π^2/8