四角数の逆数和
Σ1/n^2 =1/1^2+1/2^2+1/3^2+1/4^2+・・・
はπ^2/6に収束する.
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多角数の逆数和は
[1]三角数:n(n+1)/2→Σ2/n(n+1)=2
[2]四角数:n^2→Σ1/n^2=π^2/6
[3]五角数:(3n^2-n)/2→Σ2/n(3n-1)=3log3-π/√3
Σ2/n(3n-1) (n=1~)
=Σ2/(n+1)(3n+2) (n=0~)
=6Σ1/(n+1)(n+2/3) (n=0~)
=2Σ{1/(n+2/3)-1/(n+1)} (n=0~)
Σ{1/(n+2/3)-1/(n+1)}=-π/2√3+log12-2{-1/2・log1/2-1/2・log√3/2}
=-π/2√3+2log2+log3-log2+(1/2・log3-log2)
=-π/2√3+3/2・log3
したがって,
Σ2/n(3n-1)=3log3-π/√3
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[4]六角数:2n^2-n→Σ1/2n(2n-1)=2log2
[5]七角数:n(5n-3)/2→Σ2/n(5n-3)=π/3・(1-2/√5)^1/2+5/6・log5-√5/3・log(φ)
[6]八角数:n(6n-4)/2→Σ2/n(6n-4)π/4√3+3/4・log3
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