不等式スレッド
>128
【ingleby】
F_n ≡ n[1+x^2+x^4+････+x^(2n)] - (n+1)[x+x^3+････+x^(2n-1)] とおく.
F_1 = [1+x^2] - 2x = (x-1)^2 ≧0.
F_{n+1} - F_n = [1+x^2+････+x^(2n)+(n+1)･x^(2n+2)] - [x+x^3+････+x^(2n+1)+(n+1)･x^(2n+1)]
= (1-x){1+x^2+････+x^(2n) - (n+1)･x^(2n+1)}
= (1-x){1+x^2+････+x^(2n) - (n+1)･x^(2n) + (n+1)(1-x)x^(2n)}
= (1-x){Σ[k=0 to n-1]{x^(2k)-x^(2n)} + (n+1)･(1-x)x^(2n))}
= (1-x){Σ[j=0 to n-1](Σ[k=0 to j]1)･(1-x^2)x^(2j) + (n+1)･(1-x)x^(2n)}
= (1-x)^2{Σ[j=0 to n-1](1+j)･(1+x)x^(2j) + (n+1)･x^(2n)} ≧ 0. (∵すべての係数>0)
∴ xを固定したとき F_{n+1} ≧ F_n ≧ ･････ ≧ F_1 ≧ 0. 等号成立は x=1.（終）
【ingleby】
F_n ≡ n[1+x^2+x^4+････+x^(2n)] - (n+1)[x+x^3+････+x^(2n-1)] とおく.
F_1 = [1+x^2] - 2x = (x-1)^2 ≧0.
F_{n+1} - F_n = [1+x^2+････+x^(2n)+(n+1)･x^(2n+2)] - [x+x^3+････+x^(2n+1)+(n+1)･x^(2n+1)]
= (1-x){1+x^2+････+x^(2n) - (n+1)･x^(2n+1)}
= (1-x){1+x^2+････+x^(2n) - (n+1)･x^(2n) + (n+1)(1-x)x^(2n)}
= (1-x){Σ[k=0 to n-1]{x^(2k)-x^(2n)} + (n+1)･(1-x)x^(2n))}
= (1-x){Σ[j=0 to n-1](Σ[k=0 to j]1)･(1-x^2)x^(2j) + (n+1)･(1-x)x^(2n)}
= (1-x)^2{Σ[j=0 to n-1](1+j)･(1+x)x^(2j) + (n+1)･x^(2n)} ≧ 0. (∵すべての係数>0)
∴ xを固定したとき F_{n+1} ≧ F_n ≧ ･････ ≧ F_1 ≧ 0. 等号成立は x=1.（終）
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