08/07/10 23:05:37
>>360
(分子) = Σ[k=0→2n] C[2n,k] = (1+1)^(2n) = 2^(2n),
(分母) = Σ[k=0→n] (C[n,k])^2 = Σ[k=0→n] C[n,k]･C[n,n-k] = C[n+n,n],
より
(左辺) = {2^(2n)}/C[2n,n] = b[n]
とおく。
b[1] = 2 = √(2n),
b[n]/b[n-1] = 4*(n*n)/[(2n)(2n-1)] = n/(n - 1/2)
= √{n*n/(n - 1/2)(n - 1/2)}
= √{n/(n-1)} * √{(n-1)n/[(n-1)n + (1/4)]}
< √{n/(n-1)}.
∴ b[n]/√(2n) は単調減少。
なお、
b[n]/√(2n) → (√π)/2, (n→∞)
ｽﾚﾘﾝｸ(math板:239番)