★東大入試作問者になったつもりのスレ★ 第十五問
>>231
Σ[k=0→2n] C[2n,k] = (1+1)^(2n) = 2^(2n),
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)/(右辺) = {2^(2n)}/{(2√n)C[2n,n]} = a[n]
とおく。
a[1] =1,
a[n]/a[n-1] = 4*(n*n)/[(2n)(2n-1)] * √{(n-1)/n}
= √{(n-1)n/(n - 1/2)(n - 1/2)}
= √{(n-1)n/[(n-1)n + (1/4)]}
<1.
なお、
(√π)/2 < a[n] ≦ 1.
Σ[k=0→2n] C[2n,k] = (1+1)^(2n) = 2^(2n),
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)/(右辺) = {2^(2n)}/{(2√n)C[2n,n]} = a[n]
とおく。
a[1] =1,
a[n]/a[n-1] = 4*(n*n)/[(2n)(2n-1)] * √{(n-1)/n}
= √{(n-1)n/(n - 1/2)(n - 1/2)}
= √{(n-1)n/[(n-1)n + (1/4)]}
<1.
なお、
(√π)/2 < a[n] ≦ 1.
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