证明:C(0,n)+1/2C(1,n)+1/3C(2,n)+……+1/kC(k-1,n)+……+1/(n+1)C(n,n)=(2^n+1)/(n+1)-1/(n+1)
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证明:C(0,n)+1/2C(1,n)+1/3C(2,n)+……+1/kC(k-1,n)+……+1/(n+1)C(n,n)=(2^n+1)/(n+1)-1/(n+1)
证明:C(0,n)+1/2C(1,n)+1/3C(2,n)+……+1/kC(k-1,n)+……+1/(n+1)C(n,n)=(2^n+1)/(n+1)-1/(n+1)
证明:C(0,n)+1/2C(1,n)+1/3C(2,n)+……+1/kC(k-1,n)+……+1/(n+1)C(n,n)=(2^n+1)/(n+1)-1/(n+1)
由二项式定理,
(1+x)^n=C(0,n)+xC(1,n)+(x^2)C(2,n)+……+(x^(k-1))C(k-1,n)+...+(x^n)C(n,n).
两边对x从0到1积分,得
∫[0,1] (1+x)^n dx=C(0,n)+(1/2)C(1,n)+(1/3)C(2,n)+……+(1/k)C(k-1,n)+...+(1/(n+1))C(n,n),
而左边=(1+x)^(n+1)/(n+1)|[0,1]=2^(n+1)/(n+1)-1/(n+1).即为待证式
(1/k)C(k-1,n)=(1/k)n!/((k-1)(n-k+1))=1/(n+1)C(k,n+1)
所以左边=1/(n+1)C(1,n+1)+1/(n+1)C(2,n+1)+……+1/(n+1)C(n+1,n+1)=1/(n+1)(C(1,n+1)+C(2,n+1)+……+C(n+1,n+1))=1/(n+1)(C(0,n+1)+C(1,n+1)+C(2,n+1)+……+C(n+1,n+1)-1)=(1/(n+1))((2^n+1)-1)