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)
问题描述：
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)
答
(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)
答
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...