设n为正整数,证明1+(1/2)C(n,1)+(1/3)C(n,2)+(1/4)C(n,3)+…+(1/n+1)C(n,n)=(1/n+1)((2^(n+1))-1)母函数解题中必须涉及组合数学母函数的概念
2019-03-30
设n为正整数,证明1+(1/2)C(n,1)+(1/3)C(n,2)+(1/4)C(n,3)+…+(1/n+1)C(n,n)=(1/n+1)((2^(n+1))-1)
母函数
解题中必须涉及组合数学母函数的概念
优质解答
证明：取
Tr＝(1/(r+1))C(n,r).r=0,1,2,...,n.
则：Tr=(1/(r+1))*(n!)/[(r!)(n-r)!]
=(1/(n+1))*[(n+1)!]/[(r+1)!*(n-r)!]
=(1/(n+1))C(n+1,r+1).
∴∑Tr=(1/(n+1))∑C(n+1,r+1)[r=0,1,2,...,n]
=(1/(n+1))[2^(n+1)-1].
原等式获证!
证明：取
Tr＝(1/(r+1))C(n,r).r=0,1,2,...,n.
则：Tr=(1/(r+1))*(n!)/[(r!)(n-r)!]
=(1/(n+1))*[(n+1)!]/[(r+1)!*(n-r)!]
=(1/(n+1))C(n+1,r+1).
∴∑Tr=(1/(n+1))∑C(n+1,r+1)[r=0,1,2,...,n]
=(1/(n+1))[2^(n+1)-1].
原等式获证!