Question

怎样证明数列^n是单调有界数列

Answer 1

设x(n)=[1+(1/n)]^n利用二项式展开有x(n)=1+[n*(1/n)]+[n(n-1)/(n^2*2!)]+[n(n-1)(n-2)/(n^3*3!)]+……+[n(n-1)(n-2)……*3*2*1/(n^n*n!)]整理得x(n)=1+1+{[1-(1/n)]/2!}+{[1-(1/n)][1-(2/n)]/3!}+……+{[1-(1/n)][1-(2/n)]……[1-(n-1/n)]}/n!所以 x(n+1)=1+1+{[(1-1/n+1)]/2!}+{[1-(1/n+1)][1-(2/n+2)]/3!}+……+{[1-(1/n+1)][1-(2/n+1)]……[1-(n-1/n+1)]}/n!+{[(1-1/n+1)][(1-2/n+1)]……[(1-n/n+1)]/(n+1)!}[1-(1/n)]