Question

怎样证明数列(1+1/n)^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)]