我来个初等证明吧
(1+1/n)^n=(1+1/n)(1+1/n)...(1+1/n)*1
由均值不等式
它小于等于
{[(1+1/n)+(1+1/n)+...+(1+1/n)+1]/(n+1)}^(n+1)
=(1+1/(n+1))^(n+1)
所以(1+1/n)^n递增!
n=2时,(1+1/n)^n=2.25
所以都比2大
又(1+1/n)^n < (1+1/n)^(n+1)
且同理可证(1+1/n)^(n+1)递减
再知道n=3时,
(1+1/n)^(n+1)<3
所以n>3时
(1+1/n)^n < (1+1/n)^(n+1)
<(1+1/3)^4