Question

请帮我解决一下这道不等式+克拉美定理..

Answer 1

算术平均值大于几何平均值

[(1-1/n)n+1]/(n+1)>[(1-1/n)^n]^[1/(n+1)]

=> n/(n+1)>[(1-1/n)^n]^[1/(n+1)]

=> [n/(n-1)]^n]^[1/(n+1)]>(n+1)/n

=> [n/(n-1)]^n>[(n+1)/n]^(n+1)

当n=2,3,...,n

2^2>(3/2)^3>...>[n/(n-1)]^n>[(n+1)/n]^(n+1)

则[(n+1)/n]^(n+1)[(n+1)/n]^n