An=2^n*sin(π/3n)比较An+1（n+1是下标）和An的大小

Question

An=2^n*sin(π/3n)比较An+1（n+1是下标）和An的大小

Answer 1

An=2^n*sin(π/3n)
An+1=2^(n+1)*sin(π/3(n+1))
An+1/An = 2 sin(π/3(n+1))/sin(π/3n)
因为
π/3n 0
2π/3(n+1) 2sin(π/3(n+1))cos(π/3(n+1))
= sin(2π/3(n+1))
= sin(π/3n * 2n/(n+1))
= sin(π/3n * (n+n)/(n+1))
>= sin(π/3n * (n+1)/(n+1))
= sin(π/3n)
所以 An+1/An = 2 sin(π/3(n+1))/sin(π/3n) > sin(π/3n)/sin(π/3n) =1
An+1 >An