不等式证明,

题目

不等式证明,
1/(n+1)+1/(n+2)+1/(n+3)+..+1/3n>4n/(4n+1)

答案

缺条件吧,应该n为自然数
1/(n+1) + 1/3n -4/(4n+1) = [3n(4n+1)+(n+1)(4n+1)-12n(n+1)]/[3n(n+1)(4n+1)]
= (4n^2-4n+1)/[3n(n+1)(4n+1)]=(2n-1)^2/[3n(n+1)(4n+1)] >0
所以：1/(n+1) + 1/3n > 4/(4n+1)
1/(n+2)+1/(3n-1) -4/(4n+1)=[(3n-1)(4n+1)+(n+2)(4n+1)-4(n+2)(3n-1)]/[(n+2)(3n-1)(4n+1)]
=(2n-3)^2/[(n+2)(3n-1)(4n+1)] >0
所以：1/(n+2) + 1/(3n-1) > 4/(4n+1)
同样：1/(n+3) + 1/(3n-2) > 4/(4n+1)
.
1/2n + 1/(2n+1) > 4/(4n+1)
以上共n个不等式相加,得到：
1/(n+1)+1/(n+2)+1/(n+3)+..+1/3n>4n/(4n+1)