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

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

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