证明不等式的高二数学题n∈N+,证明: 1<1/(n+1)+1/(n+2)+1/(n+3)+……+1/(3n+1)<2

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证明不等式的高二数学题n∈N+,证明: 1<1/(n+1)+1/(n+2)+1/(n+3)+……+1/(3n+1)<2
证明不等式的高二数学题
n∈N+,证明:
1<1/(n+1)+1/(n+2)+1/(n+3)+……+1/(3n+1)<2

证明不等式的高二数学题n∈N+,证明: 1<1/(n+1)+1/(n+2)+1/(n+3)+……+1/(3n+1)<2
令f(n)=1/(n+1) + 1/(n+2) +1/(n+3) +……+1/(3n+1)
f(n+1)=1/(n+2) + 1/(n+3) +1/(n+4) +……+1/[3(n+1)+1]
f(n+1)-f(n)=1/(n+1) - 1/(3n+2)-1/(3n+3)-1/(3n+4)>0
所以函数f(n)对于n为正整数时为单调增函数
所以f(n)>1/2+1/3+1/4=13/12 >1
再来证小于2
1/(n+1)>1/(n+2)>1/(n+3)>……>1/(3n+1)
1/(n+1)+1/(n+2)+1/(n+3)+……+1/(3n+1)<2n/(n+1)=2 -2/(n+1)

一共有 2N项相加!
1/(n+1)>1/(n+2)>1/(n+3)>……>1/(3n+1)
采用放缩法! 变成 小于2N 个 1/(N+1) 所以 结果是
2N/(N+1)=2 -2/(n+1) 所以 结果是 小于 2的 !