MT299对数型数列不等式

Posted 2021-02-08 mathstudy

已知数列${dfrac{1}{n}}$的前$n$项和为$S_n$,则下面选项正确的是(      )
A.$S_{2018}-1>ln 2018$
B.$S_{2018}-1<ln 2018$
C.$ln2018<S_{1009}-1$
D.$ln2018>S_{2017}$

分析:这里主要考察$dfrac{x}{1+x}leln(1+x)le x$
令$x=dfrac{1}{n}$累加易得$dfrac{1}{2}+dfrac{1}{3}dots+dfrac{1}{n+1}<ln(n+1)<1+dfrac{1}{2}+dfrac{1}{3}dots+dfrac{1}{n}$
易得答案选B
练习:证明:当$nin N^+$时$dfrac{1}{n+1}+dfrac{1}{n+2}+dots+dfrac{1}{3n+1}<dfrac{9}{8}$
提示:$ln(1+x)gedfrac{2x}{2+x}=dfrac{1}{k}$