Question

$\stackrel{lim}{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\cdots \frac{1}{n+n}]$ is equal to

• A. 3 log 2
• B. log 2
• C. 2 log 2
• D. None of these

Answer 1

The correct option is B log 2

$\stackrel{lim}{n\to \infty }[\frac{1}{n+1}+\frac{1}{n+2}+\cdots \frac{1}{n+n}]$
$=\stackrel{lim}{n\to \infty }{\sum }_{r=1}^n\frac{1}{n+r}=\stackrel{lim}{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}.\frac{1}{1+\frac{r}{n}}$
$={\int }_0^1\frac{1}{1+x}dx$
$=\left[log\right(1+x){]}_0^1=log2.$