Question

Evaluate $\underset{n\to \infty }{lim}(\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{6n})$

• A. $\log 6$
• B. $3\log 2$
• C. $2\log 3$
• D. $\log 3$

Answer 1

Answer: (A)

$\log 6$

${lim}_{n\to \infty }(\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{6n})={\sum }_{r=1}^{5n}\frac{1}{n+r}={lim}_{n\to \infty }\frac{1}{n}{\sum }_{r=1}^{5n}\frac{1}{1+\frac{r}{n}}={\int }_0^5\frac{dx}{1+x}={\left[\log (1+x)\right]}_0^5=\log 6$