Question

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

Answer 1

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

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

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

$\underset{n\to \infty }{Lim}(\frac{6n}{6n}-\frac{n}{n+1})$

$\underset{n\to \infty }{Lim}(1-\frac{1}{1+\frac{1}{n}})$

$=1-\frac{1}{1+\frac{1}{\infty }}$

$=1-\frac{1}{1+0}$

$=0$