Question

Evaluate $\underset{n\to \infty }{lim}[\frac{1}{(n+1)(n+2)}+\frac{1}{(n+2)(n+4)}+....+\frac{1}{6n^2}]$.

Answer 1

${lim}_{n\to \infty }[\frac{1}{(n+1)(n+2)}+\frac{1}{(n+2)(n+4)}+...+\frac{1}{6n^2}]$

${lim}_{n\to \infty }\frac{1}{n^2}[\frac{1}{(1+1/n)(1+2/n)}+\frac{1}{(1+2/n)(1+4/n)}+...+\frac{1}{6}]$

as $n\to \infty ,\frac{1}{n}\to 0$

$\frac{1}{n^2}\to 0$

$=0[\frac{1}{1\times 1}+\frac{1}{1\times 1}+...]$

$=0$