Question

${lim}_{n\to \infty }(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+....+\frac{n-1}{n^2})$

Answer 1

${lim}_{n\to \infty }(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+....+\frac{n-1}{n^2})$

${lim}_{n\to \infty }\left(\frac{1+2+3+....(n-1)}{n^2}\right)$

$={lim}_{n\to \infty }\frac{(n-1)\left(n\right)}{2\times n^2}$

$[1+2+3+....+(n-1)=\frac{(n-1)\left(n\right)}{2}]$

$={lim}_{n\to \infty }\frac{n^2-n}{2n^2}\left[\frac{\infty }{\infty }form\right]$

$={lim}_{n\to \infty }\frac{1-\frac{1}{n}}{2}$

$=\frac{1-0}{2}=\frac{1}{2}$

$=\frac{1}{2}$