Question

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

Answer 1

$=\underset{x\to \infty }{lim}\left(\frac{1}{x^2}+\frac{2}{x^2}+\frac{3}{x^2}+...+\frac{x}{x^2}\right)$

$=\underset{x\to \infty }{lim}\frac{1}{x^2}\left(1+2+3+.......+x\right)$

$=\underset{x\to \infty }{lim}\frac{1}{x^2}\times \frac{x\left(x+1\right)}{2}$

$=\underset{x\to \infty }{lim}\frac{1}{x^2}\times \frac{x^2}{2}\left(1+\frac{1}{x}\right)$

$\underset{x\to \infty }{lim}\frac{1}{2}\left(1+\frac{1}{x}\right)$

$=\frac{1}{2}$