Question

Solve $\underset{n\to \infty }{lim}\left[\frac{1}{1-n^2}+\frac{2}{1-n^2}+\frac{3}{1-n^2}+\cdots +\frac{n}{1-n^2}\right]$

Answer 1

$\underset{n\to \infty }{lim}[\frac{1}{1-n^2}+\frac{2}{1-n^2}+....+\frac{n}{1-n^2}]$

$\underset{n\to \infty }{lim}\left[\frac{1+2+\mathrm{3...}+n}{1-n^2}\right]$

$\underset{n\to \infty }{lim}\left[\frac{n(n+1)}{2(1-n^2)}\right]$

$\underset{n\to \infty }{lim}\frac{(1+1/n)}{2(1/n^2-1)}$

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

$\frac{1+0}{2(0-1)}=-\frac{1}{2}$