$\lim _{n \rightarrow \infty}\left\{\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}}+\ldots+\frac{n}{1-n^{2}}\rig

Question

$\lim _{n ightarrow \infty}\left\{\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}}+\ldots+\frac{n}{1-n^{2}}ight\}$ is equal to\

Professor PC Thomas Classes · Answer

(b) : Here $\lim _{n ightarrow \infty}\left\{\begin{array}{c}\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}} \ +\ldots+\frac{n}{1-n^{2}}\end{array}ight\}$\ $=\lim _{n ightarrow \infty}\left(\frac{1+2+\ldots+n}{1-n^{2}}ight)$\ $=\lim _{n ightarrow \infty} \frac{n(n+1)}{2(1+n)(1-n)}=\lim _{n ightarrow \infty} \frac{n}{2 n\left(\frac{1}{n}-1ight)}$\ $=\lim _{n ightarrow \infty} \frac{1}{2\left(\frac{1}{n}-1ight)}=-\frac{1}{2}$\