$\lim\Bigg[ \dfrac{1}{1.3}+\dfrac{1}{3.5}+...+\dfrac{1}{(2n-1)(2n+1)}\Bigg]$
$=\dfrac{1}{2}\Bigg(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\Bigg)$
$=\dfrac{1}{2}\Bigg(1-\dfrac{1}{2n+1}\Bigg)$
$⇒ \lim\Bigg[ \dfrac{1}{1.3}+\dfrac{1}{3.5}+...+\dfrac{1}{(2n-1)(2n+1)}\Bigg]=\lim\dfrac{1}{2}\Bigg(1-\dfrac{1}{2n+1}\Bigg)=\dfrac{1}{2}$
