Limit+sum

Using partial fraction decomposition,

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(2n-1)(2n+1)}=\frac{1}{2}\sum_{n=1}^{\infty}\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)$

$\displaystyle=\lim_{n\to\infty}\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\cdots-\frac{1}{2n-1}+\frac{1}{2n-1}-\frac{1}{2n+1}\right)$

$\displaystyle=\lim_{n\to\infty}\frac{1}{2}\left(1-\frac{1}{2n+1}\right)=\boxed{\frac{1}{2}}$ .