${t_r} = \frac{1}{{4{r^2} - 1}}$
$ = \frac{1}{{(2r - 1)(2r + 1)}} = \frac{1}{2}\left( {\frac{1}{{2r - 1}} - \frac{1}{{2r + 1}}} \right)$
${S_n} = \sum\limits_{r = 1}^n {\frac{1}{2}\left( {\frac{1}{{2r - 1}} - \frac{1}{{2r + 1}}} \right)} $
${S_n} = \frac{1}{2}\left[ {\left( {\frac{1}{1} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \left( {\frac{1}{5} - \frac{1}{7}} \right) + ....... + \left( {\frac{1}{{2n - 1}} - \frac{1}{{2n + 1}}} \right)} \right]$
${S_n} = \frac{1}{2}\left[ {1 - \frac{1}{{2n + 1}}} \right] = \frac{n}{{2n + 1}}$