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${S_n} = \frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + ......... = ?$

${t_r} = \frac{1}{{4{r^2} - 1}}$

$ = \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)} $

${S_n} = \frac{1}{2}\left[ {\left( {\frac{1}{1} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \left( {\frac{1}{5} - \frac{1}{7}} \right) + ....... + \left( {\frac{1}{{2n - 1}} - \frac{1}{{2n + 1}}} \right)} \right]$

${S_n} = \frac{1}{2}\left[ {1 - \frac{1}{{2n + 1}}} \right] = \frac{n}{{2n + 1}}$