S = ${\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan ^{ - 1}}\left( {{1 \over {13}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {21}}} \right) + ....$ = ${\tan ^{ - 1}}\left( {{1 \over {1 + 1 \times 2}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {1 + 2 \times 3}}} \right) + ...$ $\therefore$ T r = ${\tan ^{ - 1}}\left( {{1 \over {1 + r \times \left( {r + 1} \right)}}} \right)$ = tan –1 (r + 1) – tan –1 r $\therefore$ T 1 = tan –1 2 – tan –1 1 T 2 = tan –1 3 – tan –1 2 T 3 = tan –1 4 – tan –1 3 . . . T 10 = tan -1 11 – tan –1 10 $\therefore$ S = tan –1 11 – tan –1 1 = ${\tan ^{ - 1}}\left( {{{11 - 1} \over {1 + 11}}} \right)$ $\therefore$ tan(S) = $\tan \left( {{{\tan }^{ - 1}}\left( {{{11 - 1} \over {1 + 11}}} \right)} \right)$ = ${{{11 - 1} \over {1 + 11}}}$ = ${{10} \over {12}} = {5 \over 6}$