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^–1r

$$ \therefore $$ T₁ = tan^–12 – tan^–11

T₂ = tan^–13 – tan^–12

T₃ = tan^–14 – tan^–13
.
.
.

T₁₀ = tan^-111 – tan^–110

$$ \therefore $$ S = tan^–111 – tan^–11 = $${\tan ^{ - 1}}\left( {{{11 - 1} \over {1 + 11}}} \right)$$

$$ \therefore $$ tan(S) = $$\tan \left( {{{\tan }^{ - 1}}\left( {{{11 - 1} \over {1 + 11}}} \right)} \right)$$

= $${{{11 - 1} \over {1 + 11}}}$$ = $${{10} \over {12}} = {5 \over 6}$$