If cot^$-$1($\alpha$) = cot^$-$1 2 + cot^$-$1 8 + cot^$-$1 18 + cot^$-$1 32 + ...... upto 100 terms, then $\alpha$ is :

$${\cot ^{ - 1}}(\alpha ) = co{t^{ - 1}}2 + {\cot ^{ - 1}}8 + {\cot ^{ - 1}}18 + {\cot ^{ - 1}}32 + ....100$$ terms<br><br>$$ = {\tan ^{ - 1}}{1 \over 2} + {\tan ^{ - 1}}{1 \over 8} + {\tan ^{ - 1}}{1 \over {18}} + {\tan ^{ - 1}}{1 \over {32}} + ....100$$ term<br><br>$= \sum\limits_{k = 1}^{100} {{{\tan }^{ - 1}}{1 \over {2{k^2}}}}$<br><br>$$ = \sum\limits_{k = 1}^{100} {{{\tan }^{ - 1}}{2 \over {4{k^2}}} = \sum\limits_{k = 1}^n {{{\tan }^{ - 1}}{{(2k + 1) - (2k - 1)} \over {1 + (2k - 1)(2k + 1)}}} } $$<br><br>$$ = \sum\limits_{k = 1}^{100} {\left( {{{\tan }^{ - 1}}(2k + 1) - {{\tan }^{ - 1}}(2k - 1)} \right)} $$<br><br>$= {\tan ^{ - 1}}201 - {\tan ^{ - 1}}1$<br><br>$= {\tan ^{ - 1}}{{200} \over {202}}$<br><br>$= {\cot ^{ - 1}}(1.01)$<br><br>Hence $\alpha = 1.01$