Answer
$-\dfrac{\pi}{4}$
Work Step by Step
Here, we have: $ s_n=(\tan ^{-1} (1) - \tan ^{-1} (2))+(\tan ^{-1} (2) - \tan ^{-1} (3))+.......+(\tan ^{-1} (n) - \tan ^{-1} (n+1))=(\tan ^{-1} (1) - \tan ^{-1} (n+1))$
Thus, we have the sum: $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} [(\tan ^{-1} (1) - \tan ^{-1} (n+1))]=\tan^{-1} (1)-\dfrac{\pi}{2}=\dfrac{\pi}{4}-\dfrac{\pi}{2}=-\dfrac{\pi}{4}$
