## University Calculus: Early Transcendentals (3rd Edition)

$-\dfrac{\pi}{4}$
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: $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} [(\tan ^{-1} (1) - \tan ^{-1} (n+1))]=\tan^{-1} (1)-\dfrac{\pi}{2}=\dfrac{\pi}{4}-\dfrac{\pi}{2}=-\dfrac{\pi}{4}$