Answer
$$\frac{\pi}{4}$$
Work Step by Step
$$\Sigma_{n=1}^{\infty}[tan^{-1}(n+1)-tan^{-1}n]=\lim\limits_{k \to \infty}\Sigma_{n=1}^{k}[tan^{-1}(n+1)-tan^{-1}n]$$$$=\lim\limits_{k \to \infty}tan^{-1}(k+1)-tan^{-1}1$$$$=\frac{\pi}{2}-\frac{\pi}{4}$$$$=\frac{\pi}{4}$$
