Answer
$\dfrac{\pi}{4}$
Work Step by Step
Let us consider an infinite series $\Sigma_{n=k}^{n=\infty} a_n$. This series will converge to the sum $S$ when the sequence of its partial sums $\{S_n\}$ converge to $S$.
That is, $S=\lim\limits_{N \to \infty} S_n$ and $S=\Sigma_{n=k}^{n=\infty} a_n$.
This series will diverge when the limit does not exist and the series will diverge to infinity when the limit is infinite.
Here, we have
$S_n= \Sigma_{k=1}^{n} [\tan^{-1} (k+1) -\tan^{-1} k]$
or, $= [\tan^{-1} (2) -\tan^{-1} (1)]+[\tan^{-1} (3) -\tan^{-1} (2)]+........+[\tan^{-1} (n) -\tan^{-1} (n-1)]+[\tan^{-1} (n+1) -\tan^{-1} (n)]$
or, $= \tan^{-1} (n+1) -\tan^{-1} (1)$
Now, $S=\lim\limits_{n \to \infty} S_n=\lim\limits_{n \to \infty} [\tan^{-1} (n+1) -\tan^{-1} (1)]=\lim\limits_{n \to \infty} [\tan^{-1} (n+1)] -\lim\limits_{n \to \infty}\tan^{-1} (1)$
or, $S=\dfrac{\pi}{2}-\dfrac{\pi}{4}$
or, $S=\dfrac{\pi}{4}$
