Question:
$\lim _{x \rightarrow \infty} \tan \left\{\sum_{r=1}^{n} \tan ^{-1}\left(\frac{1}{1+r+r^{2}}\right)\right\}$ is equal to
Solution:
$\tan \left(\lim _{n \rightarrow \infty} \sum_{r=1}^{n}\left[\tan ^{-1}(r+1)-\tan ^{-1}(r)\right]\right)$
$=\tan \left(\lim _{n \rightarrow \infty}\left(\tan ^{-1}(n+1)-\frac{\pi}{4}\right)\right)$
$=\tan \left(\frac{\pi}{4}\right)=1$
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