$\begin{aligned} & \text { } \lim _{n \rightarrow \infty} \sum_{r=1}^n \cot ^{-1}\left(r^2+\frac{3}{4}\right) \\ & =\lim _{n \rightarrow \infty} \sum_{r=1}^n \tan ^{-1}\left[\frac{1}{r^2+1-\frac{1}{4}}\right] \\ & =\lim _{n \rightarrow \infty} \sum_{r=1}^n \tan ^{-1}\left[\frac{\left(r+\frac{1}{2}\right)-\left(r-\frac{1}{2}\right)}{1+\left(r+\frac{1}{2}\right)\left(r-\frac{1}{2}\right)}\right] \\ & =\lim _{n \rightarrow \infty} \sum_{r=1}^n\left[\tan ^{-1}\left(r+\frac{1}{2}\right)-\tan ^{-1}\left(r-\frac{1}{2}\right)\right], \\ & \quad\left[\because \tan ^{-1}(a-b)=\tan ^{-1}\left[\frac{a-b}{1+a \cdot b}\right]\right]\end{aligned}$
$\begin{aligned} & =\lim _{n \rightarrow \infty}\left[\left(\tan ^{-1} \frac{3}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\right)\right]+\left[\tan ^{-1} \frac{5}{2}-\tan ^{-1} \frac{3}{2}\right] \\ & \quad+\left[\tan ^{-1} \frac{7}{2}-\tan ^{-1} \frac{5}{2}\right] \ldots \\ & +\left[\tan ^{-1}\left(n+\frac{1}{2}\right)-\tan ^{-1}\left(n-\frac{1}{2}\right)\right] \\ & =\lim _{n \rightarrow \infty}\left[\tan ^{-1}\left(n+\frac{1}{2}\right)-\tan ^{-1}\left(\frac{1}{2}\right)\right] \\ & =\lim _{n \rightarrow \infty}\left[\tan ^{-1}\left(n+\frac{1}{2}\right)-\tan ^{-1}\left(\frac{1}{2}\right)\right] \\ & =\tan ^{-1} \infty-\tan ^{-1} \frac{1}{2}=\frac{\pi}{2}-\tan ^{-1} \frac{1}{2} \\ & =\cot ^{-1} \frac{1}{2}=\tan ^{-1} 2\end{aligned}$