${\tan }^{-1}\left(\frac{4}{7}\right)+{\tan }^{-1}\left(\frac{4}{19}\right)+{\tan }^{-1}\left(\frac{4}{39}\right)+{\tan }^{-1}\left(\frac{4}{67}\right)+\mathrm{......}\infty ={\tan }^{-1}k$  then  $k=$

${\tan }^{-1}\left(\frac{4}{7}\right)+{\tan }^{-1}\left(\frac{4}{19}\right)+{\tan }^{-1}\left(\frac{4}{39}\right)+{\tan }^{-1}\left(\frac{4}{67}\right)+\mathrm{......}\infty ={\tan }^{-1}k$
$S_n=\underset{n=\mathrm{\infty }}{lim} \left(\sum _{r-1}^n {\tan }^{-1}\left(\frac{4}{4r^2+3}\right)\right)$
$S_n=\underset{n=\mathrm{\infty }}{lim} \left(\sum _{r-1}^n {\tan }^{-1}\left(\frac{1}{r^2+{\displaystyle \frac{3}{4}}}\right)\right)$
$S_n=\underset{n=\mathrm{\infty }}{lim} \left(\sum _{r-1}^n {\tan }^{-1}\left(\frac{r+{\displaystyle \frac{1}{2}}-\left(r-{\displaystyle \frac{1}{2}}\right)}{1-\left(r^2-{\displaystyle \frac{1}{4}}\right)}\right)\right)$
$\begin{array}{l}\begin{array}{l}=\underset{n\to \mathrm{\infty }}{lim} \sum _{r=1}^n \left({\tan }^{-1}\left(\left(r+\frac{1}{2}\right)\right)-{\tan }^{-1}\left(r-\frac{1}{2}\right)\right)\\ ={\tan }^{-1}(\infty )-{\tan }^{-1}\left(\frac{1}{2}\right)\\ =\frac{\mathrm{\pi }}{2}-{\tan }^{-1}\left(\frac{1}{2}\right)=co{t}^{-1}\left(\frac{1}{2}\right)\\ ={\tan }^{-1}\left(2\right)\end{array}\end{array}$