Question

The sum of the infinite series $cot^{-1}\, 2 + cot^{-1}\, 8 + cot^{-1}\, 18 + cot^{-1}\, 32 + ....$ is

• A. $\pi$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{4}$
• D. None of these

Answer 1

Answer: (C)

Let $S_{\infty}=cot^{-1}\,2+cot^{-1}\,8+cot^{-1}\,18+cot^{-1}\,32+...$
$\therefore T_{n}=cot^{-1}\,2n^{2}$
$=tan^{-1} \frac{1}{2n^{2}}$
$=tan^{-1}\left(\frac{2}{4n^{2}}\right)=tan^{-1}\left(\frac{\left(2n+1\right)-\left(2n-1\right)}{1+\left(2n+1\right)\left(2n-1\right)}\right)$
$=tan^{-1}\left(2n+1\right)-tan^{-1}\left(2n-1\right)$
$\therefore S_{n}=$ $\displaystyle \sum_{n=1}^\infty$$\left\{tan^{-1}\left(2n+1\right)-tan^{-1}\left(2n-1\right)\right\}$
$=tan^{-1}\,\infty-tan^{-1}\,1$
$=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$