Question

Find the sum of $co{t}^{-1}2+co{t}^{-1}8+co{t}^{-1}18+.....\infty$

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\pi$
• D. $\frac{3\pi }{2}$

Answer 1

The correct option is B $\frac{\pi }{4}$

Let $S={\cot }^{-1}{2.1}^2+{\cot }^{-1}2.2^2+{\cot }^{-1}{2.3}^2+{\cot }^{-1}{2.4}^2+...$
$T_n={\cot }^{-1}{2n}^2={\tan }^{-1}\frac{1}{{2n}^2}={\tan }^{-1}\left(\frac{2}{{4n}^2}\right)$
$={\tan }^{-1}\left[\frac{(2n+1)-(2n-1)}{1+({4n}^2-1)}\right]$
$={\tan }^{-1}(2n+1)-{\tan }^{-1}(2n-1)\therefore S={\tan }^{-1}3-{\tan }^{-1}1+{\tan }^{-1}5-{\tan }^{-1}3+{\tan }^{-1}7-{\tan }^{-1}5+...\infty$

$={\tan }^{-1}\infty -{\tan }^{-1}1=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}$