Question

Sum to infinite terms of the series $co{t}^{-1}3+co{t}^{-1}7+co{t}^{-1}13+....$ is

• A. $\pi /2$
• B. $co{t}^{-1}1$
• C. $ta{n}^{-1}2$
• D. none of these

Answer 1

Answer: (B)

$co{t}^{-1}1$

${\cot }^{-1}\left(3\right)+{\cot }^{-1}\left(7\right)+{\cot }^{-1}\left(13\right)+..............$

General terms in ${\tan }^{-1}$

${\tan }^{-1}\left(\frac{1}{n^2+n+1}\right)$

$={\tan }^{-1}\left[\frac{(x-1)-n}{1+x(n+1)}\right]$

$={\tan }^{-1}(n+1)-{\tan }^{-1}\left(n\right)$

$n=1,{\tan }^{-1}\left(2\right)-{\tan }^{-1}\left(1\right)$

$n=2,{\tan }^{-1}\left(3\right)-{\tan }^{-1}\left(2\right)$

$n=3,{\tan }^{-1}\left(4\right)-{\tan }^{-1}\left(3\right)$

$={\tan }^{-1}\left(\infty \right)-{\tan }^{-1}\left(1\right)$

$=\frac{\pi }{2}-{\tan }^{-1}\left(1\right)$

$={\cot }^{-1}\left(1\right)$