Question

The sum of of infinite terms of the series${\cot }^{-1}(\frac{1}{2}+2^2)+{\cot }^{-1}(\frac{1}{2^2}+2^3)+(\frac{1}{2^3}+2^4)+......\infty$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. ${\tan }^{-1}2$
• D. ${\cot }^{-1}2$

Answer 1

Answer: (D)

${\cot }^{-1}2$

$S_{\infty }=\underset{n\to \infty }{lim}\sum _{r=1}^n{\cot }^{-1}(\frac{1}{2^r}+2^{r+1})$
$=\underset{n\to \infty }{lim}\sum _{r=1}^n{\tan }^{-1}\left(\frac{2^r}{1+2^r{2}^{r+1}}\right)$
$=\underset{n\to \infty }{lim}\sum _{r=1}^n{\tan }^{-1}\left(\frac{2^{r+1}-2^r}{1+2^r{2}^{r+1}}\right)$
$=\underset{n\to \infty }{lim}\left[\sum _{r=1}^n\right({\tan }^{-1}{2}^{r+1}-{\tan }^{-1}{2}^r\left)\right]$
$=\underset{n\to \infty }{lim}\left[\right({\tan }^{-1}{2}^{n+1}-{\tan }^{-1}2]={\tan }^{-1}\infty -{\tan }^{-1}2=\frac{\pi }{2}-{\tan }^{-1}2={\cot }^{-1}2$