Question

The value of $\sum_{r=1}^{\infty} \tan ^{-1}\left(\frac{1}{r^2+5 r+7}\right)$ is equal to

• A. $\tan ^{-1} 3$
• B. $\frac{\pi}{4}$
• C. $\sin ^{-1} \frac{1}{\sqrt{10}}$
• D. $\cot ^{-1} 2$

Answer 1

Answer: (C)

$\tan ^{-1}\left(\frac{1}{r^2+5 r+7}\right)=\tan ^{-1}\left[\frac{(r+3)-(r+2)}{1+(r+3)(r+2)}\right]$
$=\tan ^{-1}( r +3)-\tan ^{-1}( r +2)$
$\displaystyle\sum_{r=1}^{\infty} \tan ^{-1}\left(\frac{1}{r^2+5 r+7}\right)=\tan ^{-1} 4-\tan ^{-1} 3+\tan ^{-1} 5-\tan ^{-1} 4+\ldots \ldots \ldots+\tan ^{-1}(\infty)$
$=\frac{\pi}{2}-\tan ^{-1} 3=\cot ^{-1} 3=\sin ^{-1}\left(\frac{1}{\sqrt{10}}\right) $