Question

The value of the expression $cot^{-1}\frac{1}{2}+cot^{-1}\frac{9}{2}+cot^{-1}\frac{2 5}{2}+cot^{-1}\frac{4 9}{2}+......$ upto $n$ terms is

• A. $tan^{- 1}2n$
• B. $\tan ^{-1}(2 n-1)$
• C. $tan^{- 1}n$
• D. $tan^{- 1}2n-tan^{- 1}1$

Answer 1

Answer: (A)

Given expression
$=tan^{-1}2+tan^{-1}\frac{2}{9}+tan^{-1}\frac{2}{2 5}+tan^{-1}\frac{2}{4 9}+....$
General term
$=\frac{2}{\left(2 n - 1\right)^{2}}=\frac{2}{4 n^{2} - 4 n + 1}=\frac{2}{1 + 4 n \left(n - 1\right)}=\frac{2 n - \left(2 n - 2\right)}{1 + 2 n \left(2 n - 2\right)}$
$ T_{n}=\tan ^{-1} 2 n-\tan ^{-1}(2 n-2) $
$\therefore$ Sum of the series
$=\tan ^{-1} 2-\tan ^{-1} 0+\tan ^{-1} 4-\tan ^{-1} 2+\tan ^{-1} 6-\tan ^{-1} 4+\ldots \tan ^{-1} 2 n-\tan ^{-1}(2 n-2)$
$=\tan ^{-1} 2 n-\tan ^{-1} 0=\tan ^{-1} 2 n$