The sum of the infinite terms of the series cot−1(12+34)+cot−1(22+34)+cot−1(32+34)+.......... is equal to :

Tn=cot−1(n2+34)=tan−1(1n2+(3/4))=tan−1(11+n2−(1/4))=tan−1(11+(n−12)(n+12))=tan−1((n+12)−(n−12)1+(n+12)(n−12))]

The sum of the infinite terms of the series cot−1(12+34)+cot−1(22+34)+cot−1(32+34)+.......... is equal to :

Tn=cot−1(n2+34)=tan−1(1n2+(3/4))=tan−1(11+n2−(1/4))=tan−1(11+(n−12)(n+12))=tan−1((n+12)−(n−12)1+(n+12)(n−12))]

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

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$\tan^{- 1} (3)$

$\tan^{- 1} (4)$

$\tan^{- 1} (2)$

$\tan^{- 1} (1)$

answer is B.

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Detailed Solution

$T_{n} = \cot^{- 1} (n^{2} + \frac{3}{4}) = \tan^{- 1} (\frac{1}{n^{2} + (3 / 4)}) = \tan^{- 1} (\frac{1}{1 + n^{2} - (1 / 4)})$

$= \tan^{- 1} (\frac{1}{1 + (n - \frac{1}{2}) (n + \frac{1}{2})}) = \tan^{- 1} (\frac{(n + \frac{1}{2}) - (n - \frac{1}{2})}{1 + (n + \frac{1}{2}) (n - \frac{1}{2})})]$

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The sum of the infinite terms of the series cot−1(12+34)+cot−1(22+34)+cot−1(32+34)+.......... is equal to :

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