The sum to infinite terms of the series cosec-1-10-cosec-1-50-cosec-1-170-cosec-1-n2-1n2-2n-2 is

Given series =cot^ 1(3)+cot^ 1(7)+cot^ 1(13)+cot^ 1(21)+.... =Tan^ 1(1/3 )+Tan^ 1(1/7 )+Tan^ 1(1/13 )+Tan^ 1(1/21 )+..... =∑_n=1^∞Tan^ 1(1/1+n(n+1) )=π/4 ...

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Byju's Answer

Standard XIII

Mathematics

Properties Derived from Trigonometric Identities

The sum to in...

Question

The sum to infinite terms of the series $c o s e c^{- 1} \sqrt{10} + c o s e c^{- 1} \sqrt{50} + c o s e c^{- 1} \sqrt{170} + . . . . . . + c o s e c^{- 1} \sqrt{(n^{2} + 1) (n^{2} + 2 n + 2)}$ is

\frac{π}{4}

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\frac{π}{2}

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\frac{π}{3}

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\frac{π}{6}

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Solution

The correct option is A $\frac{π}{4}$
Given series = $c o t^{- 1} (3) + c o t^{- 1} (7) + c o t^{- 1} (13) + c o t^{- 1} (21) + . . . . = T a n^{- 1} (\frac{1}{3}) + T a n^{- 1} (\frac{1}{7}) + T a n^{- 1} (\frac{1}{13}) + T a n^{- 1} (\frac{1}{21}) + . . . . . = \sum_{n = 1}^{\infty} T a n^{- 1} (\frac{1}{1 + n (n + 1)}) = \frac{π}{4}$

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Similar questions

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c o s e c^{- 1} \sqrt{10} + c o s e c^{- 1} \sqrt{50} + c o s e c^{- 1} \sqrt{170} + . . . . . . + c o s e c^{- 1} \sqrt{(n^{2} + 1) (n^{2} + 2 n + 2)}

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{sec}^{- 1} \sqrt{2} + {sec}^{- 1} \frac{\sqrt{10}}{3} + {sec}^{- 1} \frac{\sqrt{50}}{7} + . . . + {sec}^{- 1} \sqrt{\frac{(n^{2} + 1) (n^{2} - 2 n + 2)}{(n^{2} - n + 1)^{2}}}

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The sum to infinite terms of the series cosec−1√10+cosec−1√50+cosec−1√170+......+cosec−1√(n2+1)(n2+2n+2) is

The correct option is A π4Given series =cot−1(3)+cot−1(7)+cot−1(13)+cot−1(21)+....=Tan−1(13)+Tan−1(17)+Tan−1(113)+Tan−1(121)+.....=∑∞n=1Tan−1(11+n(n+1))=π4

The sum to infinite terms of the series $c o s e c^{- 1} \sqrt{10} + c o s e c^{- 1} \sqrt{50} + c o s e c^{- 1} \sqrt{170} + . . . . . . + c o s e c^{- 1} \sqrt{(n^{2} + 1) (n^{2} + 2 n + 2)}$ is