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Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

MathematicsInverse Trigonometric FunctionsSummation of seriesMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

For positive adjacent values $z_{0}, z_{1}, \dots, z_{n}, tan^{- 1} ((z_{r} - z_{r - 1}) / (1 + z_{r} z_{r - 1})) = tan^{- 1} z_{r} - tan^{- 1} z_{r - 1}$ , so the sum telescopes.

Find $cot^{- 1} (2^{2} + 1/2) + cot^{- 1} (2^{3} + 1/ 2^{2}) + cot^{- 1} (2^{4} + 1/ 2^{3}) + \dots$

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Answer:

Solution:

The rth term, $r \geq 1$ , is $cot^{- 1} (2^{r + 1} + 2^{- r}) = tan^{- 1} [2^{r} / (1 + 2^{r} \cdot 2^{r + 1})]$ . It equals $tan^{- 1} (2^{r + 1}) - tan^{- 1} (2^{r})$ ; the infinite sum is $π /2 - tan^{- 1} 2 = cot^{- 1} 2$ .

Stream:JEESubject:MathematicsTopic:Inverse Trigonometric FunctionsSubtopic:Summation of series

⏱ 2mℹ️ Source: QB

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