The value of tan [ tan − 1 3 1 + tan − 1 9 2 + tan − 1 33 4 + ……… .. + tan − 1 ( 1 + 2 2 n − 1 2 n − 1 ) + ……… .∞ ] is`

T n = tan − 1 [ 1 + 2 2 n − 1 2 n − 1 ] = tan − 1 [ 1 + 2 n ⋅ 2 n − 1 2 n − 2 n − 1 ] = tan ( 2 n ) − tan − 1 ( 2 n − 1 )

Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

MathematicsInverse Trigonometric FunctionsSummation of seriesMedium2 minQB

MathematicsMediuminteger

The value of $tan [tan^{- 1} \frac{1}{3} + tan^{- 1} \frac{2}{9} + tan^{- 1} \frac{4}{33} + \dots\dots\dots .. + tan^{- 1} (\frac{2 ^{n - 1}}{1 + 2 ^{2 n - 1}}) + \dots\dots\dots .\infty]$ is`

Answer:

Solution:

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

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

⏱ 2mℹ️ Source: QB

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