Get 2 FREE Instant-Explanations on Filo with code FILOAPP

The value of ^{-1} ({1}{3}) + ^{-1} ({2}{9}) + ^{-1} ({4}

Question

The value of \tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +....... $n$ terms is

tan^{- 1} 2^{n} - \frac{π}{4}

tan^{- 1} 2^{n}

cot^{- 1} 2^{n}

\frac{sin ^{- 1} 2 ^{n}}{cos ^{- 1} 2 ^{n}}

Views: 647 students

Found 4 tutors discussing this question

Asher Discussed

The value of \tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +.......

n

terms is

15 mins ago

Discuss this question LIVE

15 mins ago

Text solutionVerified

tan^{- 1} (\frac{1}{3}) + tan^{- 1} (\frac{2}{9}) + tan^{- 1} (\frac{4}{33}) + tan^{- 1} (\frac{8}{129}) + .... n

terms.

= tan^{- 1} (\frac{2 - 1}{1 + 2.1}) + tan^{- 1} (\frac{4 - 2}{1 + 4.2}) + tan^{- 1} (\frac{8 - 4}{1 + 8.4}) + tan^{- 1} (\frac{16 - 8}{1 + 16 \times 8}) + ... + tan^{- 1} (\frac{2 ^{n} - 2 ^{n - 1}}{1 + 2 ^{n} \times 2 ^{n - 1}})

we know that

tan^{- 1} x - tan^{- 1} y = tan^{- 1} \frac{x - y}{1 + x y}

= tan^{- 1} 2 - tan^{- 1} 1 + tan^{- 1} 4 - tan^{- 1} 2 + tan^{- 1} 8 - tan^{- 1} 4 + tan^{- 1} 16 - tan^{- 1} 8 + ... + tan^{- 1} 2^{n} - tan^{- 1} 2^{n - 1}

= tan^{- 1} 2^{n} - tan^{- 1} 1

= tan^{- 1} 2^{n} - \frac{π}{4}

Was this solution helpful?

150

Report

One destination for complete JEE/NEET preparation

One destination to cover all your homework and assignment needs

Learn Practice Revision Succeed

Instant 1:1 help, 24x7

60, 000+ Expert tutors

Textbook solutions

Big idea maths, McGraw-Hill Education etc

Essay review

Get expert feedback on your essay

Schedule classes

High dosage tutoring from Dedicated 3 experts

Trusted by 4 million+ students

Stuck on the question or explanation?

Connect with our maths tutors online and get step by step solution of this question.

231 students are taking LIVE classes

Question Text	The value of \tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +....... $n$ terms is
Answer Type	Text solution:1
Upvotes	150

The value of \tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +.......n terms is

Text solutionVerified

Practice questions from similar books

The value of \tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +....... $n$ terms is