Home
Class 12
MATHS
Prove that sum(r=1)^(n) tan^(-1) ((2^(r...

Prove that `sum_(r=1)^(n) tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = tan^(-1) (2^(n)) - (pi)/(4)`

Text Solution

Verified by Experts

`underset(r =1) overset(n) sum tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = underset( r=1)overset(n) sum tan^(-1) ((2^(r -1))/(1 + 2^(r).2^(r-1)))`
`= underset(r =1) overset(n)sum tan^(-1) ((2^(r) -2^(r-1))/(1 + 2^(r).2^(r -1)))`
`= underset(r=1)overset(n) sum [tan^(-1) (2^(r)) - tan^(-1) (2^(r -1))]`
`= tan^(-1) (2^(n)) - tan^(-1) (1)`
`= tan^(-1) (2^(n)) - (pi)/(4)`
Promotional Banner

Topper's Solved these Questions

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise Exercise 7.6|9 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise Exercise (Single)|80 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    CENGAGE|Exercise Exercise 7.4|12 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE|Exercise JEE Previous Year|20 Videos

  • JEE 2019

    CENGAGE|Exercise Chapter 10|9 Videos

Similar Questions

Explore conceptually related problems

sum_(r=1)^(n)tan^(-1)((2^(r-1))/(1+2^(2r-1))) is equal to

sum_(r=1)^(n)tan^(-1)((2^(r-1))/(1+2^(2r-1))) is equal to:

sum_(r=1)^9 tan^-1(1/(2r^2)) =

sum_(i=1)^(oo)tan^(-1)((1)/(2r^(2)))=

sum_(r=1)^(n)tan((x)/(2^(r)))sec((x)/(2^(r-1)));r,n in N

sum_(r=1)^(oo)tan^(-1)((2)/(1+(2r+1)(2r-1)))

Find the sum sum_(r =1)^(oo) tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) -2r + 1)))

sum_(r=1)^(n)(r)/(2^(r))=2-((1)/(2))^(2)(n+2)

sum_(r=1)^oo tan^-1((6^r)/(2^(2r+1)+3^(2r+1)))