Home
Class 12
MATHS
The value of sum(r=1)^oo tan^-1 (4/(4r^...

The value of `sum_(r=1)^oo tan^-1 (4/(4r^2 +3))=`

A

1

B

3

C

2

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
let `T_(r)=tan^(-1)((4)/(4r^(2)+3))=tan^(-1)((1)/((3)/(4)+r^(2)))`
`=tan^(-1)((1)/(1+r^(2)-(1)/(4)))`
`=tan^(-1)(((r+(1)/(2))-(r-(1)/(2)))/(1+(r+(1)/(2))(r-(1)/(2))))`
`=tan^(-1)(r+(1)/(2))-tan^(-1)(r-(1)/(2))`
`thereforesum_(r=1)^(infty)T_(r)=tan^(_1)((3)/(2))-tan^(-1)((1)/(2))`
`+tan^(-1)((5)/(2))-tan^(-1)((3)/(2))`
`+tan^(-1)((7)/(2))-tan^(-1)((5)/(2))`
`thereforesum_(r=1)^(infty)T_(r)=(pi)/(2)-tan^(-1)((1)/(2))=cot^(-1)((1)/(2))=tan^(-1)(2)`
`thereforetan(sum_(r=1)^(infty)T_(r))=2`
Promotional Banner

Similar Questions

Explore conceptually related problems

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

The value of sum_(r=2)^(oo) tan^(-1)((1)/(r^(2)-5r+7)) , is

The value of sum_(r=1)^(oo)tan^(-1)((2r-1)/(r^(4)-2r^(3)+r^(2)+1)) is equal to

Find the value of sum_(r=0)^(oo) tan^(-1) ((1)/(1 + r + r^(2)))

If sum_(r = 1)^(oo) (1)/((2r - 1)^2) = (pi^2)/(8) then the value of sum_(r = 1)^(oo) (1)/(r^2) is

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

The value of sum_(r=1)^(oo)cot^(-1)(2r^(2)) is equal to

The value of sum_(r=1)^(oo)cot^(-1)((r^(2))/(2)+(15)/(8)) is equal to

The value of sum _(r =1) ^(oo) (8r)/(4r ^(4) +1) is equal to :