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The value of tansum(r=1)^oo tan^-1 (4/(...

The value of tan`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`
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