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sum(n=1)^(prop)tan^(-1)((4n)/(n^(4)+5))=...

`sum_(n=1)^(prop)tan^(-1)((4n)/(n^(4)+5))=`

A

`(pi)/(4)+tan^(-1)2`

B

`(3pi)/(4)-tan^(-1)2`

C

`tan^(-1)3`

D

`(pi)/(4)+cot^(-1)2`

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
`sum_(n=1)^(infty)tan^(-1)((4n)/(n^(4)+5))=sum_(n=1)^(infty)tan^(-1)((4n)/(1+(n^(2)+2)^(2)-4n^(2)))`
`=sum_(n=1)^(infty)tan^(-1)((4n)/(1+(n^(2)+2n+2)(n^(2)-2n+2)))`
`sum_(n=1)^(infty)[tan^(-1)(n^(2)+2n+2)-tan^(-1)(n^(2)-2n+2)]`
`=2tan^(-1)(infty)-tan^(-1)1-tan^(-1)2=(3pi)/(4)-tan^(-1)2`
`=(pi)/(4)+tan^(-1)(1)/(2)`
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