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The expression sum(n=1)^(oo)cot^(-1)(n^(...

The expression `sum_(n=1)^(oo)cot^(-1)(n^(2)-3n+3)` simplifies to

A

`(pi)/(4)`

B

`(pi)/(2)`

C

`(3pi)/(4)`

D

`pi`

Text Solution

Verified by Experts

The correct Answer is:
C
`S=sum_(n=1)^(oo)cos^(-1)(n^(2)-3n+3)`
`=sum_(n=1)^(oo)tan^(-1)((1)/(1+n^(2)-3n+2))`
`=sum_(n=1)^(oo)tan^(-1)(((n-1)-(n-2))/(1+(n-1)(n-2)))`
`=sum_(n=1)^(oo)(tan^(-1)(n-1)-tan^(-1)(n-2))`
`=(pi)/(2)+(pi)/(4)=(3pi)/(4)`
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