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if sum(n=1)^oo tan^(-1)(1/(7-5n+n^2))=pi...

if `sum_(n=1)^oo tan^(-1)(1/(7-5n+n^2))=pi/2 +tan^(-1)c` , then c is equal to

A

2

B

3

C

17

D

`5/2`

Text Solution

Verified by Experts

The correct Answer is:
A
`sum_(n=1)^oo tan^(-1)(1/(7-5n+n^2))=sum_(n=1)^oo "tan"^(-1) ((n-2)-(n-3))/(1+(n-2)(n-3))`
`sum_(n=1)^oo[tan^(-1) (n-2)-tan^(-1)(n-3)]`
`=pi/2-tan^(-1)(-2)=pi/2+"tan"^(-1)2=(api)/b + tan^(-1)c`
a=1,b=2,c=2
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