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lim(n rarr oo)((n)/(n^(2)+1^(2))+(n)/(n^...

`lim_(n rarr oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2)) + (n)/(n^(2)+3^(2))+......+(1)/(5n))` is equal to :

A

`(pi)/(2)`

B

`tan^(-1)(3)`

C

`tan^(-1)(2)`

D

`(pi)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
C
`lim_(n rarr oo)[(n)/(n^(2)+1^(2)) + (n)/(n^(2)+2^(2)) +.....(n)/(n^(2)+(2n)^(2))], lim_(n rarr oo) sum_(r=1)^(2n) [(n)/(n^(2)+r^(2))] = lim_(n rarr oo)sum_(r = 1)^(2n)(1n)[(1)/(1+((r )/(n))^(2))]`
Let `(r )/(n) = x = int_(0)^(2)(dx)/(1+x^(2)) = [tan^(-1)x]_(0)^(2) = tan^(-1) 2`
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