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lim(ntooo) ((n)/(n^(2)+1^(2))+(n)/(n^(2)...

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

A

`tan^(-1) (3)`

B

`tan^(-1) (2)`

C

`pi//4`

D

`pi//2`

Text Solution

Verified by Experts

The correct Answer is:
B
Clearly
`underset( n to oo)(lim)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+(n)/(n^(2)+3^(2))+ . . . +(1)/(5n))`
`= underset(n to oo)(lim)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+(n)/(n^(2)+3^(2))+ . . . + (n)/(n^(2)+(2n)^(2)))`
`=underset(n to oo)(lim)sum_(r=1)^(2n)(n)/(n^(2)+r^(2))`
`=underset(n to oo)(lim)sum_(r=1)^(2n)(1)/(1+((r)/(n))^(2))*(1)/(n)=int_(0)^(2)(dx)/(1+x^(2))`
`[:' underset(n to oo)(lim)sum_(r=1)^(pn)(1)/(n)f((r)/(n))=int_(0)^(p)f(x)dx]`
`=[tan^(-1)x]_(0)^(2)=tan^(-1)2`
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