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The value of ("lim")(nvecoo)sum(r=1)^(4n...

The value of `("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2)` is equal to

A

`1/35`

B

`1/14`

C

`1/10`

D

`1/5`

Text Solution

Verified by Experts

The correct Answer is:
C
`lim_(nto oo) sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+4sqrt(n))^(2))`
`T_(r)=1/(sqrt(r/n n(3 sqrt(r/n)+4)^(2))`
`:. S=lim_(n to oo) 1/n sum_(1) ^(4n) 1/((3sqrt(r/n)+4)^(2)sqrt(r/n))=int_(0)^(4)(dx)/(sqrt(x)(3sqrt(x)+4)^(2))`
Put `3sqrt(x)+4=t`
or `3/2 1/(sqrt(x))dx=dt`
`:. S=2/3int_(4)^(10)(dt)/(t^(2))=2/3[-1/t]_(10)^(4)=1/10`
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