`IfS_n=[1/(1+sqrt(n))+1/(2+sqrt(2n))++1/(n+sqrt(n^2))],t h e n(lim)_(nvecoo)S_n` is equal to log 2 (b) log 4 log 8 (d) none of these

Options

A. `log2`
B. `log 4`
C. `log 8`
D. none of these

Answer

B

Explanation

`lim_(nto oo) S_(n)=lim_(nto oo) [1/(1+sqrt(n))+1/(2+sqrt(2n))+………..+1/(n+sqrt(n)^(2))]`
`=lim_(nto oo) 1/n [1/(1/n+1/(sqrt(n)))+1/(2/n+sqrt(2/n))+……..+1/(n/n+sqrt(n/n))]`
`=int_(0)^(1)(dx)/(sqrt(x)(sqrt(x)+1))`
Put `sqrt(x)=z` or `1/(2sqrt(x))dx=dz`
`:. lim_(n to oo) S_(n)=int_(0)^(1)(2 dz)/(z+1)=2|log(z+1)|_(0)^(1)`
`=2(log2-log1)`
`=2log2=log4`