1.

`lim_(nto oo)+(1)/(sqrt(n^(2)+n))+(1)/(sqrt(n^(2)+2n))+...(1)/(sqrt(n^(2)+(n-1)n))` is equal toA. `2+2sqrt(2)`B. `2sqrt(2)-2`C. `2sqrt(2)`D. 2

Answer» Correct Answer - B
Given , `lim_(ntoinfty)[(1)/(n)+(1)/(sqrt(n^(2)+n))+..+(1)/(sqrt(n^(2)+(n-1)n))]`
`=lim_(n to infty)[(1)/(n)+(1)/(nsqrt(1+(1)/(n)))+..+(1)/(nsqrt(1+((n-1))/(n)))]`
`=(1)/(n)lim_(ntoinfty)[1+(1)/(sqrt(1)+(1)/(n))+..+(1)/(sqrt(1+((n-1))/(n)))]`
`-lim_(ntoinfty)sum_(r=0)^(n-1)((1)/(sqrt(1+(r)/(n))))(1)/(n)`
`=int_(0)^(1)(dx)/(sqrt(1+x))=[2sqrt(1+x)]_(0)^(1)=2sqrt(2)-2`


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