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Evaluate the following limit: lim(nto ...

Evaluate the following limit:
`lim_(nto oo)(sum_(r=1)^(n) sqrt(r)sum_(r=1)^(h)1/(sqrt(r)))/(sum_(r=1)^(n)r)`

Text Solution

Verified by Experts

The correct Answer is:
`8/3`
`lim_(n to oo) (sum_(r=1)^(n)sqrt(r) sum_(r=1)^(n)1/(sqrt(4)))/(sum_(r=1)^(n)r)`
`:.` Limit `=lim_(nto oo) (1/n sum_(r=1)^(n)sqrt(4/n)(1/nsum_(r=1)^(n)sqrt(n/r)))/(1/n sum_(r=1)^(n)r/n)`
`=(int_(0)^(1)sqrt(x)dxint_(0)^(1)(dx)/(sqrt(x)))/(int_(0)^(1)x dx) =8/3`
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