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Evaluate ("lim")(nvecoo)sum(k=1)^nk/(n^2...

Evaluate `("lim")_(nvecoo)sum_(k=1)^nk/(n^2+k^2)`

Text Solution

Verified by Experts

The correct Answer is:
`1/2log2`
`lim_(nto oo) sum_(K=1)^(n)K/(n^(2)+K^(2))`
`=lim_(nto oo) sum_(K=1)^(n) 1/(n^(2))xx K/(1+(K/n)^(2))`
`=lim_(nto oo) sum_(K=1)^(n)1/n xx(K//n)/(1+(K/n)^(2))`
`=lim_(n to oo) sum_(K=1)^(n)1/nxx(K//n)/(1+(K/n)^(2))`
`=int_(0)^(1)x/(1+x^(2))dx`
`=1/2 log (1+x^(2))|_(0)^(1)=1/2log2`
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