Evaluate `lim_(ntooo) (1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2)).`

Explanation

`P_(n)=underset(ntooo)lim(1)/(1+n^(2))+(2)/(2+n^(2))+...+(n)/(n+n^(2))`
Now, `P_(n)ltunderset(ntooo)lim(1)/(1+n^(2))+(2)/(1+n^(2))+...+(n)/(1+n^(2))`
`=(1)/(1+n^(2))(1+2+3+...+n)`
`=(n(n+1))/(2(1+n^(2)))`
Also, `P_(n)gt(1)/(n+n^(2))+(2)/(n+n^(2))+(3)/(n+n^(2))+...+(n)/(n+n^(2))`
`=(n(+1))/(2(1+n^(2)))`
Thus `(n(n+1))/(2(n+n^(2)))ltP_(n)lt(n(n+1))/(2(n+n^(2)))`
or`" "underset(ntooo)lim(n(n+1))/(2(n+n^(2)))ltunderset(ntooo)limP_(n)lt(n(n+1))/(2(n+n^(2)))`
or` underset(ntooo)lim(1(1+((1))/(n)))/(2((1)/(n)+1))ltunderset(ntooo)limltP_(n)ltunderset(ntooo)lim(1(1+((1))/(n)))/(2((1)/(n^(2))+1))`
or `1/2ltunderset(ntooo)limP_(n)lt1/2`
or `underset(ntooo)limP_(n)=1/2`