# How you demonstrate the limit is pie3^.5/18 without hospital and with antiderivative in sequence n/(3n^2+1)+n/(3n^2+4)+---+n/(3n^2+n^2), n go to infinite sign?

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Student Comments

pramodpandey | Student

`lim_(n->oo){n/(3n^2+1)+n/(3n^2+4)+.....+n/(3n^2+n^2)}`

`` `=lim_(n->oo)(1/n){1/(3+(1/n)^2)+1/(3+(2/n)^2)+....+1/(3+(n/n)^2)}`

=`int_0^1f(x)dx`

where `f(x)=(3+x^2)^(-1)`

`=(1/sqrt(3)){tan^(-1)(x/sqrt(3))}_0^1`

`=(1/sqrt(3))(pi/6)`

`=(pisqrt(3))/18`