Evaluate : `lim_(n rarr oo)(1*2+2*3+3*4+...+n(n+1))/(n^3)`

Evaluate: lim_(n rarr oo)(1^(2)+2^(2)+......+n^(2))/(n^(3))

Evaluate: lim_(n rarr oo)(1^(2)+2^(2)+......+n^(2))/(n^(3))

Evaluate: lim_(n rarr oo) (n^(2)(1+2+3+4+......+n))/(n^(4)+4n^(2)) .

lim_(n rarr oo)(1+2^(4)+3^(4)+...+n^(4))/(n^(5))

The value of lim_(n rarr oo) (1 + 2^(4) + 3^(4) +…...+n^(4))/(n^(5)) - lim_(n rarr oo) (1 + 2^(3) + 3^(3) +…...+n^(3))/(n^(5)) is :

lim_ (n rarr oo) (1 + 2 + 3 + ...... + n) / (n ^ (2))

lim_ (n rarr oo) (1 + 2 + 3 * -n) / (n ^ (2))

Evaluate: lim_(n rarr oo) (n(1+4+9+16+......+n^(2)))/(n^(4)+8n^(3)) .

Evaluate: lim_(n rarr oo)((1^(2))/(n^(3))+(2^(2))/(n^(3))+(3^(2))/(n^(4))+...+(1)/(n))

Evaluate: lim_(n rarr oo) ((1^(2)+2^(2)+3^(2)+...+n^(2)))/((1+3+5+7+...+"n terms")) .