Evaluate `lim_(n->infty) (1+2+3+.....+n)/n^2`

Evaluate lim_(nrarrinfty)((1)/(2n+1)+(1)/(2n+2)+......+(1)/(6n)) .

Evaluate: lim_(n rarr infty)[1/n+1/(n+1)+1/(n+2)+ cdot cdot cdot +1/(3n)] .

Evaluate lim_(ntooo) (1^(3)+2^(3)+3^(3)+...+n^(3))/(sqrt(4n^(8)+1)).

lim_(n rarr infty ) [((n+1)(n+2)...3n)/(n^(2n))]^(1//n) is equal to

The value of lim_(n rarr infty) (1)/(n) {(n+)(n+2)(n+3)…(n+n)}^(1//n) is equal to

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

Evaluate: lim_(x rarr oo)sin^(n)((2 pi n)/(3n+1)),n in N

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

Using Sandwich theorem, evaluate lim_ (n rarr oo) ((1) / (1 + n ^ (2)) + (1) / (2 + n ^ (2)) + ...... + (n) / (n + n ^ (2)))

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