`Lim_(n->oo)((n^4)/(1^3+2^3+3^3+.....+n^3))`

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Let alpha=lim_(n->oo)((1^3-1^2)+(2^3-2^2)+.....+(n^3-n^2))/(n^4), then alpha is equal to :

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

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

The value of lim_(nto oo)(1^(3)+2^(3)+3^(3)+……..+n^(3))/((n^(2)+1)^(2))

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

lim(n->oo)(1^2+2^2+3^2+..........+n^2)/n^3

lim_(n->oo) (1.2+2.3+3.4+....+n(n+1))/n^3

Evaluate : lim_(n-> oo) (1^4+2^4+3^4+...+n^4)/n^5 - lim_(n->oo) (1^3+2^3+...+n^3)/n^5

The value of lim_(n->oo) (1^2 . n+2^2.(n-1)+......+n^2 . 1)/(1^3+2^3+......+n^3) is equal to

lim_(n -> oo) (((n+1)(n+2)(n+3).......2n) / n^(2n))^(1/n) is equal to