`underset(n→∞)lim((1+2+3+....+n)(1^(3)+2^(3)+3^(3)+....+n^(3)))/(1^(2)+2^(2)+3^(2)+....+n^(2))^(2)`

1^(3)+2^(3)+3^(3)+....+n^(3)=((n(n+1))/(2))^(2)

1^(3)+2^(3)+3^(3)+...+n^(3)=n^(2)((n+1)^(2))/(4)

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

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

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

1^(3)+2^(3)+3^(3)+.....+n^(3)=(n(n+1)^(2))/(4), n in N

The value of quad 1^(6)+2^(6)+3^(6)...n^(6)lim_(n rarr oo)(1^(6)+2^(6)+3^(6)...n^(6))/((1^(2)+2^(2)+3^(2)+...n^(2))(1^(3)+2^(3)+...n^(3)))

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

What is lim_(ntooo) (1+2+3+......+n)/(1^(2)+2^(2)+3^(2)+......n^(2))

Prove that : 1^(3)+2^(3)+3^(3)++n^(3)={(n(n+1))/(2)}^(2)