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

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

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

Prove that 1^1*2^2*3^3....n^nle((2n+1)/3)^((n(n+1))/2) .

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

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

Find the limit, when n->oo,\ of\ [((n^3+1^3)(n^3+2^3)(n^3+3^3)........(n^3+n^3))/(n^(3n))]^(1/n)

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

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

f(n)=(1^(2)n+2^(2)(n-1)+3^(2)(n-2)+...+n^(21))/(1^(3)+2^(3)+3^(3)+......+n^(3)) then (where [.] denotes greatest integer function)