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

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

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

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

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

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

1.2+2.3+3.4+.........+n(n+1)=(1)/(3)n(n+1)(n+3)

Evaluate lim _( x to oo) ((1^(2) )/(n ^(3) +1 ^(3))+(2 ^(2))/(n ^(3) +2 ^(3)) + (3 ^(2))/(n ^(3)+ 3 ^(3))+ .... + (4)/(9n)).

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