If `1/1^3 + (1+2)/(1^3+2^3)+(1+2+3)/(1^3+2^3+3^3) +.......n` terms then `lim_(n->oo) [S_n]`

If 1/1^3+[1+2]/[1^3+2^3]+[1+2+3]/[1^3+2^3+3^3]+..... upto n terms,then lim_(n->oo)[S_n], where [.] denotes the greatest integer function.

If S_n=1/1^3 +(1+2)/(1^3+2^3)+...+(1+2+3+...+n)/(1^3+2^3+3^3+...+n^3) Then S_n is not greater than

lim_ (x rarr oo) {(1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) + ...... (. 1) / (3 ^ (n))} =

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

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

lim_ (n rarr oo) [(1) / (n) + (n ^ (2)) / ((n + 1) ^ (3)) + (n ^ (2)) / ((n + 2) ^ (3)) + ...... + (1) / (8n)]

If S_(n)=3+(1+3+3^(2))/(3!)+(1+3+3^(2)+3^(3))/(4!) ………… upto n -terms Then the value of [lim_(ntooo)S_(n)] is (where [.] represent G.I.F)

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)).

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)).