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

Prove that: 1^2+2^2+3^2.....+n^2>(n^3)/3, n in N

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

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

lim_(n to infty)(1^(2)/(1-n^(3))+2^(2)/(1-n^(3))+…+n^(2)/(1-n^(3))) is equal to :

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

Match the following . {:(,"ColumnI",,"ColumnII"),((i) ,1^(2) +2^(2) +3^(2) +....+n^(2) ,(a) ,[(n(n+1))/(2)]^(2)),((ii) , 1^(3) +2^(3) +3^(3) +...+n^(3) ,(b), n(n+1)),((iii),2+4+6+...+2n,( c),(n(n+1)(2n+1))/(6)),((iv),1+2+3+...+n,(d),(n(n+1))/(2)):}

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