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Lt(1^(2)n+2^(2)(n-1)+3^(2)(n-2)+...+n^(2...

Lt(1^(2)n+2^(2)(n-1)+3^(2)(n-2)+...+n^(2)*1)/(1^(3)+2^(3)+....+n^(3))

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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)

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

S_(n)=(1)/(1^(3))+(1+2)/(1^(3)+2^(3))+(1+2+3)/(1^(3)+2^(3)+3^(3))+......+(1+2+....+n)/(1^(3)+2^(3)+......+n^(3)).100S_(n)=n then n is equal to :

Match the following . {:(,"ColumnI",,"ColumnII"),((i) ,1^(2) +2^(2) +3^(2) +....+n^(2) ,(a) ,[(n(n+1))/(2)]^(2)),((ii) , 1^(3) +2^(2) +3^(2) +...+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)):}

underset(n to oo)lim (n(1^(3)+2^(3)+...+n^(3))^(2))/((1^(2)+2^(2)+...+n^(2))^(3))=

{:(" " Lt),(n rarroo):} [(1^(2))/(n^(3)+1^(3))+(2^(2))/(n^(3)+2^(3))+......+(1)/(2n)]=

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