(a-(1)/(a))^(2)=3

((1)/(2), (2)/(2) )/(1^3) + ((2)/(2) , (3)/(2) )/( 1^3 + 2^3) + ((3)/(2) , (4)/(2) ) / (1^(3) + 2^(3) + 3^(3) ) + ..... n terms

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

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

((1)/(2) . (2)/(2))/(1^(3))+((2)/(3) . (3)/(2))/(1^(3)+2^(3)) +((3)/(2) . (4)/(2))/(1^(3)+2^(3)+3^(3))+.. . . to n terms

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 :

If (1)/(1^(3))+(1+2)/(1^(3)+2^(3))+(1+2+3)/(1^(3)+2^(3)+3^(3))+......n terms then lim_(n rarr oo)[S_(n)]

If x=(1)/(1^(2))+(1)/(3^(2))+(1)/(5^(2))+.... , y=(1)/(1^(2))+(3)/(2^(2))+(1)/(3^(2))+(3)/(4^(2))+.... and z=(1)/(1^(2))-(1)/(2^(2))+(1)/(3^(2))-(1)/(4^(2))+... then

If x=(1)/(1^(2))+(1)/(3^(2))+(1)/(5^(2))+.... , y=(1)/(1^(2))+(3)/(2^(2))+(1)/(3^(2))+(3)/(4^(2))+.... and z=(1)/(1^(2))-(1)/(2^(2))+(1)/(3^(2))-(1)/(4^(2))+... then

If x=(1)/(1^(2))+(1)/(3^(2))+(1)/(5^(2))+ . . . y=(1)/(1^(2))+(3)/(2^(2))+(1)/(3^(2))+(3)/(4^(2))+ . . . . z=(1)/(1^(2))-(1)/(2^(2))+(1)/(3^(2))-(1)/(4^(2))+ . . .then