" (7) "|[1,a,a^(2)],[a^(2),1,a],[a,a^(2),1]|=(a^(3)-1)^(2)

Using properties of determinant, prove that |{:( 1,a,a^(2)),(a^(2) , 1,a),( a,a^(2),1):}|=(a^(3) -1)^(2)

Prove: |(1,a, a^2),(a^2, 1,a),( a, a^2, 1)|=(a^3-1)^2

Prove: |1a a^2a^2 1a a a^2 1|=(a^3-1)^2

If A=[(1,0,2),(0,1,2),(1,2,0)],B=[(1,-2,3),(2,3,-1),(-3,1,2)] then

Let H_(n)=1+(1)/(2)+(1)/(3)+ . . . . .+(1)/(n) , then the sum to n terms of the series (1^(2))/(1^(3))+(1^(2)+2^(2))/(1^(3)+2^(3))+(1^(2)+2^(2)+3^(2))/(1^(3)+2^(3)+3^(3))+ . . . , is

The value of (x-1)/(x+1)+(x^(2)-1)/(2(x+1)^(2))+(x^(3)-1)/(3(x+1)^(2))+......oo equals

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

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