Show that `(1+3^(- 1))(1+3^(- 2))(1+3^(- 4))(1+3^(- 8)).........(1+3^(-2^n))=3/2(1-3^(-2^((n-1))))` .

If A=[(1,1,1),(1,1,1),(1,1,1)] then show that A^n=[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))] .

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

1.3+3.5+5.7+......+(2n-1)(2n+1)=(n(4n^(2)+6n-1))/(3)

(1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+.....+(n^(2))/ ((2n-1)(2n+1))=((n)(n+1))/((2(2n+1)))

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

(2.3^(n+1)+7.3^(n-1))/(3^(n+1)-2((1)/(3))^(1-n))=

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))/(1^(3))+(2^(2))/(2^(3))+(1^(2)+2^(2)+3^(2))/(1^(3)+2^(3)+3^(3))+ . . . , is

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

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