" 14."A=[[1,1],[1,1]]" then prove that "A^(1oo)=2^(m)A

Prove that lim_(n->oo)(1+1/n)^n=e

"If "x=1+a+a^(2)+...oo,|a|lt1,y=1+b+b^(2)+...oo,|b|lt1,"then prove that ":(xy)/(x+y-1)=1+ab+a^(2)b^(2)+...oo

If S=1+a+a^2+a^3+a^4+……….to oo and |a|<1 then prove that a= (S-1)/S

If =1+a+a^(2)+oo, where |a|< 1andy =1+b+b^(2)+oo, where |b|<1 prove that: 1+ab+a^(2)b^(2)+oo=(xy)/(x+y-1)

If S=1+a+a^2+a^3+a^4+……….to oo then prove that a= (S-1)/S

If x=1 +a + a^2+…. to oo , where |a| lt1 and y=1+b+b^2+… to oo , where |b|lt1 , prove that 1+ab+a^2b^2+… to oo=((xy)/(x+y-1))

If A=1+r^a+r^(2a)+ to oo and B=1+r^b+r^(2b)+oo , prove that r=((A-1)/A)^(1//a)=((B-1)/B)^(1//b)

If A=1+r^a+r^(2a)+ ......... to oo and B=1+r^b+r^(2b)+........... to oo , prove that r=((A-1)/A)^(1//a)=((B-1)/B)^(1//b) .

If A=1+r^(a)+r^(2a)+ to oo andB=1+r^(b)+r^(2b)+oo, prove that r=((A-1)/(A))^(1/a)=((B-1)/(B))^(1/b)

If x = 1 + a + a^(2) +… oo (-1 lt a lt 1) and y = 1 + b + b^(2)+ … oo (-1 lt b lt 1) , prove that, 1+ab + a^(2)b^(2)+ … oo = (xy)/(x+y - 1)