If x = 1 + a + `a^(2) + …. infty " , " y = 1 + b + b^(2) + …. infty, " " |a| lt 1 , |b| lt 1, ` then ` 1 + ab + a^(2) b^(2) + ….. infty = `

If |x| lt 1 ,then the sum of the series 1 + 2x + 3x^(2) + 4x^(2) + ....infty will be

Let x=1+a +a^(2) + ….. And y= 1 + b + b^(2)+ …… where |a| lt 1 and |b| lt 1 . Prove that: 1+ ab + a^(2)b^(2) +…… =(xy)/(x+y-1) .

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)

If (x^(2) + x)/(1-x) = a_(1) x + a_(2) x^(2) + ... to infty , |x| lt 1, then

If (x^(2) + x)/(1-x) = a_(1) x + a_(2) x^(2) + ... to infty , |x| lt 1, then

If x=1+a+a^(2)+...oo, a lt 1 and y=1+b+b^(2)+...oo, b lt 1, then 1+ab+a^(2)b^(2)+...oo :

If |r| gt 1, x = a + a/r + a/r^(2) + .......... infty , y = b - b/r + b/r^(2) - ......... infty and z = c + c/r^(2) + c/r^(4) + ...... infty , then the value of (xy)/z =

If 1+sin x+sin ^(2) x+….. to infty=4+2 root 3 , 0 lt x lt pi then x=