I am trying to solve the problem and but i am stuck.. Can someone please have look and help out...

Problem

if $\displaystyle x=1+a+a^2+...$ and $\displaystyle y=1+b+b^2$ where |a| <1 and |b| < 1. Prove that $\displaystyle 1 + ab + a^2b^2 + ...= \frac{xy}{x+y-1}$

Solution

$\displaystyle x=1+a+a^2+... $

GP Series

$\displaystyle x = \frac{1-a^n}{1-a} [1] $

Similarly

$\displaystyle y=\frac{1-b^n}{1-b} [2] $

taking LHS of the equation

$\displaystyle 1 + ab + a^2b^2 + ... $

using GP

$\displaystyle \frac{1-(ab)^n}{1-ab} [3] $

RHS of the equation

$\displaystyle \frac{xy}{x+y-1} [4]$

If I substitute 1 and 2 in 4,i get

$\displaystyle \frac{(1-a^n)(1-b^n)}{(1-a^n)(1-b)+(1-b^n)(1-a)-(1-a)(1-b)}$

this is no where near to equ 4. My approach might be wrong. please suggest if possible...