Official solution
(x + y) + (x2+xy+y2) + (x3+x2y + xy2+y3) + ....
By multiplying and dividing x -- y :
\[{{\left( {{x^2} - {y^2}} \right) + \left( {{x^3} - {y^3}} \right) + \left( {{x^4} - {y^4}} \right) + ...} \over {x - y}}\]
= \[{{\left( {{x^2} + {x^3} + {x^4} + ....} \right) - \left( {{y^2} + {y^3} + {y^4} + ...} \right)} \over {x - y}}\]
= \[{{{{{x^2}} \over {1 - x}} - {{{y^2}} \over {1 - y}}} \over {x - y}}\]
= \[{{\left( {{x^2} - {y^2}} \right) - xy\left( {x - y} \right)} \over {\left( {1 - x} \right)\left( {1 - y} \right)\left( {x - y} \right)}}\]
= \[{{x + y - xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}\]