Question

If `|x|<1a n d|y|<1,` find the sum of infinity of the following series: `(x+y)+(x^2+x y+y^2)+(x^3+x^2y+x y^2+y^3)+`

Answer 1

`(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+yx^3)+...oo`
`=(x-y)/(x-y)[(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+yx^3)+...oo]`
`=1/(x-y)[(x-y)(x+y)+(x-y)(x^2+xy+y^2)+(x-y)(x^3+x^2y+xy^2+yx^3)+...oo]`
`=1/(x-y)[(x^2-y^2)+(x^3-y^3)+(x^4-y^4)+...oo]`
`=1/(x-y)[(x^2+x^3+x^4..oo)-(y^2+y^3+y^4+...oo)]`
`=1/(x-y)[(x^2/(1-x)) - (y^2/(1-y))]...[As S_n = a/(1-r)]`
`:. (x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+yx^3)+...oo =1/(x-y)[(x^2/(1-x)) - (y^2/(1-y))]`