Question

If $\mid x\mid <1$ and $\mid y\mid <1$ then $(x+y)+(x^2+xy+y^2)+(x^3+x^{2}y+x{y}^2+y^3)+.....\infty$ is

• A. $\frac{x+y-xy}{(1-x)(1-y)}$
• B. $\frac{x-y-xy}{(1-x)(1-y)}$
• C. $\frac{x+y-xy}{(1+x)(1+y)}$
• D. $\frac{x-y-xy}{(1+x)(1+y)}$

Answer 1

Answer: (A)

$\frac{x+y-xy}{(1-x)(1-y)}$

The given $S=(x+y)+(x^2+xy+y^2)+.....$
$=\frac{1}{(x-y)}\left\{\right(x^2-y^2)+(x^3-y^3)+(x^4-y^4)+.....\}$
$=\frac{1}{(x-y)}\left\{\right(x^2+x^3+....)-(y^2+y^3+.....\left)\right\}$
$=\frac{1}{(x-y)}(\frac{x^2}{1-x}-\frac{y^2}{1-y})=\frac{1}{x-y}\left(\frac{x^2-y^2-x^{2}y+x{y}^2}{(1-x)(1-y)}\right)=\frac{x+y-xy}{(1-x)(1-y)}$