Why does $\lim\limits_{n \to \infty } ((1 + x)(1 + {x^2})(1 + {x^4})\ldots(1 + {x^{{2^n}}})) = \frac{1}{{1 - x}}$? Let $\left| x \right| < 1$.       
Why does $$\lim\limits_{n \to \infty } ((1 + x)\cdot(1 + {x^2})\cdot(1 + {x^4})\cdot\ldots\cdot(1 + {x^{{2^n}}})) = \frac{1}{{1 - x}}$$

$$(1+x)(1+x^2)(1+x^4)(1+x^8).....=1+x+x^2+x^3+x^4+x^5......$$
then you can use the fact of geometric series
$$\frac{1}{1-x}=1+x+x^2+x^3+x^4+x^5......$$ 
$$|x|<1$$
