every number $n\in \mathbb{Z} $ can be represented as sum of different powers of $2$

Question

Using generating function prove that every number $n\in \mathbb{Z} $ can be represented as sum of different power of $2$, I mean, that for every $n\in \mathbb{Z}$ $$n=2^{k_1} +2^{k_2} +2^{k_3} +... $$ where $k_i \neq k_j $ Please give me some advices.

Answer 1

Hint Prove that $\displaystyle \prod_{\nu\geqslant 0}(1+x^{2^{\nu}})=\frac{1}{1-x}=1+x+x^2+x^3+\cdots.$ Note that this proves existence and uniqueness.