Prove $\prod_{n=0}^{\infty}\left ( 1 + x^{2^n} \right ) = \sum_{n=0}^{\infty}x^n$

Question

I was just messing around trying to learn generating functions and a friend sent me this question.

Prove: $$ \prod_{n=0}^{\infty}\left ( 1 + x^{2^n} \right ) = \sum_{n=0}^{\infty}x^n $$

I'm not entirely sure how to begin so I'd love it if you someone could hint me to the right direction (not answers for now).

Answer 1

Hint 1. Any integer number has a unique binary representation.

Hint 2. You may also try to prove by induction that for $N\geq 1$, $$\prod_{n=0}^{N-1}\left ( 1 + x^{2^n} \right )= \sum_{k=0}^{2^{N}-1} x^k.$$

Answer 2

Hint: Note that $$\prod_{k=0}^{N}\left(1+x^{2^{k}}\right)=\prod_{k=0}^{N}\frac{\left(1-x^{2^{k+1}}\right)}{1-x^{2^{k}}}=\frac{1-x^{2^{N+1}}}{1-x}. $$